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c     File:     geom_base
c
c     Author:   Arthur L. Edwards (email:  <xenopsbird@comcast.net>)
c
c     Version:  Updated    2007 May 30 15:00.
c               Originated 1993 September 8 16:00.
c
c     Source:   Web site http://nuclear.llnl.gov/CNP/apt/.
c
c     Purpose:  File geom_base is a database for the GEOM code, which does a
c               variety of geometric, Monte Carlo, mathematical and data
c               manipulation operations, including 3-D ray tracing (geometric
c               optics) and particle tracking, and may be used to test and to
c               demonstrate the use of many subroutines in the APT Subroutine
c               Library.
c
c               The GEOM code is an interactive code with its own command
c               language, including the use of subscripted variables and the
c               equivalents of do loops, if-endif blocks, and subroutine calls.
c
c               File geom_base is in the form of a thoroughly cross-referenced
c               alphabetically organized encylopedia, with the following format:
c
c               Between entries:
c                 Columns  1-80:  Blank.
c               The first line of entries:
c                 Columns  1- 5:  A flag, indicating the type of entry.
c                 Columns  6- 6:  Blank.
c                 Columns  7-24:  The entry name, if < 20 characters.
c                 Columns  7-80:  The entry name, if > 19 characters.
c                 Columns 25-25:  Blank, if the entry name has < 20 characters.
c                 Columns 26-80:  Beginning of the entry body, containing a
c                                 description or definition of the entry name,
c                                 and sometimes commands related to the entry,
c                                 if the entry name has < 20 characters.
c               Additional lines of entries:
c                 Columns 28-80:  The rest of the entry body.
c
c               To find an entry, e.g. ENTRY, in this file using a general
c               search tool, specify the entry as "cc    ENTRY" for commands,
c               or "cg    ENTRY" for other definitions or descriptions.
c               See "search geom_base".
c
c               To extract entries that contain any desired key words, see
c               "extract entries".
c
c               File geom_defs contains an alphabetical list of all entries
c               in geom_base.
c
c               To execute GEOM, see "GEOM execution".
c
c               To display entries in this file during execution of GEOM, see
c               command "define".
c
c     Note:     In command lines, arguments shown in lower case are to be typed
c               literally as shown.  Arguments shown in UPPER CASE are place
c               holders, indicating data to be provided by the user, unless
c               otherwise stated.  Arguments in square brackets are options with
c               choices and/or defaults.  See entry "[".
c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

FLAG  ENTRY              DESCRIPTION OR DEFINITION

!-@-----------------------------------------------------------------------------

cc    !                  The exclamation point (or "bang").  The first character
cc                         of a comment line, or following a field delimiter in
cc                         a command line, the beginning of an appended comment.
cc
cc                         Ignored in command "hex".
cc
cc                         The character ";", if not bracketed by single or
cc                         double quotes, will terminate a comment, allowing a
cc                         command to follow on the same line.
cc
cc                         Do not use the character "!" at the beginning of the
cc                         name of any alias, marker, symbol or other object.
cc
cc                         See "c", "#", "*", "/".

cg    !                  In a mathematical expression, following a non-negative
cg                         integer, indicates the factorial function:
cg                         0! = 0, 1! = 1, N! = 1 * 2 * ... * (N - 1) * N.

cc    "                  In command "alias", delimits the replacement string.
cc                         Double any " in the replacement string.
cc                         Synonyms:  [", ' (in pairs)].

cc    "                  In command "define", delimits the entry.
cc                         Double any " in the entry.
cc                         Synonyms:  [", ' (in pairs)].

cc    "                  In command "marker", delimits the text string.
cc                         Double any " in the text string.
cc                         Synonyms:  [", ' (in pairs)].

cc    #                  If the first character of an input line, the entire
cc                         input line is treated as a comment, and echoed in the
cc                         output file.  See "c".
cc                         Synonyms:  [#, *, /].

cc    #                  The first character of a comment line.
cc                         Synonyms:  [#, *, /].
cc                         See "c", "!".

cc    #                  An option in command "input", to switch input to the
cc                         previous input medium, starting with the line after
cc                         the last line read from that input medium, and with
cc                         the same specification for the final line to be read.
cc                         See "%", command "return".

cc    $                  An option in command "input", in place of LINE1 or
cc                         LINE2, to indicate the last line saved in memory,
cc                         or the end-of-file of an input file.

cc    $                  An option in commands "indo" and "redo", in place of
cc                         LINE, to indicate the last line saved in memory.

cc    $                  May be used as the first character of a variable name,
cc                         as may "@", "%" or any lower case letter, or any
cc                         upper case letter followed by one or more characters.

cg    %                  In a display of data resulting from command "mcvol",
cg                         percent, as in "% estimated error".

cc    %                  An option in command "input", to indicate the current
cc                         input file, when used in place of IN_FILE.  Used to
cc                         move to a different line in the current input file.

cc    %                  May be used as the first character of a variable name,
cc                         as may "@", "$" or any lower case letter, or any
cc                         upper case letter followed by one or more characters.

cc    &                  An option in command "cluster", to indicate that
cc                         additional points are being added to an existing
cc                         cluster.
cc                         To add additional clusters to an existing cluster,
cc                         use command "cluster CLNAME cluster CLNAME ...".

cc    &                  An option in command "pdf", to indicate that
cc                         additional probability bins are being specified for
cc                         a pdf already partially specified.

cc    &                  An option in command "quadric", to respecify one or
cc                         more individual coefficients in the general implicit
cc                         equation of a specified quadric surface.  Options
cc                         CFNAME1, CFNAME2, ... may be QC, QX, QY, QZ, QXY,
cc                         QYZ, QZX, QXX, QYY or QZZ.

cc    &                  An option in command "zone", to indicate that
cc                         additional bounding surfaces are being specified for
cc                         a zone already partially specified.

cc    '                  A synonym for ", in commands "alias", "define",
cc                         "marker" and "variable".

cc    '                  In command "alias", delimits the replacement string.
cc                         Double any ' in the replacement string.
cc                         Synonyms:  [", ' (in pairs)].

cc    '                  In command "define", delimits the entry.
cc                         Double any ' in the entry.
cc                         Synonyms:  [", ' (in pairs)].

cc    '                  In command "marker", delimits the text string.
cc                         Double any ' in the text string.
cc                         Synonyms:  [", ' (in pairs)].

cg    (, )               The parenthesis characters.  See "subscripts".

cg    (xyz)              In a display of data, indicates three coordinates of a
cg                         point or three components of a vector, in Cartesian
cg                         (x, y, z) coordinates.  To convert to another
cg                         coordinate system, use the following commands:
cg
cg                         coordinate xyz
cg                         [point,vector] TEMP = {copy xyz coordinates here}
cg                         coordinate [cyl,sph]
cg                         angle [deg,rad]
cg                         [point,vector] TEMP
cg
cg                         See "cut and paste", "coordinate conversion",
cg                         "vector conversion".

cc    *                  A synonym for # in a comment line.

cc    *                  An option in commands "big", "icalc" and "variable",
cc                         argument FUNCTION, to indicate multiplication.
cc                         In the expression A * B, indicates that A is
cc                         multiplied by B.

cc    *                  An option in command "quadric", to indicate that the
cc                         coefficients of the implicit equation of the quadric
cc                         surface are to be multiplied by argument FACT.

cc    *                  An option in command "variable VARNEW = K * L mod M",
cc                         to multiply K by L.

cc    *                  An option in command "variable", to give the variable
cc                         the value of the product of the specified arguments.

cc    *                  A synonym for rel in command "vector".

cc    *                  An option in command "variable VARNEW = K * L mod M",
cc                         to multiply K by L.

cc    **                 A synonym for ^.

cc    +                  When used as a prefix (without a delimiter) to a
cc                         variable name used in place of an integer or floating
cc                         point argument, has no effect.  See "-".

cc    +                  In a command that creates, copies or renames an object,
cc                         a "+" in place of the object name indicates that the
cc                         object name should be formed by incrementing the base
cc                         name for the object type.  The base name is the name
cc                         of the last object of that type created, or the base
cc                         name specified with command "last", whichever occurs
cc                         later.
cc                         The new name becomes the current base name.
cc                         The "+" option may not be used in command "delete",
cc                         or in command "rename" with option "array".
cc                         See command "last", "increment names", "-".
cc
cc                         You must be sure that incrementing a base name
cc                         does not create a disallowed name, that could be
cc                         interpreted as a key word in a command, such as
cc                         "all", "comp.", "h", "help", "list", "random",
cc                         "thru" or "?".

cc    +                  An option in commands "big", "icalc" and "variable",
cc                         argument FUNCTION, to indicate addition.  In the
cc                         expression x + y, indicates that y is added to x.

cc    +                  An option in commands "invert", "reflect", "rotate" or
cc                         "scale".  The delimited character "+" preceding the
cc                         name of a tensor operator indicates that the
cc                         specified tensor operator will be used (the default
cc                         option), rather than its inverse.
cc                         The word "+" is the default option, and need not be
cc                         included in the command.

cc    +                  An option in command "sort", to sort in increasing
cc                         alphanumeric order.  The default option.

cc    +                  An option in command "variable", to give the variable
cc                         the value of the sum of the specified arguments.

cc    +                  An option in command "variable VARNEW = K + L mod M",
cc                         to add L to K.

cc    +                  An option in command "zone".  The delimited character
cc                         "+" preceding the name of a bounding surface
cc                         indicates that the direction out of the zone is in
cc                         the same direction as the normal vector of the
cc                         bounding surface.
cc                         If not specified, the default argument is "+".
cc
cc                         For example, if the zone is inside of a sphere,
cc                         cylinder, cone or ellipsoid that has been created
cc                         by any command other that "quadric QNAME = ...",
cc                         the corresponding argument is blank or "+".

cc    ,                  In subscripted object names, used to separate
cc                         subscripts from each other.

cg    -                  See "- |...|".

cg    -                  When the first character of a point name, indicates the
cg                         point was previously a mesh point, and was released
cg                         from the mesh.  A point with a name beginning with
cg                         a minus sign may not be assigned to the mesh.

cc    -                  When used as a prefix (without a delimiter) to a
cc                         variable name used in place of an integer or floating
cc                         point argument, results in use of the negative of the
cc                         variable value.  See "+".

cc    -                  In a command that creates, copies or renames an object,
cc                         a "-" in place of the object name indicates that the
cc                         object name should be formed by decrementing the base
cc                         name for the object type.  The base name is the name
cc                         of the last object of that type created, or the base
cc                         name specified with command "last", whichever occurs
cc                         later.  The new name becomes the current base name.
cc                         The "-" option may not be used in command "delete",
cc                         or in command "rename" with option "array".
cc                         See command "last", "increment names", "-".
cc
cc                         You must be sure that decrementing a base name
cc                         does not create a disallowed name, that could be
cc                         interpreted as a key word in a command, such as
cc                         "all", "comp.", "h", "help", "list", "random",
cc                         "thru" or "?".

cc    -                  An option in command "cluster", to remove points from
cc                         an existing cluster, either by point names or by
cc                         the names of clusters containing the points.

cc    -                  An option in commands "big", "icalc" and "variable",
cc                         argument FUNCTION, to indicate subtraction.  In the
cc                         expression x - y, indicates that y is subtracted
cc                         from x.

cc    -                  An option in commands "mesh" and "point", to release
cc                         the assignment of mesh indices to mesh points
cc                         (make the mesh indices zero).

cc    -                  An option in command "pdf", to remove probability bins
cc                         from an existing pdf.

cc    -                  An option in command "plot center", to default the
cc                         plot center to the center of the plot axes.
cc                         An option in command "plot limits", to specify that
cc                         the plot axis limits will include all plot points.

cc    -                  An option in command "rotate".  The delimited character
cc                         "-" preceding the name of a rotation operator,
cc                         indicates that the inverse of the specified rotation
cc                         operator will be used.
cc                         Do not use this option in commands "invert" or
cc                         "reflect".

cc    -                  An option in command "sort", to sort in decreasing
cc                         alphanumeric order.

cc    -                  An option in command "zone".  The delimited character
cc                         "-" preceding the name of a bounding surface
cc                         indicates that the direction out of the zone is in
cc                         the opposite direction of the normal vector of the
cc                         bounding surface.
cc                         If not specified, the default argument is "+".
cc
cc                         For example, if the zone is outside of a sphere,
cc                         cylinder, cone or ellipsoid that has been created
cc                         by any command other that "quadric QNAME = ...",
cc                         the corresponding argument is "-".

cc    -                  An option in command "brick", option "increment".
cc                         The delimited character "-" preceding the name of a
cc                         coordinate direction indicates that bricks are to be
cc                         created in the negative coordinate direction from a
cc                         base brick.

cc    -                  An option in command "variable VARNEW = K - L mod M",
cc                         to subtract L from K.

cg    - |...|            Indicates that the quantity bracketed by "|" characters
cg                         must be negative.

cg    -999999999         Sometimes used for a value when no correct value can
cg                         be found.

cg    .                  See ". (period)".

cc    . (period)         A synonym for 0, in command indo.

cc    . (period)         An option in command "redo".  When used for R1, R2, R3,
cc                         ..., means to use the original argument, with no
cc                         replacement.

cc    . (period)         Command to repeat the preceding command.
cc                         Do not use the statement separator ";" on the same
cc                         line following this command.
cc
cc                         Forms include:
cc
ccin                       . help
cc
cc                           Display the command options.
cc
ccin                       .
cc
cc                           Repeat the preceding command.
cc                           This is the same as the command "input command -1".
cc                           To repeat the preceding N commands, use command
cc                           "input command -N $".
cc
cc                         Synonyms:  [help, h].

cc    .and.              An option in command 'icalc", to find the Boolean
cc                         "and" or "intersection" of two machine words.
cc                         M .and. N means the Boolean "and" or "intersection"
cc                         of M and N.
cc                         Use 1 where M and N have a 1, otherwise 0.
cc                         See "Boolean".
cc                         Synonyms:  [.and., .int.].

cc    .and.              An option in command "cluster", to create a cluster
cc                         consisting of all points that are in both of two
cc                         specified clusters.  This is the Boolean "and" or
cc                         "intersection" of the two specified clusters.

cc    .eq.               A synonym for = in command "if".

cc    .eqv.              A synonym for .xnor. in commands "cluster" and "icalc".

cn    .exrc              A VI command file, for use when reading or editing this
cn                         file with UNIX text editor VI.  Allows a search for
cn                         a pattern starting in column 7 by beginning the VI
cn                         command with "q", followed by the pattern.

cc    .ge.               A synonym for >= in command "if".

cc    .gt.               A synonym for > in command "if".

cc    .int.              A synonym for .and. in commands "cluster" and "icalc".

cc    .le.               A synonym for <= in command "if".

cc    .lt.               A synonym for < in command "if".

cc    .nand.             A Boolean operation.  In the case of two binary digits,
cc                         returns a 0 if both are 1, otherwise returns a 1.

cc    .nand.             An option in command "icalc", to find the Boolean
cc                         "nand" of two machine words.
cc                         M .nand. N means the Boolean "nand" of M and N
cc                         (0 where both M and N have a 1-bit, otherwise 1).
cc                         See "Boolean".

cc    .nand.             An option in command "cluster", to create a cluster
cc                         consisting of all points that are not in both of
cc                         two specified clusters.  This is the Boolean "nand"
cc                         of the two specified clusters.

cc    .ne.               A synonym for /= in command "if".

cc    .nor.              A Boolean operation.  In the case of two binary digits,
cc                         returns a 1 if both are zero, otherwise returns zero.

cc    .nor.              An option in command "icalc", to find the Boolean
cc                         "nor" of two machine words.
cc                         M .nor. N means the Boolean "nor" of M and N
cc                         (1 where both M and N have a 0-bit).  See "Boolean".

cc    .nor.              An option in command "cluster", to create a cluster
cc                         consisting of all points that are in neither of
cc                         two specified clusters.  This is the Boolean "nor"
cc                         of the two specified clusters.

cc    .not.              A Boolean operation.  In the case of two binary digits,
cc                         returns a 1 if the first is 1 and the second is 0,
cc                         otherwise returns zero.

cc    .not.              An option in command "icalc", to find the Boolean
cc                         "not" of two machine words.
cc                         M .not. N means the Boolean "not" of M and N
cc                         (1 where M has a 1-bit and N has a 0-bit).
cc                         See "Boolean".

cc    .not.              An option in command "cluster", to create a cluster
cc                         consisting of all points in one specified cluster
cc                         but not in another specified cluster.

cc    .or.               An option in command 'icalc", to find the Boolean
cc                         union of two machine words.
cc                         M .or. N means the Boolean "or" or "union" of M and N
cc                         (1 where either M or N has a 1-bit).  See "Boolean".
cc                         Synonyms:  [.or., .un.].

cc    .or.               An option in command "cluster", to create a cluster
cc                         consisting of all points that are in either of
cc                         two specified clusters.  This is the Boolean
cc                         "or" or "union" of the two specified clusters.

cc    .un.               A synonym for .or. in commands "cluster" and "icalc".

cc    .xnor.             A Boolean operation.  In the case of two binary digits,
cc                         returns a 1 if both are 0 or both are 1.

cc    .xnor.             An option in command 'icalc", to find the Boolean
cc                         "xnor" of two machine words.
cc                         M .xnor. N means the Boolean "xnor" of M and N
cc                         (1 where both M and N have a 1-bit, or both M and N
cc                         have a 0-bit).  See "Boolean".

cc    .xnor.             An option in command "cluster", to create a cluster
cc                         consisting of all points which are in either both
cc                         or neither of two specified clusters.  This is the
cc                         Boolean "equivalence" of the two specified clusters.
cc                         Synonymns:  [.xnor., .eqv.].

cc    .xor.              An option in command 'icalc", to find the Boolean
cc                         exclusive or of two machine words.
cc                         M int N means the Boolean exclusive or of M and N.
cc                         See "Boolean".

cc    .xor.              An option in command "cluster", to create a cluster
cc                         consisting of all points that are in one, but not
cc                         the other, of two specified clusters.  This is the
cc                         Boolean "exclusive or" of the two specified clusters.

cc    /                  An option in commands "big", "icalc" and "variable",
cc                         argument FUNCTION, to indicate division.  In the
cc                         expression x / y, indicates that x is divided by y.

cc    /                  An option in command "quadric", to indicate that the
cc                         coefficients of the implicit equation of the quadric
cc                         surface are to be divided by argument FACT.

cc    /                  A synonym for # in a comment line.

cc    /                  An option in commands "icalc" and "variable".  When
cc                         used for FUNCTION, indicates division of the
cc                         preceding argument by the following argument.

cc    /                  An option in command "variable", to give the variable
cc                         the value of the sum of the reciprocals of the
cc                         specified arguments.

cc    /                  An option in command "variable VARNEW = K / L mod M",
cc                         to multiply K by the multiplicative inverse of L, if
cc                         it exists.

cc    /=                 An option in command "if", meaning not equal to.
cc                         Synonyms:  [/=, =/, .ne.].

cc    //                 An option in command "big", to divide one big integer
cc                         by another big integer as many times as possible, if
cc                         an exact factor.

cg    /^......STRING     A VI command to search for a character string STRING
cg                         which starts in column 7 of this file.

cc    0                  An option in command "indo".  When used for I1, I2, I3,
cc                         ..., means to use the original argument, with no
cc                         increment.
cc                         Synonyms:  [0, .].

cg    10^(-99)           Sometimes used for a value when no correct value can be
cg                         found.

cg    10^99              Sometimes used for a value when no correct value can be
cg                         found.  As an input argument, specified as "1E99".

cc    17777777777        In command "icalc", the largest allowed octal
cc                         integer on a 32-bit computer.
cc                         Same as 2147483647 decimal, 7fffffff hex.

cg    21                 A game (blackjack) which may be played with GEOM,
cg                         using input file 21.mac.

cc    2147483647         In command "icalc", the largest allowed decimal
cc                         integer on a 32-bit computer.
cc                         Same as 17777777777 decimal, 7fffffff hex.

cc    777777777777777777777
cc                         In command "icalc", the largest allowed octal
cc                         integer on a 64-bit computer.
cc                         Same as 9223372036854775807 decimal,
cc                         7FFFFFFFFFFFFFFF hex.

cc    7fffffff           In command "icalc", the largest allowed hexadecimal
cc                         integer on a 32-bit computer.
cc                         Same as 2147483647 decimal, 17777777777 octal.

cc    7fffffffffffffff   In command "icalc", the largest allowed hexadecimal
cc                         integer on a 64-bit computer.
cc                         Same as 9223372036854775807 decimal,
cc                         777777777777777777777 oct.

cc    80                 Maximum number of characters read in an input line,
cc                         including input lines expanded by substitution of
cc                         an alias.

cc    9223372036854775807
cc                         In command "icalc", the largest allowed decimal
cc                         integer on a 64-bit computer.
cc                         Same as 777777777777777777777 octal,
cc                         7FFFFFFFFFFFFFFF hex.

cc    ;                  Separates multiple commands and/or comments on one
cc                         input line.  Ignored if between single or double
cc                         quotes (as in commands "alias", "define" and
cc                         "marker").  Does not need to be delimited.
cc                         If not between single or double quotes, will
cc                         terminate a comment, allowing a command to follow on
cc                         the same line.  Ignored in command "hex".
cc                         For example:
cc
ccin                       COMMAND1;COMMAND2 !COMMENT;COMMAND3 ...
cc
cc                           Execute commands COMMAND1, COMMAND2, COMMAND3, ...
cc
cc                         Can not be used after commands "do", "enddo",
cc                         "indo", "input", "read", "redo", "return" or"undo",
cc                         or their synonyms, because chaos might ensue.
cc
cc                         WARNING:  if used in an input file, the line number
cc                         displayed in the output file will not be the same
cc                         as in the input file, but will be a statement
cc                         counter, instead.  To reexecute lines, use command
cc                         "input geom_cmd ?" to find the correct line numbers
cc                         to use, say N1 through N2, and then use command
cc                         "input geom_cmd N1 N2" to execute them.

cc    <                  An option in command "if", meaning less than.
cc                         Synonyms:  [<, .lt.].

cc    <                  Used in a UNIX execution line to redirect the standard
cc                         input.  Do NOT use any of the GEOM execution lines:
cc                         "geom < IN_FILE"           ,
cc                         "geom < IN_FILE > OUT_FILE",
cc                         "geom   IN_FILE > OUT_FILE" or
cc                         "geom           > OUT_FILE".
cc
cc                         The following forms are allowed, to prevent large
cc                         amoumts of data from being displayed on the user's
cc                         terminal:
cc
cc                         "geom   IN_FILE   OUT_FILE  >   STDOUT"
cc                         "geom   IN_FILE   OUT_FILE  >!  STDOUT"
cc
cc                         The latter form allows writing over an existing
cc                         file STDOUT.
cc
cc                         See "GEOM execution".

cg    <                  In column 1 of the output file, indicates an input
cg                         line that begins with "input", "read", "return",
cg                         "call", "goto", "undo", "redo", "indo" or a synonym
cg                         for any of these, or an alias.  See ">".

cc    <=                 An option in command "if", meaning less than or
cc                         equal to.  Also in displayed values from command
cc                         "root".
cc                         Synonyms:  [<=, =<, .le.].

cc    <return>           The <return> key must be typed at the end of every
cc                         input line typed at the user's terminal.

cc    =                  An option in some commands that create objects, between
cc                         the object name and the specification of the object.

cc    =                  An option in command "if", meaning equal to.
cc                         Synonyms:  [=, .eq.].

cc    =                  An option in command "root", to find the real
cc                         coefficients of an Nth-order polynomial equation
cc                         with N specified real roots.

cc    =                  An option in command "roots", to find the real
cc                         coefficients of a polynomial equation with from
cc                         one to four specified real and/or complex roots.

cc    =/                 A synonym for /= in command "if".

cc    =<                 A synonym for <= in command "if".

cc    =>                 A synonym for >= in command "if".

cc    >                  An option in command "if", meaning greater than.
cc                         Synonyms:  [>, .gt.].

cg    >                  In column 1 of the output file, indicates an input
cg                         line that does not begin with "input", "read",
cg                         "return", "undo", "redo", "indo" or a synonym for
cg                         any of these, or an alias.  A new input file, which
cg                         will duplicate the run that made the output file, may
cg                         be made by selecting only those lines beginning with
cg                         ">" from the input file, then deleting the ">".
cg                         See "<".

cc    >                  Used in a UNIX execution line to redirect the standard
cc                         output.  Do NOT use any of the GEOM execution lines:
cc                         "geom < IN_FILE"           ,
cc                         "geom < IN_FILE > OUT_FILE",
cc                         "geom   IN_FILE > OUT_FILE" or
cc                         "geom           > OUT_FILE".
cc
cc                         See "GEOM execution".

cc    >=                 An option in command "if", meaning greater than or
cc                         equal to.
cc                         Synonyms:  [>=, =>, .ge.].

cc    ?                  The default short input prompt, if the long prompt is
cc                         turned off.  Input may follow this prompt on the
cc                         same line.

cc    ?                  A synonym for random, except in command "input".

cc    ?                  An option in command "input", to indicate that the
cc                         contents of the specified file will be displayed
cc                         by the standard output (normally the user's
cc                         terminal) and in the output file, preceded by line
cc                         numbers, but not executed.  Is not a synonym for
c                          random in command "input".

cc    ?                  An option in command "sort", to sort in random
cc                         alphanumeric order.

cc    @                  May be used as the first character of a variable name,
cc                         as may "$", "%" or any lower case letter, or any
cc                         upper case letter followed by one or more characters.

cg    A                  The frequency of the musical note Concert A, or A4,
cg                         is 440 Hz.  Middle C (C4) is 523.2511306012 Hz.

A-------------------------------------------------------------------------------

cc    a                  A synonym for add, when responding to question about
cc                         appending to an existing output file.

cc    a                  A synonym for alias.

cc    A(N)               In command "variable", option "polynomial", the
cc                         coefficient of XARG^N in a polynomial function of
cc                         XARG.  N = 0 to whatever will fit on the command
cc                         line.  See command "roots", option "?".


cc    A(n)               In command "root", the coefficient of x^n in the
cc                         polynomial P(x) = sum(n=1,N) {A(n) * x^n} = 0.

cc    A0                 In command "roots", the constant term in the polynomial
cc                         equation to be solved.
cc                         If A0 is zero, at least one root is zero.

cc    A1                 In command "roots", the coefficient of z in the
cc                         quadratic, cubic or quartic polynomial equation to be
cc                         solved.  If A0 and A1 are zero, at least two roots
cc                         are zero.

cc    A2                 In command "roots", the coefficient of z^2 in the
cc                         quadratic, cubic or quartic polynomial equation to be
cc                         solved.  If A0 = A1 = A2 = 0, at least three roots
cc                         are zero.

cc    A3                 In command "roots", the coefficient of z^3 in the
cc                         cubic or quartic polynomial equation to be solved.
cc                         If A0 = A1 = A2 = A3 = 0, at least four roots
cc                         are zero.

cc    A4                 In command "roots", the coefficient of z^4 in the
cc                         quartic polynomial equation to be solved.

cc    abs                An option in command "variable", argument FUNCTION,
cc                         to indicate the absolute value function.
cc                         A abs B means A * abs (B).

cc    abs                An option in command "vector", to give the vector an
cc                         absolute length equal to ABSLEN (defaults to 1).
cc                         See "ABSLEN", "rel", "relative".

cf    abs                The absolute value function.  Example:  y = abs (x)
cf                         means y is positive, with the magnitude of x.

cc    ABSLEN             Length of a vector of type "absolute".  Defaults to 1.
cc                         May be integer, floating point or a variable.
cc                         Specified with command "vector".

cg    absolute           A vector specified by two points is absolute if key
cg                         word "rel" or "*" is not used.  If key word "abs" or
cg                         "unit" is used, the vector will have a length equal
cg                         to ABSLEN (defaults to 1).
cg                         Once a vector has been created, it no longer depends
cg                         on any points used to create it.
cg                         See "ABSLEN", "rel", "relative", "vector".

cc    accel              A synonym for accelerate.

cc    accelerate         Command to find the position, path length and velocity
cc                         on the parabolic trajectory of a particle with a
cc                         specified initial position, initial velocity and
cc                         constant acceleration, for a specified range of
cc                         times or at specified time intervals,
cc                         or to find the intersection of the particle path with
cc                         one or more points, a line, a plane or a quadric
cc                         surface, or to find the initial position and velocity
cc                         and constant acceleration for a particle, given two
cc                         or three positions at two or three times.
cc
cc                         Command "accelerate" relates to objects:  axisym,
cc                         cluster, cone, cylinder, ellipsoid, hyperb, line,
cc                         plane, point, quadric, sphere, symbol, variable,
cc                         vector.
cc
cc                         Forms include:
cc
ccin                       help accelerate
ccin                       accelerate [help]
cc
cc                           Display the command options.
cc
ccin                       accelerate path PINIT VINIT VACC
cc
cc                           For a particle initially at point PINIT, with
cc                           initial velocity vector VINIT and constant
cc                           acceleration vector VACC, find the path type
cc                           (simple linear, degenerate linear parabolic or
cc                           parabolic), and if parabolic, find the position,
cc                           path length and velocity at the vertex of its path.
cc
ccin                       accelerate path PINIT VINIT VACC time TMIN [DT TMAX]
cc
cc                           For a particle initially at point PINIT, with
cc                           initial velocity vector VINIT and constant
cc                           acceleration vector VACC, find the path type and
cc                           vertex data as above, and the position, path length
cc                           and velocity, for time[s] TMIN [(DT) TMAX].
cc                           Any unspecified words at the end of the command,
cc                           after TMIN, default to 0.
cc
ccin                       accelerate path PINIT VINIT VACC point PNAME
cc
cc                           For a particle initially at point PINIT, with
cc                           initial velocity vector VINIT and constant
cc                           acceleration vector VACC, find the path type and
cc                           vertex data as above, and the points on its path
cc                           that intersect or are proximal to the point PNAME,
cc                           and for each such point, the coordinates, the
cc                           velocity, the path length, the time, and the
cc                           distance to point PNAME.
cc                           If point PNAME is not on the path, find any initial
cc                           velocities VINIT' with the same magnitude as VINIT,
cc                           that will make the path intersect point PNAME.
cc
ccin                       accelerate path PINIT VINIT VACC cluster CLNAME
cc
cc                           For a particle initially at point PINIT, with
cc                           initial velocity vector VINIT and constant
cc                           acceleration vector VACC, find the path type and
cc                           vertex data as above, and the other data as in the
cc                           preceding command, for each point in cluster
cc                           CLNAME.
cc
ccin                       accelerate path PINIT VINIT VACC line ALNAME
cc
cc                           For a particle initially at point PINIT, with
cc                           initial velocity vector VINIT and constant
cc                           acceleration vector VACC, find the path type and
cc                           vertex data as above, and the points on its path
cc                           that intersect or are proximal to the line ALNAME,
cc                           and for each such point, the coordinates, the
cc                           velocity, the path length, the time, and the
cc                           distance to line ALNAME.
cc
ccin                       accelerate path PINIT VINIT VACC plane PLNAME
cc
cc                           For a particle initially at point PINIT, with
cc                           initial velocity vector VINIT and constant
cc                           acceleration vector VACC, find the path type and
cc                           vertex data as above, and the points on its path
cc                           that intersect or are proximal to the plane PLNAME,
cc                           and for each such point, the coordinates, the
cc                           velocity, the path length and the time.  For a
cc                           proximal point, find the proximal point on the
cc                           plane, and the distance from the plane to the
cc                           particle.
cc
ccin                       accelerate path PINIT VINIT VACC quadric QNAME
cc
cc                           For a particle initially at point PINIT, with
cc                           initial velocity vector VINIT and constant
cc                           acceleration vector VACC, find the path type and
cc                           vertex data as above, and any points on its path
cc                           that intersect the quadric surface QNAME, and for
cc                           each such point, the coordinates, the velocity, the
cc                           path length and the time.  To check the results,
cc                           store each such intersection point in PNAME, and
cc                           use command "side PNAME QNAME".
cc                           If no intersection points are found, find any
cc                           extrema of the quadric surface equation on the
cc                           particle path.  In general, these are NOT proximal
cc                           or distal points, but may be used as initial
cc                           guesses in finding such points.
cc
ccin                       accelerate fit P1 [T1 P2 T2 [P3 T3]]
cc
cc                           For a particle at from one to three points and
cc                           times P1 and T1, P2 and T2, and P3 and T3, find the
cc                           position and velocity at time zero, and the
cc                           constant acceleration vector for a parabolic
cc                           trajectory.  Each time specified must be different.
cc                           If all points are the same, the velocity and
cc                           acceleration will be zero.  If only two points are
cc                           specified, the acceleration will be zero.  If any
cc                           two of three specified points are the same, the
cc                           acceleration will be parallel to the velocity.
cc                           If three specified points are collinear, the
cc                           acceleration will be parallel to the velocity.
cc
cc                         Synonyms:  [accelerate, accel], [help, h],
cc                         [plane, pl], [quadric, q, quad], [time, t].

cg    acceleration       A particle initially at position P(0), with velocity
cg                         vector V(0) and constant acceleration vector A,
cg                         obeys the following equations of motion:
cg
cg                           V(t)    = V(0) + A * t
cg                           VAVG(t) = 0.5 * (V(0) + V(t)) = V(0) + 0.5 * A * t
cg                           P(t)    = P(0) + VAVG(t) * t
cg                                   = P(0) + V(0) * t + 0.5 * A * t^2
cg
cg                         The path lies on a planar parabolic curve, in the
cg                         plane containing point P(0) and vectors V(0) and A.
cg
cg                         The vertex of the parabolic path is at the point
cg                         where the velocity and acceleration are perpendicular
cg                         to each other:
cg
cg                           V(tver) dot A =  V(0) dot A + A^2 tver = 0
cg                           tver          = -V(0) dot A / A^2
cg                           V(tver)       =  V(0) - (V(0) dot A) / A^2
cg
cg                         The path length may be obtained by integrating the
cg                         magnitude of the velocity vector over time.
cg
cg                           d(t) = Integral {|V(0) + A * t'| * dt'}, t' = 0, t,
cg
cg                         where the integrand |V(0) + A * t| is
cg
cg                           sqrt (V(0)^2 + 2 * V(0) dot A * t + A^2 * t^2)
cg
cg                         The integral may be found in a math handbook.
cg
cg                         If V(0) or A is zero or if V(0) and A are parallel,
cg                         the path is linear, and the path length is
cg
cg                           d(t) = VAVG(t) * t
cg
cg                         If two [or three] points on the particle path are
cg                         known at two or three times, the initial position
cg                         and velocity, and the zero [or constant] acceleration
cg                         may be found by solving the two [or three]
cg                         simultaneous linear equations of the particle path.
cg
cg                         The distance B between the particle and a plane
cg                         through the point Q with normal vector N is given by:
cg
cg                           B = (P(t) - Q) dot N / |N|
cg                             = (P(0) - Q) dot N + (V(0) dot N) * t +
cg                               0.5 * (A dot N) * t^2
cg
cg                         Any intersections between the particle path and the
cg                         plane may be found by setting B = 0 and solving for
cg                         t.  If no intersections exist, the proximal point
cg                         between the particle path and the plane occurs when
cg
cg                           V(t) dot N = 0 = (V(0) dot N) + (A dot N) * t
cg
cg                         or everywhere on the path if (A dot N) = 0.
cg
cg                         The point on the plane nearest the particle is at
cg
cg                           Q' = P(t) + ((Q - P(t)) dot N / |N|) * N
cg
cg                         The intersection between the particle path and a
cg                         quadric surface is obtained by substituting the
cg                         time-dependent coordinates of the particle into the
cg                         equation of the quadric surface, solving the
cg                         resulting quartic equation for any real time of
cg                         intersection, and calculating the particle
cg                         coordinates at that time.
cg
cg                         See command "accelerate".

cg    acceptable         An input line is acceptable if recognized as a comment
cg                         or a command.  An input argument is acceptable if
cg                         recognized as a command argument, is in the correct
cg                         mode, and if an integer or floating point number, is
cg                         within the allowed numerical range.

cg    accuracy           See "precision", "TOL", command "tol", command "big",
cg                         "error estimate", "significant figures".

cc    acos               An option in command "variable", argument FUNCTION,
cc                         to indicate the trigonometric inverse cosine
cc                         function.  A acos B means A * acos (B).

cf    acos               The trigonometric inverse cosine function.
cf                         Example:  y = acos (x) means y is the angle whose
cf                         cosine is x, and x = cos (y).  The result is in
cf                         radians, between 0 and pi.

cc    acosh              An option in command "variable", argument FUNCTION,
cc                         to indicate the inverse hyperbolic cosine function.
cc                         A acosh B means A * acosh (B).

cf    acosh              The inverse hyperbolic cosine function.
cf                         Example:  y = acosh (x) means y is the argument for
cf                         which the hyperbolic cosine is x, and x = cosh (y).
cf                         The range of x is x >= 1.
cf
cf                         acosh (x) = ln (x + sqrt (x^2 - 1)) (principal val)
cf                         acosh (x) = ln (x - sqrt (x^2 - 1)) (secondary val)

cg    add                See "add vectors".

cg    add                To add real or integer numbers, see commands
cg                         "variable, "big", "icalc".

cc    add                A synonym for sum in commands "quadric", "vector.
cc                         See commands "icalc", "variable", argument
cc                         FUNCTION, and command "big"..

cc    add                Following command "output", if the specified output
cc                         file already exists, and the current input is from
cc                         the user's terminal, the user must respond "add" to
cc                         allow GEOM to append to the existing output file.
cc                         This is the default if the current input is not from
cc                         the user's terminal.  To write over the existing
cc                         output file, the response must be "yes".
cc
cc                         Synonyms:  [add, a].

cg    add                See "add vectors".

cg    add vectors        See "vector sum".  See option "move" in commands:
cg                         axisym, cone, cylinder, ellipsoid, plane, point,
cg                         quadric, sphere.

cc    al                 A synonym for alias.

cg    alias              An alias may be used to replace all or part of a
cg                         frequently used long command line with a much shorter
cg                         substitute, for easier input.
cg
cg                         An alias is a delimited word up to 24 characters
c                          long, including any subscripts.
cg                         Its replacement is any character string up to 72
cg                         characters long or whatever will fit on the input
cg                         line of command "alias" (limited to 80 characters)
cg                         used to create the alias.  In any input line, an
cg                         initial word that is an alias will be replaced by the
cg                         replacement string.  This is done before symbol
cg                         replacement.  The alias replacement may contain
cg                         symbols, which will be replaced.
cg                         Only the first word of an input line containing
cg                         multiple statements can be replaced by an alias.
cg                         The combined number of characters resulting from the
cg                         replacement may not exceed 80.
cg
cg                         See "definition".
cg                         Also see "symbol", "synonym", "marker", command ".".
cg
cg                         The replacement string, if it is a single word that
cg                         contains no field delimiter, may itself be aliased to
cg                         another replacement string, but there may be no more
cg                         than 10 sequential replacements to find the final
cg                         string.  The maximum number of aliases is now 256.
cg
cg                         To repeat the preceding command many times, use
cg                         command ".", or create alias ANAME:
cg
cg                         alias ANAME = 'input command -1"
cg
cg                         Then after the command to be repeated, type
cg                         ANAME<return> as many times as desired.
cg
cg                         To repeat a single command "COMMAND ..." many times,
cg                         create alias ANAME (single characters near the
cg                         <return> key are convenient):
cg
cg                         alias ANAME = "COMMAND ..."
cg
cg                           Create alias ANAME, so that every time you
cg                           type "ANAME<return>", command "COMMAND ... "
cg                           will be executed.
cg
cg                         Then type ANAME<return> as many times as desired.
cg
cg                         To repeat a block of commands many times, execute
cg                         the block of commands once.  Then execute command
cg                         "input command ?" to find the indices N1 and N2 of
cg                         the first and last commands of the block of commands
cg                         saved in memory.
cg
cg                         Then create alias ANAME:
cg
cg                         alias ANAME = 'input command N1 N2'
cg
cg                           Create alias ANAME, so that every time you
cg                           type "ANAME<return>", stored commands N1 through
cg                           N2 are executed.
cg
cg                         Then type ANAME<return> as many times as desired.
cg
cg                         The following commands may affect or display aliases:
cg                         alias, copy, delete, help, last, list, rename,
cg                         repack, search, sort, tables.
cg
cg                         Synonyms:  [alias, al, a]. [command, c, cmd],
cg                         [input, call, i, in, r, rd, read].

cc    alias              An option in command "debug", to display GEOM internal
cc                         parameters and variables for aliases:  naliasm,
cc                         nalias, alias1, alias2, lalias2, alias1s, lalias1s.
cc                         Synonyms:  [alias, al, a].

cc    alias              Command to display one or more aliases or to create or
cc                         delete an alias.  An alias is a delimited word up to
cc                         24 characters long which, when the initial word of an
cc                         input line, is replaced by a specified character
cc                         string up to 72 characters long.
cc                         The combined number of characters resulting from the
cc                         replacement may not exceed 80.
cc                         Alias replacement is done before symbol replacement.
cc                         The alias replacement may contain symbols,
cc                         which will be replaced before executing the line.
cc                         Only the first word of an input line containing
cc                         multiple statements can be replaced by an alias.
cc                         See "do loop use".
cc
cc                         Note:  no replacement of symbols will be done
cc                         in this command.
cc
cc                         NOTE!  A warning message will be displayed if an
cc                         alias has the same name as a preset synonym or a
cc                         symbol or its replacement.
cc
cc                         Aliases may also be displayed with command
cc                         "synonymn".
cc
cc                         Forms include:
cc
ccin                       help alias
ccin                       alias [help]
cc
cc                           Display the command options.
cc
ccin                       alias [all,list]
cc
cc                           Display all alias pairs.
cc
ccin                       alias list ANAME1 ANAME2 ANAME3 ...
cc
cc                           Display the alias pairs for aliases ANAME1, ANAME2,
cc                           ANAME3, ..., including any aliases with subscripts
cc                           following the specified names.
cc
ccin                       alias ANAME
cc
cc                           Display the alias pair for alias ANAME.
cc
ccin                       alias ANAME = ''
cc                         delete alias ANAME
cc
cc                           Delete the alias pair for alias ANAME.
cc
ccin                       alias ANAME = 'STRING'
cc
cc                           Create alias ANAME with replacement string STRING.
cc                           If STRING is bracketed by single quotes, repeat any
cc                           internal single quotes.  If STRING is bracketed by
cc                           double quotes, repeat any internal double quotes.
cc                           ANAME may not contain the current field delimiter,
cc                           and should not contain any field delimiter that
cc                           will be in use when alias ANAME is used in a
cc                           command.  This might include any of the characters:
cc                           (blank) ,  :  <  >  [  \  ]  ^  _  `  {  |  }
cc                           STRING may contain blanks.
cc
cc                           Use "+" or "-" instead of ANAME to form the name
cc                           by incrementing or decrementing the base name for
cc                           aliases.  See command "last", "increment names".
cc
cc                           Note:  no replacement of symbols will be done
cc                           in this command.
cc
cc                           A warning message will be displayed if an alias has
cc                           the same name as a preset synonym.
cc
cc                         Synonyms:  [", ' (in pairs)], [alias, al, a],
cc                         [help, h].

cc    alias              An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on an alias.
cc                         Synonyms:  [alias, al, a].

cg    alias arrays       Aliases may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    aliases            Some suggested aliases, to shorten long "mesh" and
cg                         "plot" commands, are:
cg
cg                         alias dma   = "mesh - all"
cg                         alias dmbl  = "mesh - block"
cg                         alias dmcl  = "mesh - cluster"
cg                         alias dmp   = "mesh - point"
cg                         alias mbla  = "mesh block all"
cg                         alias mexbl = "mesh exchange block"
cg                         alias mins  = "mesh insert"
cg                         alias minv  = "mesh invert"
cg                         alias mlin  = "mesh linear"
cg                         alias mlst  = "mesh list"
cg                         alias mmv   = "mesh move"
cg                         alias mrx   = "mesh relax"
cg                         alias msz   = "mesh size"
cg                         alias mtst  = "mesh test"
cg                         alias pla   = "plot array"
cg                         alias plax  = "plot axis"
cg                         alias plcen = "plot center"
cg                         alias dpla  = "plot delete all"
cg                         alias dpli  = "plot delete index"
cg                         alias dplm  = "plot delete marker"
cg                         alias plim  = "plot limits"
cg                         alias plma  = "plot marker all"
cg                         alias plm   = "plot marker"
cg                         alias plpa  = "plot point all"
cg                         alias plpi  = "plot point index"
cg                         alias plpm  = "plot point marker"
cg                         alias plsz  = "plot size"
cg                         alias plst  = "plot status"
cg                         alias plt   = "plot title"
cg                         alias zoom  = "plot zoom"
cg                         alias prpa  = "project point all"
cg                         alias whpa  = "where point all"

cg    align              To move a quadric surface of type QTYPE (plane, sphere,
cg                         cylinder, cone, ellipsoid, axisym, quadric, hyperb),
cg                         with name QNAME, to the origin, and/or align its
cg                         symmetry axes with the major axes, execute the
cg                         following:
cg
cg                         point pcen QTYPE QNAME
cg
cg                           Create point pcen, the center of QNAME.
cg
cg                         vector vmove QTYPE QNAME
cg
cg                           Create vector vmove, pointing from the center of
cg                           QNAME to the origin.
cg
cg                         operator oprot QTYPE QNAME
cg
cg                           Create rotation operator oprot, to rotate the
cg                           symmetry axes of QNAME to the major axes.
cg
cg                         Move QNAME to the origin, then align it:
cg
cg                         move QTYPE QNAME vmove
cg
cg                           Translate QNAME to the origin.
cg
cg                         rotate QTYPE QNAME oprot
cg
cg                           Rotate QNAME (around the origin) to align its axes
cg                           with the major axes.
cg
cg                         Or, without first moving QNAME to the origin:
cg
cg                         rotate QTYPE QNAME oprot pcen
cg
cg                           Rotate QNAME (around point pcen) to align its axes
cg                           with the major axes.
cg
cg                         move QTYPE QNAME vmove
cg
cg                           Translate QNAME to the origin.
cg
cg                         See "center".

cc    all                An option in command "help", to display all help
cc                         messages.

cc    all                An option in commands, in place of an object name, to
cc                         indicate that all of the specified type of object are
cc                         to be processed.
cc
cc                         Do not name any object "all", but if you do, use
cc                         command "rename" to rename it, or command "delete"
cc                         to delete it.
cc
cc                         Commands that may affect all named objects include:
cc                         delete, last, list, repack, search, sort.

cc    all                An option in command "plot", to indicate that the
cc                         plot axis limits will include all plot points.

cc    ALNAME             The name of a line between two points.  May have up to
cc                         24 characters, and may be ASCII, integer or floating
cc                         point.  No line name may be "+", "-", "all", "h",
cc                         "help", "list" or "thru", begin with "!" or contain
cc                         ";".
cc                         Also referred to as ALNAME1, ALNAME2, ALNAME3, ...
cc                         Specified with command "line".
cc
cc                         Lines may appear in commands:
cc                         accelerate, cone, copy, delete, distance, help, last,
cc                         line, list, rename, repack, search, sort, track,
cc                         vector.

cc    alph               A synonym for commands.

cc    alph               An option in command "help", to display an alphabetic
cc                         list of command words, including synonyms.

cg    alphabetic         See "alphabetic order".

cg    alphabetic order   To list all commands and their synonyms in alphabetic
cg                         order, use commands "commands" and "synonyms".

cg    altitude           An altitude of a triangle is a line from a vertex
cg                         perpendicular to the opposite edge.  The three
cg                         altitudes intersect at the orthocenter.
cg                         See "median", "bisector", "trig".

cc    AMNAME             In commands "marker" and "plot", any single character,
cc                         except that no marker name may be "!", ";", "+",
cc                         "-" or "h".
cc                         Also referred to as AMNAME1, AMNAME2, AMNAME3, ...
cc                         May include numbers, upper and lower case letters,
cc                         control characters, tabs, etc, but not the current
cc                         field delimiter.

cg    amu                Atomic mass units.
cg                         1 amu = 9.3149432E8 eV (+/- 0.3 ppm).

cc    ANAME              In command "alias", any delimited string up to
cc                         24 characters long, except that no alias name may be
cc                         "+", "-", "all", "h", "help", "list" or "thru",
cc                         begin with "!" or contain ";".
cc                         Also referred to as ANAME1, ANAME2, ANAME3, ...
cc                         May include numbers, upper and lower case letters,
cc                         control characters, tabs, etc, but not the current
cc                         field delimiter.

cg    and                The and or ampersand character, "&".

cc    ang                A synonym for angle.

cc    ang                A synonym for angles in command "angles".

cc    ANG1               See "ANG1, ANG2".

cc    ANG1, ANG2         In command "cone", the first and last vertex
cc                         half-angles of a family of nested cones with the same
cc                         vertex point and axis.

cc    ANG1, ANG2         In command "vector", option "angle", the angles between
cc                         the unit vector being created and vectors VNAME1
cc                         and VNAME2, respectively, in the current angle units.

cc    ANG2               See "ANG1", ANG2.

cc    ANGLE              In command "angles", an angle, specified in degrees,
cc                         radians or grads.
cc                         Also referred to as ANGLE1, ANGLE2, ANGLE3, ...,
cc                         ANGLEN.
cc                         May be integer, floating point or a variable.
cc                         Absolute value in degrees may not exceed the largest
cc                         machine integer.

cc    ANGLE              In command "cone", the half-angle of the cone at
cc                         the vertex.
cc                         See "angles", "DANGLE".

cc    ANGLE              In command "operator", option "axial", the angle
cc                         (counterclockwise, with the axis pointed at the
cc                         observer) of rotation around the axis vector VAXIS.
cc                         See "angles", "DANGLE".
cc                         Note:  angle ANGLE may be sampled randomly.

cc    ANGLE              In command "vector", options "random" and "angle",
cc                         the fixed scattering angle from a specified axis.
cc                         May be integer, floating point or a variable.
cc                         Specified in the current angle units.
cc                         See "angles", "DANGLE".
cc                         Note:  angle ANGLE may be sampled randomly.

cc    angle              A synonym for angles in command "angles".

cc    angle              An option in command "trig", indicating that the next
cc                         argument is a positive vertex angle of a triangle,
cc                         in the current angle units, and less than 180
cc                         degrees or pi radians.
cc                         Synonyms:  [angle, ang].

cc    angle              An option in command "variable", to find the angle
cc                         between two vectors, in the current angle units.
cc                         Synonyms:  [angle, ang].

cc    angle              An option in command "vector", to create a unit vector
cc                         at specified angles from two specified vectors.
cc
cc                         Given a tetrahedron with the three vertex points P1,
cc                         P2 and P3, to find the fourth vertex point P4 such
cc                         that the edge from P1 to P4 makes angle ANG12 with
cc                         the edge from P1 to P2, makes angle ANG13 with the
cc                         edge from P1 to P3, and has length EDGE14, use the
cc                         following commands:
cc
cc                         vector V12 point P1 P2
cc                         vector V13 point P1 P3
cc                         vector V14 angle ANG12 V12 ANG13 V13
cc                         copy point P1 P4
cc                         move point P4 V14 EDGE14

cc    angle              An option in command "vector", to have the vector
cc                         randomly sampled from a uniform distribution of
cc                         azimuth angles around a specified axis vector VAXIS,
cc                         at a fixed angle ANGLE from VAXIS.
cc                         Note:  angle ANGLE itself may be sampled randomly.

cg    angle              See "angle of incidence", "angle of rotation",
cg                         "angle units", "angle, central", "angle, dihedral",
cg                         "angles", "scatter angle".

cg    angle              To find the angle between two vectors, use command
cg                         "variable", option "angle".  The angle is that
cg                         measured from one vector to the other, at the vertex
cg                         formed by placing the tails (the initial points) of
cg                         the two vectors together, so is always between 0 and
cg                         180 degrees (between 0 and pi radians.)

cg    angle of incidence
cg                       The angle of incidence between a linear track and a
cg                         line is the angle between the track and the line at
cg                         the point of intersection.
cg                         The angle of incidence between a linear track and a
cg                         surface is the angle between the track and the vector
cg                         normal to the surface at the point of intersection.

cg    angle of rotation
cg                       A rotation operator OPNAME, specified with command
cg                         "operator", has associated with it an angle
cg                         of rotation around a particular axis, and three
cg                         sequential angles of rotation around the x, y and z
cg                         axes.  These may be displayed with command
cg                         "operator OPNAME".

cg    angle units        See "angles".

cg    angle, central     The central angle is the angle between the two lines
cg                         from a central point to any other two points.

cg    angle, dihedral    The dihedral angle is the angle between two
cg                         intersecting planes.

cc    angles             An option in command "debug", to display GEOM internal
cc                         variables for the coordinate system and angles:
cc                         asys, acoordu, acoordv, acoordw, angunit.
cc                         Synonyms:  [angles, angle, ang].

cc    angles             Command to display or specify the units for angles,
cc                         degrees (default) or radians or to convert angles
cc                         from one unit to another.
cc                         Display of point coordinates and vector components
cc                         will be in the specified angle units, unless
cc                         specifically labeled otherwise.
cc                         See "coordinate conversion", "vector conversion".
cc                         Note:  variables are not affected by changes
cc                         in the angle units, so their values and units remain
cc                         as intended by the user when originally specified.
cc                         Variables may be assigned a descriptive tag to
cc                         specify the units or other data.
cc                         See command "variable", option "tag".
cc
cc                         Command "angles" relates to objects:  symbol,
cc                         variable.
cc
cc                         Forms include:
cc
ccin                       help angles
ccin                       angles help
cc
cc                           Display the command options.
cc
ccin                       angles
cc
cc                           Display the current angle units, and
cc                           display the command options.
cc
ccin                       angles [degrees,radians]
cc
cc                           Set the angle units to [degrees, radians].
cc
ccin                       angles UNIT1 UNIT2 ANGLE1 ANGLE2 ANGLE3 ...
cc
cc                           Convert from UNIT1 ("degrees", "grads" or
cc                           "radians") to UNIT2 ("degrees", "grads", "radians"
cc                           or "dms"):  ANGLE1, ANGLE2, ANGLE3, ...
cc                           The absolute value of ANGLE1, ANGLE2 or ANGLE3, ...
cc                           in degrees must not exceed the largest machine
cc                           integer.
cc
ccin                       angles dms UNIT2 DEGREES MINUTES SECONDS
cc
cc                           Convert from degrees, minutes, seconds to UNIT2
cc                           ("degrees", "grads" or "radians").
cc                           The absolute values of DEGREES, MINUTES and SECONDS
cc                           in degrees must not exceed the largest machine
cc                           integer.
cc
ccin                       angles UNIT1 UNIT2 ANGLE1 thru ANGLEN
cc
cc                           Convert from UNIT1 ("degrees", "grads" or
cc                           "radians) to UNIT2 ("degrees", "grads", "radians"
cc                           or "dms"):  all stored variables with names in the
cc                           range from ANGLE1 thru ANGLEN.
cc                           The absolute values of ANGLE1 thru ANGLEN
cc                           in degrees must not exceed the largest machine
cc                           integer.
cc
cc                         Conversion between degrees and radians:
cc                           degrees = radians * 180 / pi
cc                           radians = degrees * pi / 180
cc                           See "deg/rad", "pi", "rad/deg".
cc
cc                         See command "coordinate" to specify the coordinate
cc                         system.
cc
cc                         Synonyms:  [angles, angle, ang],
cc                         [degrees, deg, degree], [grads, grad].
cc                         [help, h], [radians, rad, radian].

cc    ANGUNIT            The name of the unit for measuring angles, degrees
cc                         (default) or radians.  Specified with command
cc                         "angles".

cg    annular            See "annulus", "annular disk", "disk".

cg    annular disk       See "disk".

cg    annulus            An annulus is the space between two surfaces at
cg                         different radii from a central point or an axis.
cg                         See "disk".
cg                         To create zones consisting of cylindrical or
cg                         spherical annuli, use command "cylinder" or "sphere",
cg                         option "concentric" or "scale", and command "zone".

cg    appended           See "appended comment".

cg    appended comment   To append a comment to any input line, precede the
cg                         comment with the field delimiter and the character
cg                         "!".  Do not begin the name of any alias, marker or
cg                         other object with the character "!".

cg    approximation      A good approximation to the decimal part of a number
cg                         may be obtained by finding the first few terms of
cg                         the continued fraction for the number (see
cg                         "continued fraction", command "contfr"), or the first
cg                         few terms of the number expressed as a series of
cg                         reciprocals of integers (see "reciprocal series",
cg                         command "iris").

cg    APT                The All-Particle Tracking subroutine library.
cg                         APT subroutines currently called directly by GEOM:
cg                         aptaxis, aptbang, aptbite, aptbrkn, aptbrkp, aptbrks,
cg                         aptbrkv, aptcang, aptchai, aptchap, aptchfp, aptchfs,
cg                         aptchia, aptchin, aptchmv, aptchrn, aptchrp, aptchsq,
cg                         aptcinc, aptcirc, aptcirk, aptcirp, aptclis, aptcris,
cg                         aptcsys, aptcsyv, aptcube, aptcubs, aptcycy, aptdode,
cg                         aptetrn, aptetrp, aptetru, aptetrw, aptexpl, aptffip,
cg                         aptgrat,
cg                         apticos, aptintq, aptlnlc, aptlnln, aptlnpl, aptlocd,
cg                         aptlocs, aptmaxw, aptmopv, aptnewt, aptnint, aptnorm,
cg                         aptocta, aptparb, aptparh, aptparl, aptparp, aptparq,
cg                         aptpars, aptpart, aptparx, aptpers, aptpfit, aptplcy,
cg                         aptplis, aptplpl, aptplqu, aptplsp, aptpolf, aptpoly,
cg                         aptprcy, aptprop, aptprsp, aptptln, aptptpl, aptqexc,
cg                         aptqext, aptqexv, aptqfit, aptqhyp, aptqnor, aptqper,
cg                         aptqprr, aptqprt, aptqrts, aptquar, aptqupr, aptrabc,
cg                         aptradv, aptrcut, aptrefq, aptrefs, aptrext, aptrich,
cg                         aptrins, aptripl, aptripq, aptrips, aptripv, aptrkis,
cg                         aptrlop, aptrois, aptrota, aptrotq, aptrots, aptrott,
cg                         aptrotv, aptscan, aptscat, aptscav, aptsciz, aptscll,
cg                         aptsclr, aptslid, aptsolv, aptspcy, aptspha, aptsphk,
cg                         aptsphp, aptspod, aptspsp, aptsver, apttetd, apttetr,
cg                         apttlod,
cg                         apttran, apttrig, apttrip, apttris, aptvadd, aptvang,
cg                         aptvaxb, aptvdis, aptvdot, aptvpln, aptvplp, aptvsum,
cg                         aptvunb, aptvunz, aptvusz, aptvxun, aptwhis, aptwirl,
cg                         aptwist, aptxnup, ranf.
cg
cg                         Many others are called by these.  See file apt.link
cg                         in directory ~edwards/work/apt/doc .
cg                         See files geom_apt_calls, geom_apt_called_by .

cn    apt.link           A file listing all links between APT subroutines.
cn                         In directory ~edwards/work/apt/doc .

cn    aptflibe           The TAR library of APT subroutine source files.
cn                         in ~edwards/work/apt/src on the open YANA Cluster.
cn                         in /users/u47/edwards/apt/src on FAST archive.
cn                         See "libapt.a".

cc    AQU                See "AQU, AQV, AQW".

cc    AQU, AQV, AQW      In command "zone", the name of the initial quadric
cc                         surface in a family of quadric surfaces.

cc    AQV                See "AQU, AQV, AQW".

cc    AQW                See "AQU, AQV, AQW".

cc    arc                Command to find, at a point on one of a family of
cc                         quadric surfaces, the normal vector; the number of
cc                         curves passing through the point, and lying in the
cc                         surface, and in a plane containing the normal vector,
cc                         having zero or extreme curvature; and for each such
cc                         curve, the corresponding radius of curvature, center
cc                         of curvature, and unit direction vector along the
cc                         curve u = (ux, uy, uz).  See "vector field",
cc                         "principal radius", command "extrema".
cc
cc                         Command "arc" relates to objects:  axisym, cone,
cc                         cylinder, ellipsoid, hyperb, plane, point, quadric,
cc                         sphere, symbol.
cc
cc                         Forms include:
cc
ccin                       help arc
ccin                       arc [help]
cc
cc                           Display the command options.
cc
ccin                       arc PNAME QNAME
cc
cc                           Find, at point PNAME on a member of the family of
cc                           quadric surfaces represented by quadric surface
cc                           QNAME: the normal vector; the number of curves
cc                           passing through point PNAME, and in a plane
cc                           containing the normal vector, having zero or
cc                           extreme curvature; and for each such curve,
cc                           the corresponding radius of curvature, center of
cc                           curvature, and unit direction vector along the
cc                           curve through point PNAME.
cc                           If any of the surface curves has zero curvature,
cc                           the surface is a ruled surface.
cc                           If QNAME is a plane, four straight lines, at 45
cc                           degree intervals, will be returned.
cc
cc                         Synonyms:  [help, h]

cg    Archimedean        See "archimedean spiral".

cg    Archimedean spiral
eg                       An Archimedean spiral is a curve in a plane
cg                         perpendicular to an axis, with its radial distance
cg                         from the axis a linear function of the angular
cg                         coordinate around the axis, with the angle ranging
cg                         continuously from zero to infinity.
cg                         To generate a family of points arrayed along an
cg                         Archimedean spiral, use command "point ... move" to
cg                         create a family of points along a straight line
cg                         perpendicular to the axis, then use command "cluster"
cg                         to create a cluster consisting of the points, and
cg                         then operate on the cluster with command "twist",
cg                         option "radial".

cn    archive            A file containing a list of all files last saved in
cn                         the archives.

cg    area               See "area of brick", "area of disk", "area of polygon",
cg                       "area of quadric", "area of revolution", "area units".

cc    area               Command to find the projected area of a general or
cc                         regular polygon, the average edge length, and the
cc                         average of the vertex points.
cc
cc                         Command "area" relates to objects: cluster, point,
cc                         symbol, vector.
cc
cc                         Forms include:
cc
ccin                       help area
ccin                       area [help]
cc
cc                           Display the command options.
cc
ccin                       area VNORM point PNAME1 PNAME2 PNAME3 ...
cc
cc                           Find the area projected perpendicularly onto a
cc                           plane with normal vector VNORM, of the general
cc                           polygon with three or more vertex points PNAME1,
cc                           PNAME2, PNAME3, ..., which need not be coplanar.
cc                           Also display the average of the vertex points, and
cc                           the average edge length.
cc
ccin                       area VNORM cluster CLNAME
cc
cc                           Find the area projected perpendicularly onto a
cc                           plane with normal vector VNORM, of the general
cc                           polygon formed by the points in the cluster CLNAME,
cc                           which need not be coplanar.  Also display the
cc                           average of the vertex points, and the average edge
cc                           length.
cc
cc                         Synonyms:  [cluster, cl], [help, h],
cc                         [point, p, pnt, pt].

cg    area               To find the area of the parallelogram or triangle
cg                         between two vectors, use command "cross".
cg                         The parallelogram or triangle is that formed by
cg                         placing the tails (the initial points) of the two
cg                         vectors together, and completing the figure.
cg                         To find the area of the triangle specified by three
cg                         points, use command "triangle".
cg                         To find the area of the triangle specified by three
cg                         of its parts (three edges and three angles), use
cg                         command "trig".
cg                         To find the area of a regular polygon, use command
cg                         "polygon".
cg                         To find the projected area of a regular or general
cg                         polygon, use command "area".
cg                         To find the area of a circle, use command "circle",
cg                         "disk" or "intcirc".
cg                         To find the area of an annular disk, use command
cg                         "disk".
cg                         To find the surface area of a sphere or the cross
cg                         section area of a circular cylinder, use command
cg                         "cylinder" or "sphere".
cg                         To find the areas of the faces of a "brick", use
cg                         command "brick".
cg                         To find the area of the faces of a regular
cg                         polyhedron, use command "polyhedron".
cg
cg                         To find other areas, use command "area".

cg    area of brick      The area AREA(u) of a surface with a fixed value of one
cg                         of the coordinates u of an orthogonal coordinate
cg                         system (u, v, w), and bounded by two fixed values of
cg                         each of the other two coordinates v and w, is given
cg                         below.
cg                         Cartesian coordinates (x, y, z):
cg                           AREA(x)     = (Y2 - Y1) * (Z2 - Z1)
cg                           AREA(y)     = (Z2 - Z1) * (X2 - X1)
cg                           AREA(z)     = (X2 - X1) * (Y2 - Y1)
cg                         Cylindrical coordinates (rcyl, theta, z)
cg                           (cylindrical radius, azimuthal angle, z axis)
cg                           (rcyl => 0, 0 <= theta <= 2*pi):
cg                           AREA(rcyl)  = rcyl * (Z2 - Z1) * (theta2 - theta1)
cg                           AREA(theta) = (rcyl2 - rcyl1) * (Z2 - Z1)
cg                           AREA(z)     = (rcyl2^2 - rcyl1^2) *
cg                                         (theta2 - theta1) / 2
cg                           Angle theta = arctan (y / x), in radians, and
cg                           pi = 3.141592653589793...
cg                         Spherical coordinates (rsph, theta, phi)
cg                           (spherical radius, azimuthal angle, polar angle)
cg                           (rsph => 0, 0 <= theta <= 2*pi, 0 <= phi <= pi):
cg                           AREA(rsph)  = rsph^2 * (cos (phi1) - cos (phi2)) *
cg                                         (theta2 - theta1)
cg                           AREA(theta) = (rsph2^2 - rsph1^2) *
cg                                         (phi2 - phi1) / 2
cg                           AREA(phi)   = (rsph2^2 - rsph1^2) * sin (phi) *
cg                                         (theta2 - theta1) / 2
cg
cg                         See "brick".

cg    area of disk       The annular area of an annular disk with inner radius
cg                         RADKIN and outer radius RADKOUT is:
cg                         AREADK = pi * (RADKOUT^2 - RADKIN^2)
cg                         The inner area (the "hole") is:
cg                         AREAH  = pi * RADKIN^2
cg                         pi = 3.141592653589793...

cg    area of polygon    The area of a polygon in the plane z = 0, with N
cg                         vertices, is given by the absolute value of:
cg                         area = 0.5 * sum(i = 1, N) {(x(i) + x(i+1)) *
cg                         (y(i) - y(i+1))}, where N + 1 -> 1.
cg                         See command "area".
cg
cg                         The area of any plane figure, projected onto a plane,
cg                         is equal to the original area multiplied by the
cg                         cosine of the angle between the normal vectors of the
cg                         two planes.  See commands "dot" and "variable".
cg
cg                         To find the area of the faces of a regular
cg                         polyhedron, use command "polyhedron".

cg    area of quadric    See "ellipsoid", "sphere".

cg    area of revolution
cg                       An area of revolution, formed by rotating the line
cg                         with end points (rcyl1, z1) and (rcyl2, z2) around
cg                         the z axis from azimuthal angle theta1 to azimuthal
cg                         angle theta2:
cg                         AREA = ((rcyl1 + rcyl2) / 2) *
cg                                (z2 - z1) * (theta2 - theta1)

cg    area units         See "conv.mac", "conversion factors".

cc    ARGUMENT           An argument in an input line.  Always delimited on the
cc                         left by the beginning of the line or a field
cc                         delimiter, and on the right by a field delimiter or
cc                         the end of the line (80 characters).  If the field
cc                         delimiter is a blank character, initial and trailing
cc                         blanks are ignored, and multiple blanks are treated
cc                         as a single blank.  If the field delimiter is not a
cc                         blank character, trailing null fields are ignored,
cc                         but an initial null field is an error.  Do not begin
cc                         an input line with a non-blank field delimiter.
cc                         The field delimiter is initially a blank character,
cc                         but may be changed with command "delimiter".
cc                         The argument may be ASCII, integer, floating point,
cc                         a variable whose value has the same mode as the
cc                         argument, or a symbolic word which, after symbol
cc                         replacement, has the same mode as the argument.
cc                         See command "variable".

cg    arguments          Command arguments are displayed in lower case for
cg                         arguments to be typed literally as shown, and in
cg                         UPPER CASE for arguments for which the user must
cg                         provide a value, which may be ASCII, an integer or
cg                         floating point variable name, an integer or floating
cg                         point value, or a word which when modified by symbol
cg                         replacement, is one of the above.
cg                         The first argument of a command, the command word
cg                         itself, is always in lower case.
cg
cg                         With certain exceptions (see "symbol"), any argument,
cg                         may be replaced by a symbolic word which, after
cg                         symbol replacement, must be an acceptable value for
cg                         the argument, including the name of an integer or
cg                         floating point variable or a literal argument.
cg
cg                         Any argument which allows a floating point value may
cg                         also be replaced by the name of an integer or
cg                         floating point variable, or by a symbolic word which,
cg                         after symbol replacement, is an integer or floating
cg                         point value.  See "VARNAME".
cg
cg                         Any argument which only allows an integer value may
cg                         also be replaced by an integer variable, or by a
cg                         symbolic word which, after following symbol
cg                         replacement, is an integer value, except for the
cg                         argument LINE in command "indo" or "redo".
cg
cg                         With certain exceptions, arguments are always
cg                         delimited from other arguments by the current
cg                         field delimiter (see command "delimiter"), so cannot
cg                         contain internal field delimiters.  The exceptions
cg                         are the argument STRING in command "alias",
cg                         the argument TEXT in command "marker", and the
cg                         argument ENTRY in command "define", which must also
cg                         be bracketed by single or double quotes.

cc    arguments          Numerical values that are arguments in commands, and
cc                         may be replaced by a variable or a symbolic word
cc                         which, after symbol replacement, is a variable,
cc                         include:
cc
cc                         Integer only     Commands
cc                         --------------   --------
cc                         I1, I2, I3       indo
cc                         INC              variable, vector, point, plane
cc                         INC              sphere, cylinder, cone, ellipsoid
cc                         INC              axisym, quadric, brick, tetrahedron
cc                         INC              zone, increment
cc                         INCR             symbol
cc                         INT1, INT2, ...  big
cc                         IP1,IP2,IP3,IP4  tetrahedron
cc                         IQ1, IQ2, ...    zone
cc                         IQU,IQV,IQW      zone
cc                         IVAR             variable
cc                         IZU,IZV,IZW      zone
cc                         K, L, M          mesh, variable
cc                         K1, K2           mesh
cc                         KMAX,LMAX,MMAX   mesh
cc                         KMOVE,LMOVE      mesh
cc                         L1, L2           mesh
cc                         M1, M2           mesh
cc                         M1, M2, M3       factor
cc                         MINUTES          angles
cc                         MMOVE            mesh
cc                         N                icalc, spin
cc                         N1, N2           mesh, plot
cc                         N3               plot
cc                         NCOLUMNS         plot
cc                         NITMAX           root
cc                         NLINES           plot
cc                         NQU,NQV,NQW      zone
cc                         NSAMP            sample
cc                         NUMAX            axisym
cc                         NUMBR            brick
cc                         NUMCONE          cone
cc                         NUMCYL           cylinder
cc                         NUMDIR           proximal
cc                         NUMELL           ellipsoid
cc                         NUMLINE          line
cc                         NUMPL            plane
cc                         NUMPT            point
cc                         NUMQ             quadric
cc                         NUMSPH           sphere
cc                         NUMSTR           increment
cc                         NUMT             ratio
cc                         NUMTET           tetrahedron
cc                         NUMTR            triangle
cc                         NUMV             vector
cc                         NUMVAR           variable
cc                         NUMZN            zone
cc                         NVER             polygon, polyhedron
cc
cc                         Integer or
cc                         floating point   Commands
cc                         --------------   --------
cc                         A                variable
cc                         A(0), A(1), ...  root
cc                         A0, A1, A2, ...  roots
cc                         ABSLEN           vector
cc                         ANG1, ANG2       cone
cc                         ANGLE            cone, operator, quadric
cc                         ANGLE1-ANGLEN    angles
cc                         AQU, AQV, AQW    zone
cc                         AVAR             variable
cc                         AXQ(1), AXQ(2)   axisym
cc                         B                variable
cc                         BTEMP            bin
cc                         DANGLE           plane
cc                         DEGREES          angles
cc                         DEV              bin
cc                         DMAX             walk
cc                         DMEAN            walk
cc                         DPR              bin
cc                         DPRL, DPRR       bin
cc                         DT               accelerate
cc                         DU, DV, DW       vector, plane, slice
cc                         DX               root
cc                         F1, F2, ..., FN  variable
cc                         FACT             quadric
cc                         FADD             variable
cc                         FMULT            brick, variable
cc                         FU, FV, FW       point
cc                         FVAR             variable
cc                         PATHMAX          walk
cc                         PHI              point
cc                         PITCH            twist
cc                         POW              bin
cc                         POWER            bin
cc                         PRTOT            bin
cc                         PTL, PTR         project
cc                         QC to QZZ        quadric
cc                         R1, R2, R3       redo
cc                         RAD1, RAD2       sphere, cylinder, kiss
cc                         RAD3, RAD4       kiss
cc                         RADIUS           point, sphere, cylinder
cc                         RADIUS1, RADIUS2 intcirc
cc                         RADKIN           disk
cc                         RADKOUT          disk
cc                         RATIO            bin, operator
cc                         RCYL             point
cc                         RELLEN           vector
cc                         RINV             twist
cc                         RSPH             point
cc                         SAX, SAY, SAZ    ellipsoid
cc                         SCENH, SCENV     plot
cc                         SECONDS          angles
cc                         SIGMA            bin
cc                         SMAX, SMIN       plot
cc                         SUM              ratio
cc                         T1, T2, T3       accelerate
cc                         TERM1            ratio
cc                         THETA            point, vector, plane
cc                         TMAX, TMIN       accelerate
cc                         TOL              tol
cc                         U, V, W          vector, plane, intcirc
cc                         U1, U2           intcirc
cc                         UANGLE           operator
cc                         UMAX, UMIN       brick
cc                         V1, V2           intcirc
cc                         VACC             accelerate
cc                         VAL              quadric
cc                         VALUE            variable
cc                         VANGLE           operator
cc                         VINIT            accelerate
cc                         VMAX, VMIN       brick
cc                         VMULT            move
cc                         VMULT1, VMULT2   vector
cc                         VNORM            area, disk, plane, project
cc                         VRAN             bin
cc                         VRANL, VRANR     bin
cc                         W1,W2,W3,W4      point
cc                         WANGLE           operator
cc                         WMAX, WMIN       brick
cc                         X(1), X(2), ...  root
cc                         X, Y, Z          point
cc                         X1,X2,X3,X4      roots (real part)
cc                         XMAX, XMIN       root
cc                         XROOT            rootf
cc                         Y1,Y2,Y3,Y4      roots (imaginary part)
cc                         ZOOMULT          plot
cc                         ZVOL             zone

cc    arith              A synonym for arithmetic.

cc    arithmetic         An option in command "debug", to display GEOM internal
cc                         parameters and variables for integer and floating
cc                         point arithmetic:  idmax, amode, abind, iemax, tol.
cc                         These include the maximum size of integers and
cc                         floating point machine words, the number base for
cc                         input to command "icalc", and the option for display
cc                         of the binary form of output from command "icalc".
cc                         See commands "icalc", "tol", "big".
cc                         Synonyms:  [arithmetic, arith].

cg    arithmetic         See "arithmetic series".

cg    arithmetic         Arithmetic may be done on floating point and integer
cg                         values with command "variable".
cg                         Arithmetic may be done on integer values with
cg                         commands "icalc", "big".
cg
cg                         The real and/or complex roots of a quadratic, cubic
cg                         or quartic polynomial equation with real coefficients
cg                         may be found with command "roots".
cg
cg                         The real coefficients of a polynomial equation with
cg                         from two to four specified real and/or complex roots
cg                         may be found with command "roots", option "=".
cg
cg                         The real coefficients of an Nth-order polynomial
cg                         equation with N specified real roots may be found
cg                         with command "root", option "=".
cg
cg                         To search for real roots, extrema and inflection
cg                         points of a polynomial equation using Newtonian
cg                         iteration, use command "root".
cg
cg                         The value of a quadratic, cubic or quartic polynomial
cg                         equation with real coefficients may be found for any
cg                         real or complex argument with command "roots",
cg                         option "?".
cg
cg                         The value of a polynomial with specified coefficients
cg                         and a specified argument may be found with command
cg                         "variable", option "polynomial".
cg
cg                         The ratio for a geometric series with a given sum
cg                         first term and number of terms may be found using
cg                         command "ratio".
cg
cg                         To find the continued fraction for a decimal number,
cg                         or vice versa, use command "contfr".
cg
cg                         To find the equivalent of a decimal number, expressed
cg                         as an integer plus a sum of reciprocals of integers,
cg                         use command "iris".

cg    arithmetic series
cg                       An arithmetic series is one in which the value of each
cg                         term after the first term differs from the preceding
cg                         term by an additive constant.  To create a family of
cg                         variables whose values form an arithmetic
cg                         series:
cg
cg                         variable VAR(1) = VALUE
cg
cg                           Create variable VAR(1), with the value of the
cg                           first term.
cg
cg                         variable VAR(2) series NUMVAR INC VAR(1) FADD 1
cg
cg                           Create a series of NUMVAR variables VAR(2),
cg                           ..., with names incremented by INC characters
cg                           (or INC digits, if an integer name or a name with
cg                           one or more integer subscripts), each equal to the
cg                           preceding variable plus FADD, starting from
cg                           variable VAR(1).
cg                           See "increment names".
cg
cg                         Alternately, using subscripts and a do loop:
cg
cg                         variable VAR(1) = VALUE
cg                         do NL 2 NUMVAR
cg                           variable NLM = NL - 1
cg                           variable VAR(NL) = VAR(NLM) + FADD
cg                         enddo
cg
cg                         To create a family of points, planes or quadric
cg                         surfaces in which the distance of each from an
cg                         invariant point, axis or plane forms an arithmetic
cg                         series, use option "move" of commands "point",
cg                         "sphere", "cylinder", "cone", "ellipsoid", "axisym"
cg                         or "quadric", or use command "plane", option
cg                         "parallel".

cg    array              See "array shape", 'array sizes".

cc    array              An option in command "rename", to replace the stem name
cc                         of all members of an array with a new stem name.  A
cc                         stem name without a subscript will also be replaced.

cc    array              An option in command "plot", to display the plot array,
cc                         and all of the plot parameters.

cg    array              An object with a name of the form OBJNAME(N),
cg                         OBJNAME_N. or OBJNAME.N, where N may have more than
cg                         one value, is a member of an array.
cg                         The form using parentheses, with a variable name
cg                         for N, may be used for any object.

cg    array shape        A two-dimensional array x(i,j), i = 1, imax,
cg                         j = 1, jmax, is equivalent to a one-dimensional
cg                         array y(n), n = 1, kmax,
cg                         kmax = imax * jmax,
cg                         n    = j + jmax * (i - 1),
cg                         i    = 1 + (n - 1) / jmax,
cg                         j    = 1 + mod (n - 1, jmax).
cg
cg                       A three-dimensional array x(i,j,k), i = 1, imax,
cg                         j = 1, jmax, k = 1, kmax, is equivalent to a
cg                         one-dimensional array y(n), n = 1, nmax, where
cg                         nmax = imax * jmax * kmax,
cg                         n    = k + kmax * (j - 1) + kmax * jmax * (i - 1),
cg                         i    = 1 + (n - 1) / (jmax * kmax),
cg                         j    = 1 + mod ((n - 1) / kmax, jmax),
cg                         k    = 1 + mod ((n - 1), kmax).
cg
cg                       See "shape".

cg    array sizes        See "maximum number", commands "tables", "mesh".

cg    arrays             See "subscript", "subscripted names", "do loops",
cg                         commands "do" and "enddo".

cg    ASCII              See "ASCII variable", "ASCII word".

cg    ASCII              When used to describe arguments in commands, ASCII
cg                         means symbolic, i.e., not explicitly integer or
cg                         floating point.  See "integer input",
cg                         "floating point".
cg
cg                         ASCII characters are The set of characters used for
cg                         data of type character.
cg
cg                         The order of available keyboard characters is:
cg
cg                         (blank)  !  "  #  $  %  &  '  (  )    *  +  ,  -
cg                         .  /   0-9  :  ;  <  =  >  ?  @  A-Z  [  \  ]  ^
cg                         _  `   a-z  {  |  }  ~
cg
cg                         See "character set".
cg
cg                         Objects may be sorted into ASCII order of their
cg                         names, using command "sort", with options to sort
cg                         into increasing, decreasing or random order.
cg
cg                         For the order of entries in this file, case is
cg                         ignored.

cg    ASCII variable     See "symbol", command "symbol".

cg    ASCII word         See "ASCII", "commands", "key words".

cc    asin               An option in command "variable", argument FUNCTION,
cc                         to indicate the trigonometric inverse sine
cc                         function.  A asin B means A * asin (B).

cf    asin               The trigonometric inverse sine function.
cf                         Example:  y = asin (x) means y is the angle whose
cf                         sine is x, and x = sin (y).  The result is in
cf                         radians, between -pi and pi.

cc    asinh              An option in command "variable", argument FUNCTION,
cc                         to indicate the inverse hyperbolic sine function.
cc                         A asinh B means A * asinh (B).

cf    asinh              The inverse hyperbolic sine function.
cf                         Example:  y = asinh (x) means y is the argument for
cf                         which the hyperbolic sine is x, and x = sinh (y).
cf
cf                         asinh (x) =  ln ( x + sqrt (x^2 + 1)), x >= 0,
cf                         asinh (x) = -ln (-x + sqrt (x^2 + 1)), x <= 0.

cg    ASSIGNED           In the display for a point, indicates a mesh point,
cg                         assigned a set of mesh indices (k, l, m).

cg    asymptotes         The asymptotes of a hyperbola, a hyperbolic cylinder,
cg                         or a hyperboloid are the intersecting lines,
cg                         intersecting planes, or the cone, respectively,
cg                         resulting from changing the constant term in the
cg                         equation of the quadric curve or surface to zero.

cg    at symbol          The "at" symbol, "@".

cc    atan               An option in command "variable", argument FUNCTION,
cc                         to indicate the trigonometric inverse tangent
cc                         function.  A atan B means A * atan (B).

cf    atan               The trigonometric inverse tangent function.
cf                         Example:  y = atan (x) means y is the angle whose
cf                         tangent is x, and x = tan (y).  The result is in
cf                         radians, between -pi and pi.

cc    atan2              An option in command "variable", argument FUNCTION,
cc                         to indicate the trigonometric inverse tangent
cc                         function.  A atan2 B means atan2 (A, B)

cf    atan2              The trigonometric inverse tangent function, with two
cf                         arguments.  Example:  z = atan2 (x, y) means z is the
cf                         angle whose tangent is x / y, or tan (z) = x / y.
cf                         The result is in radians, between -pi and pi, in the
cf                         quadrant for which sin (z) = x, cos (z) = y.

cc    atanh              An option in command "variable", argument FUNCTION,
cc                         to indicate the inverse hyperbolic tangent function.
cc                         A atanh B means A * atanh (B).

cf    atanh              The inverse hyperbolic tangent function.
cf                         Example:  y = atanh (x) means y is the argument for
cf                         which the hyperbolic tangent is x, and x = tanh (y).
cf                         The range of x is -1 < x < 1.
cf
cf                         atanh (x) = 0.5 * ln ((1 + x) / (1 - x)).

cc    avg                An option in command "variable", argument FUNCTION,
cc                         to indicate the average function.
cc                         A avg B means (A + B) / 2.

cg    axial              See "axial symmetry".

cc    axial              An option in command "operator", to specify a rotation
cc                         by an angle ANGLE (counterclockwise, with the axis
cc                         pointed at the observer) around an axis VNAME.
cc                         See "planar", "serial", "triple", "vector".

cc    axial              An option in command "twist", to specify a rotation of
cc                         one or more points around an axis in proportion to
cc                         the axial distance of each point from a fixed point
cc                         on the axis.

cg    axial symmetry     Axial symmetry exists when an object may be rotated
cg                         by any amount around a particular geometric axis,
cg                         without changing shape.  Any section through the
cg                         object in a plane perpendicular to the symmetry axis
cg                         must contain only circles concentric around the axis.
cg                         Any section through the object in a plane containing
cg                         the symmetry axis must be the same for all such
cg                         planes.  See "axisymmetric".

cg    axially            See "axially symmetric".

cg    axially symmetric
cg                       See "axisymmetric".

cg    axis               See "axis of rotation", "axis of symmetry",
cg                         "principal axis".

cg    axis               An axis is a direction in 3-D space, usually associated
cg                         with a point through which the axis passes.
cg                         In Cartesian or rectangular coordinates, the major
cg                         axes are the x, y and z axes, passing through the
cg                         origin, and forming a right-hand orthogonal triple,
cg                         parallel to the thumb, first and middle fingers,
cg                         respectively.  In any other coordinate system, the
cg                         principal axes may be specified relative to the x, y
cg                         and z axes of a Cartesian or rectangular coordinate
cg                         system.
cg
cg                         A vector specifies an axis in 3-D space, if bound to
cg                         a point, or an infinite set of parallel axes, if not.
cg                         A vector and a point specify an axis in 3-D space.
cg
cg                         Any quadric surface has a symmetry with respect to
cg                         a central point, and three orthogonal axes, one, two
cg                         or three of which may be arbitrary.  The principal
cg                         axis transformation rotates these three axes, the x',
cg                         y' and z' axes, to the x, y and z axes,
cg                         and translates the central point to the origin.
cg                         The direction of the axis vector at a point on the
cg                         axis is parallel to the normal vector of the quadric
cg                         surface at that point.
cg
cg                         To find the axis associated with a scaling,
cg                         reflection or rotation operator, use command
cg                         "operator".
cg
cg                         See "transverse", "conjugate".

cc    axis               An option in command "plot", to display the plot axis
cc                         coordinates and labels [and specify the horizontal or
cc                         vertical axis coordinate and label].

cg    axis of rotation   A rotation operator OPNAME, specified with command
cg                         "operator", has associated with it an axis
cg                         and angle of rotation, which may be displayed with
cg                         command "operator OPNAME".

cg    axis of symmetry   A geometric object has an axis of symmetry if the
cg                         projection of the object in any plane through that
cg                         axis is the same.  In cylindrical coordinates, with
cg                         the z axis the axis of symmetry, the description of
cg                         the object is independent of the theta coordinate.
cg                         See "axisymmetric".

cg    axisym             See "axisymmetric", "quadric surface",
cg                         "axisym zone".

cc    axisym             In a command, means an axially symmetric quadric
cc                         surface (except in command "axisym", option "scale",
cc                         when the scaling operator is linear or radial, with
cc                         an axis not through the center of the base sphere, or
cc                         not parallel to that of the base axially symmetric
cc                         quadric surface).
cc                         Axially symmetric quadric surfaces include all types
cc                         of planes, spheres, circular cylinders, circular
cc                         cones, circular ellipsoids (oblate and prolate
cc                         spheroids), circular paraboloids, and circular
cc                         hyperboloids of one or two sheets.
cc                         See "axisymmetric", "axisym zone".

cc    axisym             Command to display or create one or more axially
cc                         symmetric quadric surfaces (except that linear or
cc                         radial scaling may create axially unsymmetric quadric
cc                         surfaces).  The normal vectors will be radially
cc                         outward.  Creating a quadric surfaces replaces any
cc                         existing quadric surface having the same name.
cc                         See "quadric" for other options.
cc                         See "do loop use".
cc
cc                         Command "axisym" relates to objects:  axisym, cone,
cc                         cylinder, ellipsoid, operator, plane, point, quadric,
cc                         sphere, symbol, variable, vector.
cc
cc                         Forms include:
cc
ccin                       help axisym
ccin                       axisym [help]
cc
cc                           Display the command options.
cc
ccin                       axisym [all,list]
cc
cc                           Display all axially symmetric quadric surfaces
cc                           (short display).
cc
ccin                       axisym list AXNAME1 AXNAME2 AXNAME3 ...
cc
cc                           Display axially symmetric quadric surfaces AXNAME1,
cc                           AXNAME2, ..., including any with the specified
cc                           names followed by one or more subscripts (short
cc                           display).
cc
ccin                       axisym AXNAME
cc
cc                           Display axially symmetric quadric surface AXNAME
cc                           (long display).
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         AXNAME or AXQ(2) to form the name by incrementing or
cc                         decrementing the base name for axially symmetric
cc                         quadric surfaces.
cc                         See command "last", "increment names".
cc
ccin                       axisym AXNAME fit PCEN VAXIS PNAME2 PNAME3
cc
cc                           Create axially symmetric quadric surface AXNAME,
cc                           centered at point PCEN, with axis vector VAXIS, and
cc                           passing through the two points PNAME2 and PNAME3.
cc
ccin                       axisym AXNAME focus PFOC VAXIS PNAME2 PNAME3
cc
cc                           Create axially symmetric quadric surface AXNAME,
cc                           with focus at point PFOC, with axis vector VAXIS,
cc                           and passing through the two points PNAME2 and
cc                           PNAME3.
cc                           See "planetary orbits".
cc
ccin                       axisym AXQ(2) move NUMAX INC AXQ(1) VMOVE
cc
cc                           Create a family of NUMAX axially symmetric quadric
cc                           surfaces AXQ(2), ..., with names incremented by INC
cc                           characters or digits, and spaced at intervals of
cc                           vector VMOVE, starting from axially symmetric
cc                           quadric AXQ(1).  See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       axisym AXQ(2) rotate NUMAX INC AXQ(1) OPNAME PINV
cc
cc                           Create a family of NUMAX axially symmetric quadric
cc                           surfaces AXQ(2), ..., with names incremented by INC
cc                           characters, by rotating the preceding axially
cc                           symmetric quadric surface with operator OPNAME and
cc                           invariant point PINV, starting from axially
cc                           symmetric quadric surface AXQ(1).
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       axisym AXQ(2) scale NUMAX INC AXQ(1) OPNAME PINV
cc
cc                           Create a family of NUMAX quadrics AXQ(2), ..., with
cc                           names incremented by INC characters, by scaling the
cc                           preceding quadric surface with operator OPNAME and
cc                           invariant point PINV, starting from axially
cc                           symmetric quadric surface AXQ(1).
cc                           WARNING:  a linear or radial scaling axis not
cc                           parallel to that of AXQ(1) will produce axially
cc                           unsymmetric quadric surfaces; a linear or radial
cc                           scaling will change spheres to ellipsoids.
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
cc                         Synonyms:  [help, h], [move, mv, trans, translate],
cc                         [rotate, rot].

cc    axisym             An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on an axially symmetric quadric
cc                         surface.

cg    axisym zone        A zone is axially symmetric if it is bounded by
cg                         surfaces that are all axially symmetric around the
cg                         same axis.
cg                         If all of those surfaces are planes, cylinders or
cg                         cones, the projection of the zone in a plane through
cg                         the axis is a polygon bounded entirely by straight
cg                         lines, and the volume of the zone and the area of the
cg                         projected polygon can be calculated.  See commands
cg                         "volume" and "area".  If the vertices of the polygon
cg                         are known, the bounding surfaces can be specified
cg                         with commands "plane", "cylinder" and "cone", and
cg                         the zone specified with command "zone".
cg
cg                         To find the volume of a body of revolution with a
cg                         polygonal cross section, use command "volume".

cg    axisymmetric       A surface or volume is axially symmetric if it is a
cg                         surface or body of revolution around a specified
cg                         axis, or the same for all angles around the axis.
cg                         Real quadric surfaces that are axially symmetric
cg                         include the following, in their standard forms (each
cg                         coefficient must have the preceding sign):
cg
cg                         Simple plane:                                x = 0
cg                         Coincident planes:                         x^2 = 0
cg                         Real parallel planes:            - 1 + QXX*x^2 = 0
cg                         Sphere:          - 1 + QXX * (x^2 + y^2 + z^2) = 0
cg                         Circular cylinder:     - 1 + QXX * (x^2 + y^2) = 0
cg                         Circular cone:         x^2 + y^2 - |QZZ| * z^2 = 0
cg                         Circular ellipsoid:
cg                                    - 1 + QXX * (x^2 + y^2) + QZZ * z^2 = 0
cg                         Circular paraboloid:    - |QZ| * z + x^2 + y^2 = 0
cg                         Circular hyperboloid of 1 sheet:
cg                                  - 1 + QXX * (x^2 + y^2) - |QZZ| * z^2 = 0
cg                         Circular hyperboloid of 2 sheets:
cg                                    1 + QXX * (x^2 + y^2) - |QZZ| * z^2 = 0
cg
cg                         See command "quadric".
cg
cg                         The following commands relate to
cg                         axisymmetric quadric surfaces:
cg                         accelerate, arc, axisym, cone, copy, cylinder,
cg                         debug, delete, distance, ellipsoid, extrema, help,
cg                         invert, last, list, move, operator, plane, point,
cg                         project, proximal, quadric, reflect, rename, repack,
cg                         rotate, scale, search, side, slice, sort, sphere,
cg                         symbol, synonym, tables, track, triple, vector, zone.

cg    axisymmetric arrays
cg                         Axisymmetri quadric surfaces may be created as array
cg                         with subscripted names.  See "subscript",
cg                         "subscripted names", commands "do" and "enddo".

cg    axisymmetrics      A family of axially symmetric quadric surfaces may be
cg                         created with command "axisymmetric", options "move",
cg                         "rotate" or "scale" or as follows:
cg
cg                         p pinv (options)
cg
cg                           Create an invariant point, if needed.
cg
cg                         axisym AXQ(1) (options)
cg
cg                           Create a base axially symmetric quadric surface.
cg
cg                         op opr (options)
cg
cg                           Create a tensor operator, to reflect, rotate,
cg                           invert or scale, if needed.
cg
cg                         v vmove (options)
cg
cg                           Create a vector for use as a translation operator,
cg                           if needed.
cg
cg                         cp AXQ(1) AXQ(2)
cg
cg                           Copy axially symmetric quadric surface AXQ(1) to
cg                           the first axially symmetric quadric surface.
cg
cg                         Repeat the following commands as needed:
cg
cg                         mv axisym AXQ(1) vmove
cg
cg                           Move the base axially symmetric quadric surface by
cg                           amount vmove, if needed.
cg
cg                         [inv, refl, rot, scale] axisym AXQ(1) opr [pinv]
cg
cg                           Invert, reflect, rotate or scale the base axially
cg                           symmetric quadric surface with tensor operator opr,
cg                           if needed.
cg
cg                         cp axisym AXQ(1) +
cg
cg                           Copy the base axially symmetric quadric surface to
cg                           the next axially symmetric quadric surface.

cc    AXNAME             The name of an axially symmetric quadric surface.
cc                         May have up to 24 characters, including any
cc                         subscripts, and may be ASCII, integer or floating
cc                         point.
cc                         No axially symmetric quadric name may be "+", '-',
cc                         "all", "h", "help", "list" or "thru" or begin with
cc                         "!".
cc                         May not be the same as any other quadric surface
cc                         name QNAME.  Also referred to as AXNAME1, AXNAME2,
cc                         ..., AXQ(1), AXQ(2).  Specified with command "axisym"
cc                         or "quadric".
cc
cc                         Quadric surfaces may appear in commands:
cc                         accelerate, axisym, cone, copy, cylinder, delete,
cc                         distance, ellipsoid, extrema, invert, move, name,
cc                         operator, plane, point, proximal, quadric, reflect,
cc                         rename, rotate, scale, side, slice, sphere, surface,
cc                         track, vector.
cc                         Also see commands:  last, list, repack, search, sort.

cc    AXQ(1)             In command "axisym", the name of a base axially
cc                         symmetric quadric surface, used to create a family of
cc                         axially symmetric quadric surfaces.

cc    AXQ(2)             In command "axisym", the name of the first of a family
cc                         of axially symmetric quadric surfaces to create.

cg    azimuth            The azimuth angle, theta, of a point is the angle of
cg                         the line from the origin to the point around the z
cg                         axis, measured counterclockwise from the y plane,
cg                         and limited to the range (in different applications)
cg                         from 0 to 360 degrees (0 to 2 * pi radians), or
cg                         from -180 to 180 degrees (-pi to pi radians).
cg                         See "polar", "phi".
cg
cg                         The azimuthal angle around any axis is the angle
cg                         measured in a plane perpendicular to the axis, and
cg                         counterclockwise looking in the negative direction
cg                         along the axis.  The zero point must be specified.

B-------------------------------------------------------------------------------

cg    back               See "back up".

cg    back up            There are several ways to recover from an error.
cg                         Commands may be undone or may be redone with
cg                         modified arguments.
cg                         See commands "undo", "redo", "indo".
cg                         If an object is copied before modifying it, and an
cg                         error is made in the modification, the copy may be
cg                         used to replace the original object.

cg    backslash          The backslash or escape character, "\".

cg    backup             If an object is copied before modifying it, and an
cg                         error is made in the modification, the copy may be
cg                         used to replace the original object.

cg    ballistic          See "accelerate", "trajectory".

cg    bang               The exclamation point, "!".

cg    base               An integer or non-integer floating point value may be
cg                         specified in any number base.  For value x and base
cg                         n, the digits of x are the coefficients k(m) in the
cg                         expression:
cg
cg                         x = k(m) * n^m + k(m-1) * n^(m-1) + ... + k(0) * n^0
cg                           + k(-1) * n^(-1) + k(-2) * n^(-2) + ...
cg                           + k(-j) * n^(-j)
cg
cg                         where the coefficients k(m) through k(0) define the
cg                         integer part of x, and the coefficients k(-1) through
cg                         k(-j) define the fractional part of x.
cg                         See command "base".

cg    base               See "base axisym", "base brick", "base cone",
cg                         "base cylinder", "base ellipsoid", "base line",
cg                         "base name", "base plane", "base point",
cg                         "base quadric", "base sphere", "base tetrahedron",
cg                         "base variable", "base vector", "base zone",
cg                         command "base".

cc    base               Command to convert a decimal integer or floating point
cc                         number to a specified number base.
cc                         See commands "icalc", "big", "base".
cc
cc                         Forms include:
cc
ccin                       help base
ccin                       basew [help]
cc
cc                           Display the command options.
cc
ccin                       base NBASE VALUE1 VALUE2 VALUE3 ...
cc
cc                           Convert to number base NBASE, an integer, the
cc                           decimal integer or non-integer floating point
cc                           values VALUE1, VALUE2, VALUE3, ....

cg    base axisym        A base axially symmetric quadric surface is an axially
cg                         symmetric quadric surface specified in command
cg                         "axisym", to serve as the initial axially symmetric
cg                         quadric surface for creating a family of axially
cg                         symmetric quadric surfaces translated, rotated or
cg                         scaled from the preceding axially symmetric quadric
cg                         surface.

cg    base brick         A base brick is a brick specified in command "brick",
cg                         to serve as the initial brick for creating a family
cg                         of bricks adjacent to a specified face of the
cg                         preceding brick, with thicknesses scaled from that of
cg                         the preceding brick.

cg    base cone          A base cone is a cone specified in command "cone", to
cg                         serve as the initial cone for creating a family of
cg                         cones with the same vertex and axis, but translated,
cg                         rotated or scaled from the preceding cone.

cg    base cylinder      A base cylinder is a cylinder specified with command
cg                         "cylinder", to serve as the initial cylinder for
cg                         creating a family of cylinders translated, rotated or
cg                         scaled from the preceding cylinder.

cg    base ellipsoid     A base ellipsoid is a ellipsoid specified with command
cg                         "ellipsoid", to serve as the initial ellipsoid for
cg                         creating a family of ellipsoids translated, rotated
cg                         or scaled from the preceding ellipsoid.

cg    base line          A base line is a line specified in command "line", to
cg                         serve as the initial line for creating a family of
cg                         lines with vertex point names incremented by a
cg                         specified numbers of characters from those of the
cg                         preceding line.

cg    base name          The base name for naming objects of a given type is the
cg                         last name used for creating or renaming an object of
cg                         that type, or the name specified with command "last",
cg                         whichever occurs later.  All base names may
cg                         be displayed with command "last".
cg                         The next name after or before the base name, in a
cg                         limited ASCII sequence may be used in a command to
cg                         create, copy, or rename an object of the same type,
cg                         by using the character "+" or "-" in place of the
cg                         new object name, but not in command "rename" with
cg                         option "array".
cg                         See "+", "-", "increment names", command "last".

cg    base plane         A base plane is a plane specified in command "plane",
cg                         to serve as the initial plane for creating a family
cg                         of planes at specified vector displacements, or a
cg                         family of planes rotated at specified angular
cg                         increments around an axis, or a family of planes
cg                         scaled from the preceding plane by a specified scale
cg                         factor.

cg    base point         A base point is a point specified in command "point",
cg                         to serve as the initial point for creating a family
cg                         of points translated, rotated or scaled from the
cg                         preceding point.

cg    base quadric       A base quadric surface is a quadric surface specified
cg                         with command "quadric", to serve as the initial
cg                         quadric surface for creating a family of concentric
cg                         quadric surfaces translated, rotated or scaled from
cg                         the preceding quadric surface.

cg    base sphere        A base sphere is a sphere specified in command
cg                         "sphere", to serve as the initial sphere for creating
cg                         a family of spheres rotated, scaled or translated
cg                         from the preceding sphere.

cg    base tetrahedron   A base tetrahedron is a tetrahedron specified in
cg                         command "tetrahedron", to serve as the initial
cg                         tetrahedron for creating a family of tetrahedrons
cg                         with vertex point names incremented by specified
cg                         numbers of characters from those of the preceding
cg                         tetrahedron.

cg    base variable      A base variable is a variable specified in command
cg                         "variable", to serve as the initial variable for
cg                         creating a family of variables forming a series by
cg                         incrementing and/or multiplying the preceding
cg                         variable.

cg    base vector        A base vector is a vector specified in command
cg                         "vector", to serve as the initial vector for creating
cg                         a family of vectors rotated from the preceding
cg                         vector.

cg    base zone          A base zone is a zone specified in command "zone", to
cg                         served as the initial zone for creating a family of
cg                         zones with the names of the bounding surfaces
cg                         incremented by specified numbers of characters from
cg                         those of the preceding zone.

cg    basic              See "basic commands".

cg    basic commands     See "starting up".

cg    BETA               The random variable in a relativistic Maxwellian
cg                         probability distribution function is BETA, the ratio
cg                         of the particle velocity, v, to the velocity of
cg                         light, c = 299,792,458 m / s.
cg                         A related variable is the relativistic function
cg                         GAMMA = 1 / sqrt (1 - BETA^2).
cg                         For RATIO << 1, where RATIO is the dimensionless
cg                         ratio of gas temperature to particle rest mass
cg                         energy, the expected value of BETA is approximately
cg                         1.60 * sqrt (RATIO), with a standard deviation of
cg                         approximately 0.67 * sqrt (RATIO), and the expected
cg                         value of BETA^2 is 3 * RATIO.
cg                         For RATIO >> 1, the expected value of BETA is
cg                         approximately 1 - 0.23 / RATIO^2, with a standard
cg                         deviation of approximately 1.26 / RATIO^2,
cg                         the expected value of BETA^2 is approximately
cg                         1 - 0.46 / RATIO^2, the expected value of GAMMA is
cg                         3 * RATIO, with a standard deviation of approximately
cg                         1.73 * RATIO, and the expected value of GAMMA^2 is
cg                         12 * RATIO^2.

cc    big                An option in command "debug", to display GEOM internal
cc                         parameters and variables for big integers:
cc                         abig, abigtag, idig, nbigm, ndig, ndigm.

cc    big                Command to display or create big integers, with up to
cc                         1001 digits, and with names a single capital letter
cc                         from "A" to "Z".
cc                         Big integers may be used in calculations related to
cc                         cryptography and combinatorial mathematics.
cc                         See commands "icalc" and "variable".
cc
cc                         Forms include:
cc
ccin                       help big
ccin                       big [help]
cc
cc                           Display the command options.
cc
ccin                       big all
cc
cc                           Display all big integers that are not zero.
cc
ccin                       big all zero
cc
cc                           Zero out all big integers.
cc
ccin                       big BIGNAME
cc
cc                           Display big integer BIGNAME.  BIGNAME must be a
cc                           single upper case alphabetic character, from "A" to
cc                           "Z".
cc                           Included in the display will be the one-word
cc                           integer and floating point values of BIGNAME, if
cc                           possible, and the log of BIGNAME to the base 10.
cc
ccin                       big BIGNAME [binary,octal,hex]
cc
cc                           Display big integer BIGNAME in number base
cc                           [2, 8, 16].
cc
ccin                       big BIGNAME base NBASE
cc
cc                           Display big integer BIGNAME in number base NBASE.
cc                           The stored value of BIGNAME will not be changed.
cc
ccin                       big BIGNAME = 0
cc
cc                           Set big integer BIGNAME to zero.
cc
ccin                       big BIGNAME [N|1] = INT1 INT2 INT3 ...
cc                           Replace the digits in BIGNAME, starting with the
cc                           N'th digit (N defaults to 1 if unspecifed, or if
cc                           BIGNAME is initially zero), with the digits of the
cc                           successive integer groups INT1, INT2, INT3, ...,
cc                           which may be positive integers or integer
cc                           variables, or ASCII strings containing only the
cc                           digits 0-9, but may not be the names of big
cc                           integers.  If N is not specified, BIGNAME will
cc                           initially be set to 0.
cc                           If BIGNAME is not zero, and N exceeds the initial
cc                           length of BIGNAME, all digits in between will be
cc                           zero.  Any leading zeros in BIGNAME will be
cc                           ignored.
cc
cc                           For example, the commands:
cc                             variable n = 77777
cc                             big Q = 000123 n 000456 789000
cc                           define big integer Q = 12377777000456789000
cc
ccin                       big BIGNAME tag 'BIGTAG'
cc
cc                           Give big integer BIGNAME the label or tag BIGTAG
cc                           (up to 32 characters), which may be used to define
cc                           or describe BIGNAME.  This tag will be changed by
cc                           GEOM if BIGNAME is recalculated using one of the
cc                           available functions described below.
cc
ccin                       big BIGNAME = big BIGNAME2
cc
cc                           Replace big integer BIGNAME with big integer
cc                           BIGNAME2.
cc
ccin                       big BIGNAME = INT1 [+,-,*,/] INT2
cc
cc                           Replace big integer BIGNAME with the
cc                           [sum, difference, product, quotient] of integers
cc                           INT1 and INT2, which, in this and the following
cc                           commands, may be positive integers or integer
cc                           variables or big integers.
cc                           The quotient is rounded to an integer.
cc                           See "reciprocal".
cc
cc                           Integer INT1 or INT2 or both may be the result of
cc                           multiplying a number with digits after the decimal
cc                           point by a sufficiently large power of ten to
cc                           shift the deciml point to the right of those
cc                           digits.
cc
cc                           Numbers to be added or subtracted must be
cc                           multiplied by the same power of ten.
cc                           Numbers to be multiplied may each be multiplied by
cc                           a different power of tem.
cc                           When dividing, the numerator may be multiplied by
cc                           a much larger power of ten than the denomenator,
cc                           to produce more significant figures in the result.
cc
cc                           The correct placement of the decimal point in the
cc                           result should be obvious.  If necessary, the same
cc                           calculation may be done with ordinary floating
cc                           point variables as a check.

ccin                       big BIGNAME = INT1 mod INT2
cc
cc                           Replace big integer BIGNAME with the modulus of
cc                           integer INT1 with respect to integer INT2, with
cc                           INT1 and INT2 specified as above.
cc                           This is the remainder or residual of the division
cc                           in the preceding command.
cc
ccin                       big BIGNAME = INT1 // INT2
cc
cc                           Replace big integer BIGNAME with the result of
cc                           dividing INT1 by the highest possible power of
cc                           INT2, if the latter is an exact factor of INT1.
cc
ccin                       big BIGNAME = INT1 cat INT2
cc
cc                           Replace big integer BIGNAME with the concatenation
cc                           of nonzero INT1 and nonzero INT2.
cc
ccin                       big BIGNAME = INT1 perm INT2
cc
cc                           Replace big integer BIGNAME with the number of
cc                           permutations of INT1 things taken INT2 at a time.
cc                           The largest allowed value of INT1 is 450.
cc
ccin                       big BIGNAME = INT1 comb INT2
cc
cc                           Replace big integer BIGNAME with the number of
cc                           combinations of INT1 things taken INT2 at a time.
cc                           The largest allowed value of INT1 is 450.
cc
ccin                       big BIGNAME = INT1 ^ NPOW
cc
cc                           Replace big integer BIGNAME with the NPOW'th power
cc                           of integer INT1.  If INT1 has had its decimal point
cc                           shifted N digits to the right, then BIGNAME must
cc                           have its decimal point shifted N * NPOW digits to
cc                           the left.
cc
ccin                       big BIGNAME = INT1 root NROOT
cc
cc                           Replace big integer BIGNAME with the NROOT'th root
cc                           of integer INT1, specified as above.
cc                           i.e., BIGNAME = BIGNAME2^(1/NROOT).
cc                           The root is rounded to the nearest integer.
cc                           If INT1 has had its decimal point shifted N digits
cc                           to the right, then BIGNAME must have its decimal
cc                           point shifted N / NROOT digits to the left, so
cc                           N must be an exact multiple of NROOT.
cc                                      
ccin                       big BIGNAME = INT1 fact
cc
cc                           Replace big integer BIGNAME with the factorial of
cc                           integer INT1  The largest value of INT1 that can be
cc                           used without exceeding 1001 digits is 450.
cc
ccin                       big BIGNAME = INT1 exp
cc
cc                           Replace big integer BIGNAME with the exponential
cc                           function of integer INT1.  The largest value of
cc                           INT1 that can be used without exceeding 1001 digits
cc                           is 2302.  Accuracy is to the nearest digit, but
cc                           for INT1 = 35 or more, 16 or fewer significant
cc                           figures.
cc
ccin                       big BIGNAME random N
cc
cc                           Replace big integer BIGNAME with N random digits.
cc                           The result may have leading zeros, which may be
cc                           removed by the command "big BIGNAME = 1 * BIGNAME".
cc
cc                         WARNING:  avoid conflicts between the names of
cc                         aliases and symbols and the names of big integers.
cc
cc                         Synonyms:  [help, h], [^, **], [0, zero],
cc                         [binary, bin], [octal, oct].

cg    big integer        Command "big" allows specification of up to 26 big
cg                         integers, each with up to 1001 digits, each stored
cg                         in a separate machine word.
cg                         Allowed operations include zeroing, replacement,
cg                         addition, subtraction, multiplication, division,
cg                         modulus, powers, roots, factorials, combinations,
cg                         permutations, and exponentiation.
cg                         These operations may be of use in cryptographic
cg                         applications, including encryption and decryption.
cg
cg                         The name of a big integer must be a single upper
cg                         case letter (A to Z), in order to avoid confusion
cg                         with the names of variables, symbols, aliases and
cg                         synonyms.  Each big integer is allowed to have up to
cg                         1001 digits, each stored in a separate machine word,
cg                         and in normal order, with the units digit last.
cg
cg                         The names of big integers may be incremented or
cg                         decremented by use of command "symbol", with option
cg                         "increment", or by the use of "+" or "-" in place of
cg                         the big integer name.
cg
cg                         Digits to the right of a decimal point may be
cg                         included in calculations by moving the decimal point
cg                         to the right of those digits, doing the calculations,
cg                         then deciding where to put the decimal point.
cg                         Numbers to be added or subtracted must have the
cg                         decimal point moved by equal amounts.
cg                         To find a root, multiply by a power of 10 which is
cg                         a multiple of the number of the root.

cf    big.mac            A file with 50 significant figures in the values of
cf                         pi, the golden ratio, log_e(10), log_10(ebase), and
cf                         ebase, and 21 for Euler's constant.

cc    BIGNAME            In command "big", the name of a  big integer.
cc                         Must be a single upper case letter (A to Z).
cc                         Also referred to as BIGNAME1, BIGNAME2, BIGNAME3, ...

cg    bin                A probability bin specifies the relative probability
cg                         of a discrete event or object, a discrete value of a
cg                         random variable, or a range of values of a random
cg                         variable.  In the latter case, the probability may be
cg                         a uniform, linear, power-law, exponential, normal,
cg                         relativistic Maxwellian, Planck or Wien function of
cg                         the random variable value.  A discrete object may be
cg                         any of the objects used in GEOM, including a
cg                         probability distribution function (pdf).
cg                         The maximum number of bins is now 1000.
cg
cg                         A set of probability bins may be combined to form a
cg                         probability distribution function (pdf) over a set of
cg                         discrete events, discrete values of a random
cg                         variable, or a continuous or discontinuous set of
cg                         ranges of values of a random variable.
cg
cg                         The following commands relate to bins:
cg                         bin, copy, debug, delete, help, last, list, pdf,
cg                         rename, repack, search, sort, symbol, synonym,
cg                         tables.

cc    bin                An option in command "debug", to display GEOM internal
cc                         parameters and variables for probability bins:
cc                         nbinm, nbins, abin, atypeb, probl, probr,
cc                         vranl, vranr, prbin, prbint, cosbin, conbin, vexbin,
cc                         nsampb, abins, labins.

cc    bin                Command to display one or more probability bins or to
cc                         create a probability bin.  Creating a probability bin
cc                         replaces any existing probability bin having the same
cc                         name.  All probabilities must be positive, and
cc                         the bin widths VRANR - VRANL must be positive.
cc
cc                         Command "bin" relates to objects:  bin, pdf, symbol,
cc                         variable.
cc
cc                         Forms include:
cc
ccin                       help bin
ccin                       bin [help]
cc
cc                           Display the command options.
cc
ccin                       bin [all,list]
cc
cc                           Display all bins (short display).
cc
ccin                       bin list BINAME1 BINAME2 BINAME3 ...
cc
cc                           Display bins BINAME1, BINAME2, BINAME3, ..., with
cc                           or without subscripts (short display).
cc
ccin                       bin BINAME
cc
cc                           Display bin BINAME (long display).
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         BINAME to form the name by incrementing or
cc                         decrementing the base name for probability bins.
cc                         See "base name", command "last", "increment names".
cc
ccin                       bin BINAME event [PRTOT|1]
cc
cc                           Create probability bin BINAME for a discrete event
cc                           with relative total probability PRTOT.
cc
ccin                       bin BINAME value VRAN [PRTOT|1]
cc
cc                           Create probability bin BINAME for a discrete value
cc                           VRAN of a random variable, with relative total
cc                           probability PRTOT.
cc
ccin                       bin BINAME uniform VRANL VRANR [DPR|1]
cc
cc                           Create probability bin BINAME with a uniform
cc                           differential probability DPR from value VRANL to
cc                           value VRANR of a random variable (a histogram).
cc
ccin                       bin BINAME linear VRANL VRANR [DPRL|1] [[DPRR|DPRL]]
cc
cc                           Create probability bin BINAME with linear
cc                           differential probability, from DPRL at value
cc                           VRANL, to DPRR at value VRANR of a random
cc                           variable.  DPRL defaults to 1.  DPRR defaults
cc                           to DPRL.
cc
cc                           For DPR a known function of VRAN, use the command
cc                           above between commands "do" and "enddo", with
cc                           a calculation of DPRL and DPRR for each of a range
cc                           of values of VRANL and VRANR.
cc
ccin                       bin BINAME exp VRANL VRANR SIGMA [PRTOT|1]
cc
cc                           Create probability bin BINAME with relative total
cc                           probability PRTOT, from value VRANL to value VRANR,
cc                           with exponentially varying differential
cc                           probability, with decay constant SIGMA.
cc                           For relative differential probabilities PL at VRANL
cc                           and PR at VRANR, DVAL  = VRANR - VRANL,
cc                           SIGMA = log (PL / PR) / DVAL, and
cc                           PRTOT = (PL - PR) / SIGMA.
cc                           If SIGMA = 0, PRTOT = PL * DVAL.
cc                           See "mean free path", "decay time".
cc
ccin                       bin BINAME power VRANL VRANR POW [PRTOT|1]
cc
cc                           Create probability bin BINAME with relative total
cc                           probability PRTOT, from value VRANL to value VRANR,
cc                           with power-law differential probability, with power
cc                           POW.  For relative differential probabilities PL at
cc                           VRANL and PR at VRANR,
cc                           POW = log (PR / PL) / log (VRANR / VRANL),
cc                           PRTOT = (VRANR * PR - VRANL * PL) / (POW + 1), or
cc                           if POW = -1:  PRTOT = PL * VRANL * log (PL / PR).
cc
ccin                       bin BINAME normal VMEAN DEV [PRTOT|1]
cc
cc                           Create probability bin BINAME with relative total
cc                           probability PRTOT, with a normal probability
cc                           distribution with mean value VMEAN and standard
cc                           deviation DEV.  The random variable ranges from
cc                           minus infinity to plus infinity.
cc
ccin                       bin BINAME maxwell RATIO [PRTOT|1]
cc
cc                           Create probability bin BINAME with relative total
cc                           probability PRTOT, for a relativistic Maxwellian
cc                           probability distribution, with a dimensionless
cc                           ratio RATIO of gas temperature to particle rest
cc                           mass energy.  The random variable beta is the ratio
cc                           of particle velocity to the velocity of light,
cc                           v / c, ranging from 0 to 1.  c = 299,792,458 m / s.
cc                           1 K = 8.617385E-5 eV (+/- 8.5 ppm),
cc                           1 amu = 9.3149432E8 eV (+/- 0.3 ppm).
cc
ccin                       bin BINAME [planck,wien] BTEMP [PRTOT|1]
cc
cc                           Create probability bin BINAME with relative total
cc                           probability PRTOT, with a probability distribution
cc                           of a [Planck, Wien] spectrum at temperature BTEMP.
cc                           The random variable is a frequency XNU (same units
cc                           as BTEMP) from that spectrum, ranging from 0 to
cc                           infinity.  1 K = 8.617385E-5 eV (+/- 8.5 ppm).
cc                           1 Hz = 4.1356692E-15 eV (+/- 0.3 ppm).
cc
cc                         Synonyms:  [help, h], [maxwell, Maxwell],
cc                         [normal, norm], [planck, Planck], [power, pow],
cc                         [wien, Wien].

cc    bin                An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more bins.

cc    bin                A synonym for binary in commands "icalc", "big".

cg    bin arrays         Bins may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    bin types          The types of probability bins currently include:
cg
cg                         Type                               Key word
cg
cg                         discrete event                     event
cg                         discrete value                     value
cg                         uniform distribution (histogram)   uniform
cg                         linear distribution                linear
cg                         power-law distribution             power,   pow
cg                         exponential distribution           exp
cg                         normal (Gaussian) distribution     normal,  norm
cg                         relativistic Maxwellian spectrum   maxwell, Maxwell
cg                         Planck spectrum                    planck,  Planck
cg                         Wien spectrum                      wien,    Wien

cc    BINAME             The name of a probability bin.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.
cc                         No bin name may be "+", "-", "all", "h", "help",
cc                         "list" or "thru", begin with "!" or contain ";".
cc                         Specified with command "bin".
cc                         Also referred to as BINAME1, BINAME2, BINAME3, ...
cc
cc                         Probability bins may appear in commands:
cc                         bin, copy, delete, rename, pdf.
cc                         Also see commands:  last, list, repack, search, sort.

cc    binary             An option in command "big", to display a big integer
cc                         in binary.

cc    binary             An option in command "icalc", to turn on the binary
cc                         mode display of the results.
cc                         Synonyms:  [binary, bin].

cg    binary             The binary form of an integer M, specified in the
cg                         current integer mode (see commands "icalc", "big"),
cg                         may be displayed with commands:
cg
cg                         icalc binary
cg
cg                           Turn on display of results of command "icalc" in
cg                           binary mode.
cg
cg                         binary BIGNAME binary
cg
cg                           Display big integer BIGNAME in number base 2.
cg
cg                         icalc M
cg
cg                           Specify integer M in the current integer mode, and
cg                           display M in decimal, hexadecimal, octal and binary
cg                           mode.

cg    bisect             To find the bisector of an angle, use command "bisect".
cg
cg                         To find the midpoint of a line between two points,
cg                         PNAME1 and PNAME2:
cg
cg                         vector VNAME = PNAME1 PNAME2
cg
cg                           Create the vector from point PNAME1 to point
cg                           PNAME2.
cg
cg                         copy p PNAME1 PMID
cg
cg                           Create a point PMID at PNAME1.
cg
cg                         move p PMID VNAME 0.5
cg
cg                           Move point PMID to the midpoint.  To create a plane
cg                           at the midpoint, perpendicular to the line from
cg                           point PNAME1 to PNAME2:
cg
cg                         pl PLNAME v PMID VNAME
cg
cg                           Create a plane at the midpoint, with a normal
cg                           vector parallel to the line from point PNAME1 to
cg                           point PNAME2.
cg
cg                         To find one of the two planes bisecting the space
cg                         between two other planes, use command "distance" to
cg                         find the intersection of the two planes, then use
cg                         command "plane" to create a plane through a point on
cg                         the line of intersection, with a normal vector equal
cg                         to either the sum or the difference of the normal
cg                         vectors of the two planes.  The command "vector",
cg                         with option "sum", may used to find that sum or
cg                         difference.

cc    bisect             Command to bisect the angle formed by the lines
cc                         connecting three points.
cc
cc                         Command "bisect" relates to objects:  point, symbol.
cc
cc                         Forms include:
cc
ccin                       help bisect
ccin                       bisect [help]
cc                           Display the command options.
cc
ccin                       bisect PNAME1 PNAME2 PNAME3
cc
cc                           Find the bisector of the angle formed by the three
cc                           points PNAME1, PNAME2 and PNAME3, and display the
cc                           intercept of the bisector on the line joining
cc                           points PNAME3 and PNAME1, the vector from point
cc                           PNAME2 to the intercept and the length of that
cc                           vector.
cc
cc                         Synonyms:  [help, h].

cg    bisector           The perpendicular bisectors of the three edges of a
cg                         triangle meet at the center of the circumscribed
cg                         circle.  The bisectors of the three vertex angles of
cg                         a triangle meet at the center of the inscribed
cg                         circle.  See "altitude", "median", "trig".

cc    bl                 A synonym for block.

cg    blackjack          See "21".

cg    blank              The initial field delimiter is a blank character,
cg                         but may be changed with command "delimiter".
cg                         If the field delimiter is a blank character, any
cg                         number of leading and trailing blank characters will
cg                         be ignored, and multiple blank characters between
cg                         words are treated as a single field delimiter.

cg    blank              See "blank line".

cg    blank line         Blank input lines will be ignored.

cg    block              See "mesh block".

cc    block              An option in commands "mesh" and "cluster", to operate
cc                         on a specified mesh block.
cc                         Synonyms:  [block, bl].

cg    Boolean            Boolean operations are on individual bits of one or
cg                         two machine words.  For each bit or pair of bits,
cg                         the various Boolean operations available with
cg                         commands "icalc" and "cluster", return the following
cg                         results:
cg
cg                           Bit A:                   | 0     0     1     1
cg                           Bit B:                   | 0     1     0     1
cg                           -------------------------|--------------------
cg                              comp. A (complement)  | 1     1     0     0
cg                              comp. B (complement)  | 1     0     1     0
cg                           -------------------------|--------------------
cg                           A .not.  B (not)         | 0     0     1     0
cg                           B .not.  A (not)         | 0     1     0     0
cg                           -------------------------|--------------------
cg                           A .and.  B (intersection)| 0     0     0     1
cg                           A .nand. B (comp. .and.) | 1     1     1     0
cg                           -------------------------|--------------------
cg                           A .or.   B (union)       | 0     1     1     1
cg                           A .nor.  B (comp. .or.)  | 1     0     0     0
cg                           -------------------------|--------------------
cg                           A .xor.  B (exclusive or)| 0     1     1     0
cg                           A .xnor. B (comp. .xor.) | 1     0     0     1
cg                           ----------------------------------------------
cg
cg                         To do Boolean operations on clusters of points, see
cg                         command "cluster".
cg
cg                         To do Boolean operations on machine words, see
cg                         command "icalc".

cg    bound              See "bound point".

cg    bound point        The point in 3-D space at which a vector is specified.
cg                         Required for converting vectors between rectangular,
cg                         cylindrical and spherical coordinates.  It is only in
cg                         rectangular coordinates that the components of all
cg                         parallel vectors of the same magnitude are equal.
cg                         May be specified with command "vector VNAME  = ... ",
cg                         and changed with command "vector VNAME bound PNAME".
cg                         See "coordinate conversion", "vector conversion".

cc    box                An option in command "sudoku", to assign up to 9
cc                         digits to a 3 x 3 Sudoku box.

cg    box                To create a zone with the shape of a box, or a right
cg                         parallelepiped, use commands "plane" and "zone", or
cg                         command "brick".

cc    br                 A synonym for brick.

cg    brick              See "brick (cylindrical)", "brick (rectangular)",
cg                         "brick (spherical)".

cg    brick              A brick is a six-faced solid geometric object in 3-D
cg                         space, bounded by three specified pairs of coordinate
cg                         surfaces, one pair for each of the three coordinates
cg                         (u, v, w) of a specified orthogonal coordinate
cg                         system, with angles measured in specified units.
cg                         See "coordinate", "angles".
cg                         A brick has 8 vertices, 12 edges, and 6 faces.
cg                         The maximum number of bricks is now 1000.
cg                         See "briquette".
cg
cg                         If one pair of coordinates are equal, the brick is
cg                         a plane, or a section of a circular cylinder, sphere
cg                         or circular cone, and has zero volume.
cg                         See "tile", "sheet".
cg
cg                         If two pairs of coordinates are equal, the brick is
cg                         a line segment or an arc of a circle, and has zero
cg                         volume and zero areas on the faces.  See "filament".
cg
cg                         If all three pairs of coordinates are equal, the
cg                         brick is a point, and has zero volume, zero areas on
cg                         the faces, and zero edge lengths.
cg
cg                         A brick is a convenient object for use in specifying
cg                         a limited geometric region for carrying out other
cg                         geometric operations.  See commands "mcvol", "point",
cg                         "plot".
cg
cg                         To create and assign mesh points uniformly
cg                         distributed within a brick, use command
cg                         "point PMESH brick BRNAME mesh block [volume]".
cg
cg                         See "zone (rectangular)", "zone (cylindrical)",
cg                         "zone (spherical)".
cg                         See "x plane", "y plane", "z plane", "rcyl cylinder",
cg                         "rsph sphere", "theta plane", "phi cone".
cg
cg                         To create a brick containing a set of points,
cg                         first create a cluster of the points, and then use
cg                         the displayed minimum and maximum coordinates to
cg                         create the brick.
cg
cg                         To find if a point is in a brick, use command
cg                         "distance" or "where".
cg
cg                         To find the distances from a point to the six
cg                         surfaces bounding a brick, use command "distance".
cg
cg                         To create a family of vertex points, distributed over
cg                         the u, v and w coordinate surfaces so as to divide a
cg                         brick BRNAME up into equal volume elements, each with
cg                         6 faces, 12 edges and 8 vertex points, use command:
cg                         "point PMESH brick BRNAME mesh block volume".
cg
cg                         The following commands relate to bricks:
cg                         brick, cluster, copy, debug, delete, distance, help,
cg                         last, list, mcvol, point, rename, repack, search,
cg                         sort, symbol, synonym, tables, where.

cc    brick              An option in command "debug", to display GEOM internal
cc                         parameters and variables for bricks:  nbrickm,
cc                         nbricks, abrick, asysbr, angbr, ubr, dubr, vbr, dvbr,
cc                         wbr, dwbr, volbr, abricks, labricks.
cc                         Synonyms:  [brick, br].

cc    brick              An option in command "point", to create a family of
cc                         points in a brick, at equal-volume values between
cc                         the minimum and maximum u, v and w coordinates of
cc                         the brick, each with a unique name.

cc    brick              An option in commands "cluster", "copy", "delete",
cc                         "last", "list", "rename", "repack" and "sort", to
cc                         perform the specified operation on one or more
cc                         bricks.
cc                         Synonyms:  [brick, br].

cc    brick              Command to display or create bricks.  Creating a brick
cc                         replaces any existing brick having the same name.
cc                         The display for a brick includes the coordinate
cc                         system, the angle units, the minimum, increment and
cc                         maximum values of the coordinates of the bounding
cc                         surfaces in each of the three coordinate directions,
cc                         the edge lengths, the face areas, and the volume.
cc                         See "do loop use".
cc
cc                         Command "brick" relates to objects:  brick, symbol,
cc                         variable.
cc
cc                         Forms include:
cc
ccin                       help brick
ccin                       brick [help]
cc
cc                           Display the command options.
cc
ccin                       brick [all,list]
cc
cc                           Display all bricks (short display).
cc
ccin                       brick list BRNAME1 BRNAME2 BRNAME3 ...
cc
cc                           Display bricks BRNAME1, BRNAME2, BRNAME3, ...,
cc                           with or without subscripts (short display).
cc
ccin                       brick BRNAME
cc
cc                           Display brick BRNAME (long display).
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         BRNAME or BR(2) to form the name by incrementing or
cc                         decrementing the base name for bricks.
cc                         See command "last", "increment names".
cc
ccin                       brick BRNAME = UMIN UMAX VMIN VMAX WMIN WMAX
cc
cc                           Create brick BRNAME, bounded by the coordinate
cc                           surfaces at UMIN and UMAX, VMIN and VMAX, WMIN and
cc                           WMAX, in the current coordinate system (u, v, w),
cc                           which may be rectangular (x, y, z), cylindrical
cc                           (rcyl, theta, z) or spherical (rsph, theta, phi),
cc                           with the current units for angles (degrees or
cc                           radians).  One, two or three of the coordinate
cc                           pairs may be equal, to create a surface, a segment
cc                           of a line or arc of a circle, or a point, resp.
cc                           This command could be inside a triple do loop
cc                           over the U, V and W indices, to create a family
cc                           of bricks.
cc
ccin                       brick BR(2) incr NUMBR INC BR(1) [-] CDIR [FMULT|1]
cc
cc                           Create a family of NUMBR bricks BR(2), ..., with
cc                           names incremented by INC characters, each sharing
cc                           a coordinate surface in the [negative] coordinate
cc                           direction CDIR (x, y, z, rcyl, theta, rsph, phi),
cc                           with the preceding brick, starting from brick
cc                           BR(1), and with a thickness in direction CDIR equal
cc                           to FMULT times the thickness of the preceding
cc                           brick.  See "increment names".
cc
cc                           This command can be replaced by using commands
cc                           "do" and "enddo", with any of the "brick" commands,
cc                           to create a do loop over bricks with subscripted
cc                           names, with more general variation of properties.
cc                           For example, creating a family of bricks with
cc                           constant thickness increments in the U direction:
cc
cc                           variable UMIN    = UMINVAL
cc                           variable UMAX    = UMAXVAL
cc                           variable UTHICK  = UMAX - UMIN
cc                           variable NB      = NBINIT
cc                           brick BRNAME(NB) = UMIN UMAX VMIN VMAX WMIN WMAX
cc                           do n = 1 NUMBR
cc                             variable NB      = NB + INC
cc                             variable UMIN    = UMAX
cc                             variable UMAX    = UMAX + UTHICK
cc                             brick BRNAME(NB) = UMIN UMAX VMIN VMAX WMIN WMAX
cc                           enddo
cc
cc                         Synonyms:  [brick, br], [help, h], [increment, incr].

cg    brick (cylindrical)
cg                         A brick defined in cylindrical coordinates, bounded
cg                         by RCYLMIN, RCYLMAX, THETAMIN, THETAMAX, ZMIN and
cg                         ZMAX has the bounding surfaces:
cg                           -RCYLMIN + x^2 + y^2 = 0
cg                           -RCYLMAX + x^2 + y^2 = 0
cg                           sin (THETAMIN) * x - cos (THETAMIN) * y = 0
cg                           sin (THETAMAX) * x - cos (THETAMAX) * y = 0
cg                           -ZMIN + z = 0
cg                           -ZMAX + z = 0
cg
cg                         The distances D from a point P1 = (X1, Y1, Z1) to
cg                         each of these surfaces are:
cg                           D = RCYLMIN - sqrt (X1^2 + Y1^2)
cg                           D = RCYLMAX - sqrt (X1^2 + Y1^2)
cg                           D = sin (THETAMIN) * X1 - cos (THETAMIN) * Y1
cg                           D = sin (THETAMAX) * X1 - cos (THETAMAX) * Y1
cg                           D = ZMIN - Z1
cg                           D = ZMAX - Z1
cg
cg                         If P1 = (RCYL, THETA, Z1), substitute
cg                           X1 = RCYL * cos (THETA)
cg                           Y1 = RCYL * sin (THETA)
cg                           sqrt (X1^2 + Y1^) = RCYL
cg
cg                         See "proximal".

cg    brick (rectangular)
cg                         A brick defined in rectangular coordinates, bounded
cg                         by XMIN, XMAX, YMIN, YMAX, ZMIN and ZMAX has the
cg                         bounding planes:
cg                           -XMIN + x = 0      -XMAX + x = 0
cg                           -YMIN + y = 0      -YMAX + y = 0
cg                           -ZMIN + z = 0      -ZMAX + z = 0
cg
cg                         The distances D from a point P1 = (X1, Y1, Z1) to
cg                         each of these planes are:
cg                           D = XMIN - X1      D = XMAX - X1
cg                           D = YMIN - Y1      D = YMAX - Y1
cg                           D = ZMIN - Z1      D = ZMAX - Z1
cg
cg                         See "proximal".

cg    brick (spherical)
cg                       A brick defined in spherical coordinates, bounded by
cg                         RSPHMIN, RSPHMAX, THETAMIN, THETAMAX, PHIMIN and
cg                         PHIMAX has the bounding surfaces:
cg                           -RSPHMIN + x^2 + y^2 + z^2 = 0
cg                           -RSPHMAX + x^2 + y^2 + z^2 = 0
cg                           sin (THETAMIN) * x - cos (THETAMIN) * y = 0
cg                           sin (THETAMAX) * x - cos (THETAMAX) * y = 0
cg                           (cos (PHIMIN))^2 * (x^2 + y^2) -
cg                             (sin (PHIMIN))^2 * z^2 = 0
cg                           (cos (PHIMAX))^2 * (x^2 + y^2) -
cg                             (sin (PHIMAX))^2 * z^2 = 0
cg
cg                         The distances D from a point P1 = (X1, Y1, Z1) to
cg                         each of these surfaces are:
cg                           D = RSPHMIN - sqrt (X1^2 + Y1^2 + Z1^2)
cg                           D = RSPHMAX - sqrt (X1^2 + Y1^2 + Z1^2)
cg                           D = sin (THETAMIN) * X1 - cos (THETAMIN) * Y1
cg                           D = sin (THETAMAX) * X1 - cos (THETAMAX) * Y1
cg                           D = -sqrt (X1^2 + Y1^2) * cos (PHIMIN) +
cg                               abs (Z1) * sin (PHIMIN)
cg                           D = -sqrt (X1^2 + Y1^2) * cos (PHIMAX) +
cg                               abs (Z1) * sin (PHIMAX)
cg
cg                         If P1 = (RSPH, THETA, PHI), substitute
cg                           X1 = RSPH * sin (PHI) * cos (THETA)
cg                           Y1 = RSPH * sin (PHI) * sin (THETA)
cg                           Z1 = RSPH * cos (PHI)
cg                           sqrt (X1^2 + Y1^2 + Z1^2) = RSPH
cg
cg                         See "proximal".

cg    brick arrays       Bricks may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    bricks             A family of bricks may be created with command "brick",
cg                         option "increment".

cg    briquette          A briquette is a small volume element of a brick,
cg                         bounded by six coordinate surfaces through eight
cg                         vertex points created by command
cg                         "point PMESH brick BRNAME mesh block volume".
cg                         All such briquettes have equal volume.

cc    BRNAME             The name of a brick.  May have up to 24 characters,
cc                         including any subscripts, and may be ASCII, integer
cc                         or floating point.
cc                         No brick name may be "+", "-", "all", "h", "help",
cc                         "list" or "thru", begin with "!" or contain ";".
cc                         Specified with command "brick".
cc                         Also referred to as BRNAME1, BRNAME2, BRNAME3, ...
cc                         BR(1), BR(2), ...
cc
cc                         Bricks may appear in commands:
cc                         brick, copy, delete, mcvol, plot, point, rename,
cc                         where.
cc                         Also see commands:  last, list, repack, search, sort.

cc    BTEMP              In command "bin", the black-body temperature of a
cc                         Planck or Wien spectrum.

C-------------------------------------------------------------------------------

cc    C                  A synonym for c in a comment line.

cc    C                  In command "delimiter", a character to be used as a
cc                         field delimiter, which must be one of the following:
cc                         (blank) (  )  ,  :  <  >  [  \  ]  ^  _  `  {  |  }

cg    C                  The musical note middle C has a frequency of
cg                         523.2511306012 Hz on an equally-tempered scale.

cc    c                  A synonym for command in command "input".

cc    c                  The first word of a comment line.
cc                         See "!", "#", "*", "/".
cc                         Synonyms: [c, C, c...., C...., c++++, C++++, c----,
cc                         C----, c____, C___, !].

cc    C++++              A synonym for c++++ in a comment line.

cc    c++++              A synonym for c.

cc    C----              A synonym for c---- in a comment line.

cc    c----              A synonym for c.

cc    C....              A synonym for c.... in a comment line.

cc    c....              A synonym for c.

cg    C1                 See "C1, C2, ...".

cc    C1, C2, ...        In command "contfr", coefficients of a continued
cc                         fraction.

cg    C2                 See "C1, C2, ...".

cg    calculate          To do simple calculations with real numbers, use
cg                         command "variable".
cg                         To do simple calculations with integers, in decimal,
cg                         hexadecimal or octal mode, use command "icalc".
cg                         To do simple calculations with big integers, use
cg                         command "big".

cc    call               A synonym for input.

cg    calls              For subroutine calls, see files geom_link, geom_to,
cg                         geom_from, geom_aptcomms, geom_commapts.

cn    card.mac           An input file that sets up probability bins and a
cn                         pdf representing a deck of playing cards, to
cn                         demonstrate how to work with bins and pdfs.

cc    cart               A synonym for cartesian.

cc    Cartesian          A synonym for cartesian.

cg    Cartesian          A 3-D coordinate system using the coordinates
cg                         (x, y, z).  A rectangular coordinate system.
cg                         To create a volume element bounded by surfaces of a
cg                         rectangular coordinate system, use command "brick".
cg                         To create zones partially or completely bounded by
cg                         planar surfaces, use command "plane", options
cg                         "parallel", "move", "rotate", "scale", and
cg                         command "zone".
cg
cg                         To use a Cartesian coordinate system (the default),
cg                         use command:
cg
cg                         coord rect
cg
cg                           Make the coordinate system Cartesian.

cc    cartesian          An option in command "coordinate", to use a 3-D
cc                         Cartesian coordinate system with x, y and z
cc                         coordinates.
cc                         Synonyms:  [cartesian, cart, Cartesian,
cc                         rectangular, rect, xyz].

cg    case               The case of command arguments displayed in geom_base,
cg                         and in the "help" messages in GEOM, are lower case
cg                         for arguments to be typed literally as shown, and
cg                         UPPER CASE for arguments that are to be provided by
cg                         the user.  Arguments provided by the user
cg                         may be either ASCII strings (using any case or
cg                         mixture of cases), explicit numerical values,
cg                         variables or symbols.
cg                         See commands "alias", "symbol", "variable".

cc    cat                An option in command "big", to create a big integer by
cc                         concatenating two nonzero big integers, making it
cc                         easy to add digits at the beginning or end of a big
cc                         integer.

cc    CC                 A synonym for cc in a comment line.

cc    cc                 In columns 1-2, the first two characters of a
cc                         comment line.
cc                         Synonyms: [cc, CC].

cg    cc                 In columns 1-2 of geom_base, indicates a description of
cg                         a command or a description of a word that is part of
cg                         a command.

cg    ccin               In columns 1-4 of geom_base, indicates a command.

cg    cd                 In columns 1-2 of geom_base, indicates a directory.

cg    cell               An option in command "sudoku" to assign a digit to
cg                         a cell, then to display all of the digits permitted
cg                         in each cell.

cg    cen                Indicates the center of a geometric object.
cg                         In a brick, the centroid or center of volume.

cg    center             The center of a quadric surface is its center of
cg                         symmetry.  To move the center of a quadric surface of
cg                         type QTYPE (plane, sphere, cylinder, cone, hyperb,
cg                          ellipse, axisym, quadric), with name
cg                         QNAME, to the origin, execute the following:
cg
cg                         point pcen QTYPE QNAME
cg
cg                           Create point pcen, the center of QNAME.
cg
cg                         vector vmove QTYPE QNAME
cg
cg                           Create vector vmove, pointing from the center of
cg                           QNAME to the origin.
cg
cg                         move QTYPE QNAME vmove
cg
cg                           Translate QNAME to the origin.  QTYPE may be
cg                           "quadric" for all types of simple planes and
cg                           quadric surfaces.
cg
cg                         move point pcen vmove
cg
cg                           Check the translation.  The new pcen should be at
cg                           the origin.
cg
cg                         Vector vmove may also be used for the reverse
cg                         translation.
cg
cg                         See "align".
cg
cg                         Argument PCEN is the name of a point at the center of
cg                         an annular disk, quadric surface, regular polygon or
cg                         a regular polyhedron.  See commands "axisym", "disk",
cg                         "ellipsoid", "polygon", "polyhedron", "project",
cg                         "sphere".

cg    central            See "angle, central".

cg    centroid           In a brick, a point at the center of volume of the
cg                         brick.  Also refers to the distances through the
cg                         brick in the coordinate directions at the centroid,
cg                         and the areas of the coordinate surfaces through the
cg                         centroid.

cg    centroid           When a triangle is created or displayed, the centroid
cg                         is displayed.  The coordinates of the centroid are
cg                         the average of the coordinates of the vertices.
cg                         The centroid is at the intersection of the
cg                         medians, the lines from each vertex to the midpoint
cg                         of the opposite edge.
cg                         See "circumscribed circle", "inscribed circle",
cg                         "orthocenter", commands "triangle", "trig".

cg    cents              A logarithmic measure of musical pitches, 1200 to an
cg                         octave.  100 cents corresponds to a semitone on an
cg                         equally-tempered scale.  The difference between two
cg                         frequencies F1 and F2 in cents is given by:
cg                           cents = 1200 * log (F1/F2) / log (2).
cg                         See "equally-tempered".

cc    cf                 A synonym for contfr.

cg    cf                 In columns 1-2 of geom_base, indicates a function.

cc    CFNAME1            An option in command "quadric", to specify a new value
cc                         CFVAL1 of an individual coefficient in the general
cc                         implicit equation of a specified quadric surface.
cc                         May be QC, QX, QY, QZ, QXY, QYZ, QZX, QXX, QYY or
cc                         QZZ.  Likewise for CFNAME2, CFNAME3, ...

cc    CFVAL1             A new value of coefficient CFNAME1 in the general
cc                         implicit equation of a specified quadric surface.
cc                         Likewise for CFVAL2, CFVAL3, ...

cg    cg                 In columns 1-2 of geom_base, indicates a general
cg                         definition.

cg    chance             See "probability", "flip a coin", "spin the bottle".

cg    change             See "change name".

cg    change name        To change the name of an object, or an array of
cg                         objects, use command "rename".

cg    changes            To repeat a GEOM run with changes, make a new input
cg                         file from the output file from the GEOM run.
cg                         See "input file".

cg    character          A character is a single ASCII 8-bit symbol, included in
cg                        the ASCII character set.  See "character set".

cg    character          See "character data", "character set".

cg    character data     Character data is data which has been assigned type
cg                         character in GEOM, with a specified number of
cg                         characters.  On execution, GEOM fills all such data
cg                         with blanks.  See "data types".

cg    character set      The available keyboard character set on the systems
cg                         that GEOM is currently running on includes the
cg                         ASCII characters:
cg
cg                         (blank)  !  "  #  $  %  &  '  (  )    *  +  ,  -
cg                         .  /   0-9  :  ;  <  =  >  ?  @  A-Z  [  \  ]  ^
cg                         _  `   a-z  {  |  }  ~
cg
cg                         Other characters may be obtained with the <Alt>
cg                         or <Ctrl> keys, but may produce unexpected results.
cg
cg                         The inline function char(n) equals the characters:
cg
cg                         char(0)   =  (null)
cg
cg                         char(1)     to   char(31)  =  (control)
cg
cg                         char(32)  =  (blank)              char(33)  =  !
cg                         char(34)  =  "   char(35)  =  #   char(36)  =  $
cg                         char(37)  =  %   char(38)  =  &   char(39)  =  '
cg                         char(40)  =  (   char(41)  =  )   char(42)  =  *
cg                         char(43)  =  +   char(44)  =  ,   char(45)  =  -
cg                         char(46)  =  .   char(47)  =  /
cg
cg                         char(48)    to   char(57)  =  0-9
cg
cg                         char(58)  =  :   char(59)  =  ;   char(60)  =  <
cg                         char(61)  =  =   char(62)  =  >   char(63)  =  ?
cg                         char(64)  =  @
cg
cg                         char(65)    to   char(90)  =  A-Z
cg
cg                         char(91)  =  [   char(92)  =  \   char(93)  =  ]
cg                         char(94)  =  ^   char(95)  =  _   char(96)  =  `
cg
cg                         char(97)    to   char(122) =  a-z
cg
cg                         char(123) =  {   char(124) =  |   char(125) =  }
cg                         char(126) =  ~
cg
cg                         char(127) =  (control)
cg
cg                         char(128)   to   char(255) =  (non-ASCII)

cg    characteristics    Each type of quadric surface has certain unique
cg                         characteristics, which are displayed when the
cg                         quadric surface is displayed.  These include:
cg
cg                         alignment:  all types;
cg                         axis intercepts:  all types;
cg                         axially symmetry:  all types;
cg                         center(s) of symmetry:  all types;
cg                         coefficients of implicit equation:  all types;
cg                         distance between planes:  parallel planes;
cg                         eccentricity:  hyperbolic and elliptic cylinders,
cg                           hyperboloids of one or two sheets, ellipsoids;
cg                         extrema:  all types;
cg                         foci:  parabolic, hyperbolic and elliptic cylinders,
cg                           paraboloids, hyperboloids, ellipsoids;
cg                         general type:  all types;
cg                         half-angles:  intersecting planes, cones, hyperbolic
cg                           cylinders and hyperboloids;
cg                         invariants:  all types;
cg                         latus rectum:  parabolic and hyperbolic cylinders,
cg                           circular and elliptic paraboloids, hyperboloids;
cg                         normal vector:  simple planes;
cg                         plane type:  simple planes;
cg                         radius, circumference and circular area:
cg                           circular cylinders, spheres;
cg                         rotation tensor to align the major axes with the
cg                           coordinate axes:  all types;
cg                         semiaxes:  elliptic cylinders, ellipsoids;
cg                         simple planes:  coincident planes, parallel planes
cg                           and intersecting planes;
cg                         surface area:  sphere;
cg                         symmetry axis vectors:  all types;
cg                         transverse and conjugate semiaxes:
cg                           hyperbolic cylinders,
cg                           hyperboloids of one or two sheets;
cg                         vertex to focus distance:  parabolic cylinders.
cg                           circular paraboloids;
cg                         volume:  ellipsoid, sphere.

cg    charge             To find the machine time charged to the problem, use
cg                         command "time".

cg    chi.group.mac      A command file for finding the number of ways to select
cg                         two or more groups of items with specified lengths
cg                         from an anagram of all the items in those groups,
cg                         when the sequence of items in each group is unchanged
cg                         in the anagram.  See "chimera"

cg    chimera            An anagram of two or more groups of items, with the
cg                         sequence of items in each group unchanged in the
cg                         anagram.  See "chi.group.mac".

cg    Chimera Puzzle     A word puzzle, in which a chimera is give, and the
cg                         puzzle solver must find the phrase for which the
cg                         chimera is an anagram in which the sequence of
cg                         letters in each word of the phrase is unchanged.
cg                         See "chimera", test problem "chi.test", command file
cg                         "chi.group.mac".

cg    Chinese            See "Chinese Remainder".

cg    Chinese Remainder
cg                       The Chinese Remainder Theorem tells how to find the
cg                         least integer that has a specified set of remainders,
cg                         when divided by a specified set of divisors:
cg
cg                           REM1 = mod (I, IDIV1)
cg                           REM2 = mod (I, IDIV2)
cg                           REM3 = mod (I, IDIV3)
cg                           ...
cg                           REMN = mod (I, IDIVN)
cg
cg                         No pair of divisors may have a common factor, which
cg                         is easily satisfied with only prime divisors.
cg
cg                         Let IP = IDIV1 * IDIV2 * IDIV3 * ... * IDIVN.
cg                         There is a unique solution I (modulo IP), found as
cg                         follows:
cg
cg                         Let M1 = IP / IDIV1
cg                         Let M2 = IP / IDIV2
cg                             ...
cg                         Let MN = IP / IDIVN
cg
cg                         Let K1 be defined such that K1*M1 mod IDIV1 = 1
cg                             K1 = mod (M1^(IDIV1 - 2), IDIV1)
cg
cg                         Let K2 be defined such that K2*M2 mod IDIV2 = 1
cg                             K2 = mod (M2^(IDIV2 - 2), IDIV2)
cg                             ...
cg                         Let KN be defined such that KN*MN mod IDIVN = 1
cg                             KN = mod (MN^(IDIVN - 2), IDIVN)
cg
cg                         The K1, K2, ..., KN values may also be found just by
cg                         trying all of the integers until one works.
cg                         This avoids generating very large integers, as may
cg                         happen using the equations above.
cg
cg                         Then the solution is given by:
cg
cg                           I = K1*M1*IREM1 + K2*M2*IREM2 + ... + KN*MN*IREMN
cg                           I (minimum) = mod (I, IP)
cg
cg                         Other larger solutions are I + K * IP, where K is
cg                         any positive integer.
cg
cg                         See command "crt".
cg    choose randomly    See commands "bin", "pdf", "sample".
cg                         See "spin the bottle".

cc    circ               A synonym for circle.

cg    circle             A circle is a curve in a plane, everywhere equidistant
cg                         from a central point.  For a radius R, the
cg                         circumference is 2 * pi * R, and the area is
cg                         pi * R^2.
cg                         To find a circle through three points, use command
cg                         "circle".
cg                         To find two circles, each tangent to three mutually
cg                         tangent circles with specified radii, use command
cg                         "kiss".  See "tangent circles".
cg                         To find any intersection between two circles, use
cg                         command "intcirc".
cg                         To find the circular intersection between a plane
cg                         and a sphere or between two spheres, use command
cg                         "distance".

cc    circle             Command to find a circle through three specified
cc                         points or to find two circles each tangent to three
cc                         tangent circles with specified radii.
cc
cc                         Command "circle" relates to objects:  point, symbol.
cc
cc                         Forms include:
cc
ccin                       help circle
ccin                       circle [help]
cc
cc                           Display the command options.
cc
ccin                       circle point PNAME1 PNAME2 PNAME3
cc
cc                           Find the circle through the three points PNAME1,
cc                           PNAME2 and PNAME3, and display the radius, center,
cc                           unit normal vector, and area.
cc
cc                         Use command "disk" to create circular or annular
cc                         disks representing the circles.  Use command
cc                         "cylinder" to create cylinders parallel to the z
cc                         axis and through the circles.
cc                         Use commands "move" and "rotate", to move and orient
cc                         the disks or cylinders anywhere in 3-D space.
cc                         Use command "kiss" to find the center coordinates
cc                         of three tangent circles with specified radii, and
cc                         the radii and center coordinates of two additional
cc                         circles tangent to the first three.
cc
cc                         Synonyms:  [circle, circ], [help, h].

cg    circular           See "circular cone", "circular cylinder",
cg                         "circular ellipsoid", "circular paraboloid".

cg    circular cone      A circular cone is an axially symmetric quadric surface
cg                         for which the standard equation is:
cg                           x^2 + y^2 - |QZZ|*z^2 = 0  (QZZ < 0),
cg                         An imaginary circular cone is a quadric surface for
cg                         which the standard equation is:
cg                           x^2 + y^2 + QZZ*z^2 = 0,       (QZZ > 0).
cg                         This last equation is also satisfied at the real
cg                         point (0,0,0).

cg    circular cylinder
cg                       A circular cylinder is an axially symmetric quadric
cg                         surface for which the standard equation is:
cg                          - 1 + QXX * (x^2 + y^2) = 0
cg                         An imaginary circular cylinder is a quadric surface
cg                         for which the standard equation is:
cg                            1 + QXX * (x^2 + y^2) = 0,     (QXX > 0).

cg    circular ellipsoid
cg                       A circular ellipsoid is an axially symmetric quadric
cg                         surface for which the standard equation is:
cg                          - 1 + QXX*x^2 + QYY*y^2 + QZZ*z^2 = 0,
cg                           (QXX => QYY => QZZ > 0),
cg                         with two of the coefficients QXX, QYY, QZZ equal.
cg
cg                         A circular ellipsoid may be either an oblate spheroid
cg                         (QYY = QZZ) or a prolate spheroid (QXX = QYY).

cg    circular paraboloid
cg                       A circular paraboloid is an axially symmetric quadric
cg                         surface for which the standard equation is:
cg                               -QZ*z + x^2 + y^2 = 0.
cg                         This is the shape of a telescope reflector or the
cg                         reflector behind a beamed light source.
cg
cg                         To find the distance from a point to a circular
cg                         paraboloid, use command "distance", "proximal" or
cg                         "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and a circular paraboloid, use command "distance".

cg    circum             Abbreviation for circumference.

cg    circumference      To find the circumference of a circle or a circular
cg                         cylinder, use command "circle", "cylinder", "disk"
cg                         or "intcirc".

cg    circumscribed      See "circumscribed circle", "circumscribed sphere".

cg    circumscribed circle
cg                       When a triangle is created or displayed, the center
cg                         and radius of the circumscribed circle is displayed.
cg                         The center is at the intersection of the
cg                         perpendicular bisectors of the edges.
cg                         See "inscribed circle", "centroid", "orthocenter",
cg                         commands "circle", "kiss", "trig".
cg                         Any circumscribed circle tangent externally to each
cg                         of three mutually tangent circles may be found with
cg                         command "kiss".


cg    circumscribed sphere
cg                       A sphere may be circumscribed outside a regular
cg                         polyhedron, such as a tetrahedron, a cube, an
cg                         octahedron, a dodecahedron or an icosahedron,
cg                         so that it passes through every face of the regular
cg                         polyhedron.  See "inscribed sphere".
cg                         Any circumscribed sphere, tangent internally to each
cg                         of four mutually tangent spheres may be found with
cg                         command "kiss".

cc    cl                 A synonym for cluster.

cc    CLNAME             The name of a cluster of points.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.
cc                         No cluster name may be "+", "-", "all", "h", "help",
cc                         "list", "point" or "thru", begin with "." or "!" or
cc                         contain ";".
cc                         Specified with command "cluster".
cc                         Also referred to as CLNAME1, CLNAME2, CLNAME3, ...
cc
cc                         Clusters may appear in commands:
cc                         accelerate, area, cluster, copy, delete, distance,
cc                         invert, mesh, move, project, reflect, rename, rotate,
cc                         scale, twist, where.
cc                         Also see commands:  last, list, repack, search, sort.

cc    closest            See "minimum", "nearest", "proximal".

cg    cluster            A cluster is a named set of points.  Points may be
cg                         added or removed by name or by the names of clusters
cg                         containing the points, added by mesh status, or by
cg                         presence in a tetrahedron, brick or zone.
cg                         Any duplicate point names in a cluster will be
cg                         deleted.  Any point may be in any number of clusters.
cg                         The maximum number of clusters is now 100.
cg                         The maximum number of points assigned to clusters is
cg                         now 1000, counting points assigned to more than one
cg                         cluster.
cg                         Clusters may be copied, deleted, renamed, and listed.
cg                         Clusters may be translated, rotated, reflected,
cg                         inverted, scaled and projected.
cg                         The points in a cluster may be deleted.
cg
cg                         No cluster name may be "+", "-", "all", "comp.", "h",
cg                         "help", "list", "point" or "thru", begin with "!" or
cg                         contain ";".
cg
cg                         Clusters may be sorted by name with command "sort".
cg
cg                         The following commands relate to clusters:
cg                         accelerate, area, cluster, copy, debug, delete,
cg                         distance, help, invert, last, list, mesh, move,
cg                         point, project, reflect, rename, repack, rotate,
cg                         scale, search, sort, symbol, synonym, tables, twist,
cg                         where.

cc    cluster            An option in command "debug", to display GEOM internal
cc                         parameters and variables for clusters of points:
cc                         nclustm, nclusts, aclust, laclust, aclusts, laclusts,
cc                         npairm, npairs, apaircl, apairpt.
cc                         Synonyms:  [cluster, cl].

cc    cluster            Command to display or create clusters of existing
cc                         points, or to modify an existing cluster by adding or
cc                         removing points.  Creating a cluster replaces any
cc                         existing cluster having the same name.  The display
cc                         will include the coordinates of each point, and the
cc                         minimum, maximum, average and standard deviation of
cc                         each of the coordinates.
cc                         See "do loop use".
cc
cc                         Command "cluster" relates to objects:  brick,
cc                         cluster, mesh, point, symbol, tetrahedron, zone.
cc
cc                         Forms include:
cc
ccin                       help cluster
ccin                       cluster [help]
cc
cc                           Display the command options.
cc
ccin                       cluster [all,list]
cc
cc                           Display all clusters.
cc
ccin                       cluster list CLNAME1 CLNAME2 CLNAME3 ...
cc
cc                           Display clusters CLNAME1, CLNAME@, CLNAME3, ...,
cc                           with or without subscripts.
cc
ccin                       cluster CLNAME
cc
cc                           Display cluster CLNAME.
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         CLNAME to form the name by incrementing or
cc                         decrementing the base name for clusters.
cc                         See command "last", "increment names".
cc
ccin                       cluster CLNAME all
cc
cc                           Create new cluster CLNAME, consisting of all
cc                           points.
cc
ccin                       cluster CLNAME = PNAME1 ... PNAME2 [thru] PNAME3 ...
cc
cc                           Create new cluster CLNAME, consisting of points
cc                           PNAME1, ..., PNAME2 [, through] PNAME3, ...
cc                           If option "thru" is used, the adjacent points need
cc                           not exist.  Any error cancels entire command.
cc
ccin                       cluster CLNAME & PNAME1 ... PNAME2 [thru] PNAME3 ...
cc
cc                           Add to existing cluster CLNAME (do not use "+" or
cc                           "-"), points PNAME1, ..., PNAME2 [, through]
cc                           PNAME3, ...
cc                           If option "thru" is used, the adjacent points need
cc                           not exist.  A warning message is displayed if the
cc                           cluster CLNAME does not already exist.  Any error
cc                           cancels entire command.
cc
cc                           This command could be placed between commands "do"
cc                           and "enddo", to create a do loop over subscripted
cc                           point names, to add them to a cluster.
cc
ccin                       cluster CLNAME - PNAME1 ... PNAME2 [thru] PNAME3 ...
cc
cc                           Remove from existing cluster CLNAME (do not use
cc                           "+" or "-"), points PNAME1, ..., PNAME2 [, through]
cc                           PNAME3, ...  If option "thru" is used, the adjacent
cc                           points need not exist.  WARNING:  an error in one
cc                           word does not cancel the entire command.  For a
cc                           backup, copy the cluster before modifying.
cc
cc                           This command could be placed between commands "do"
cc                           and "enddo", to create a do loop over subscripted
cc                           point names, to remove them from a cluster.
cc
ccin                       cl CLNAME cl CLNAME1 ... CLNAME2 [thru] CLNAME3 ...
cc
cc                           Create new cluster CLNAME, consisting of existing
cc                           clusters CLNAME1, ..., CLNAME2 [, through] CLNAME3,
cc                           ...  If option "thru" is used, the adjacent
cc                           clusters need not exist.  Add to existing cluster
cc                           CLNAME if CLNAME is in the list of cluster names.
cc                           Any error cancels entire command.
cc                           Note:  "cl" is a synonym for "cluster".
cc
ccin                       cl CLNAME cl CLNAME ... CLNAME2 [thru] CLNAME3 ...
cc
cc                           Add to existing cluster CLNAME, (do not use "+" or
cc                           "-") the existing clusters CLNAME1, ..., CLNAME2 [,
cc                           through] CLNAME3, ...  If option "thru" is used,
cc                           the adjacent clusters need not exist.  Any error
cc                           cancels entire command.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                           This command could be placed between commands "do"
cc                           and "enddo", to create a do loop over subscripted
cc                           cluster names, to add them to a cluster.
cc
ccin                       cl CLNAME cl - CLNAME1 ... CLNAME2 [thru] CLNAME3 ...
cc
cc                           Remove from existing cluster CLNAME (do not use
cc                           "+" or "-"), all points in existing clusters
cc                           CLNAME1, ..., CLNAME2 [, through] CLNAME3, ...
cc                           If option "thru" is used, the adjacent points need
cc                           not exist.  A warning message is displayed if the
cc                           cluster CLNAME does not already exist.
cc                           WARNING:  an error in one word does not cancel the
cc                           entire command.  For a backup, copy the cluster
cc                           before modifying.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                           This command could be placed between commands "do"
cc                           and "enddo", to create a do loop over subscripted
cc                           cluster names, to remove them from a cluster.
cc
cc                         cluster CLNAME cluster comp. CLNAME1
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           not in cluster CLNAME1.  This is the Boolean
cc                           "complement" of the cluster CLNAME1.
cc
cc                         cluster CLNAME cl CLNAME1 .and. CLNAME2
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           that are in both clusters CLNAME1 and CLNAME2.
cc                           This is the Boolean "and" or "intersection" of
cc                           clusters CLNAME1 and CLNAME2.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                         cluster CLNAME cl CLNAME1 .nand. CLNAME2
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           that are not in both clusters CLNAME1 and CLNAME2.
cc                           This is the Boolean "complement" of the "and" or
cc                           "intersection" of clusters CLNAME1 and CLNAME2.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                         cluster CLNAME cl CLNAME1 .nor. CLNAME2
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           that are not in cluster CLNAME1 and not in cluster
cc                           CLNAME2.  This is the Boolean "complement" of the
cc                           "intersection" of clusters CLNAME1 and CLNAME2.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                         cluster CLNAME cl CLNAME1 .not. CLNAME2
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           that are in cluster CLNAME1 but not in cluster
cc                           CLNAME2.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                         cluster CLNAME cl CLNAME1 .or. CLNAME2
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           that are in cluster CLNAME1 or in cluster CLNAME2.
cc                           This is the Boolean "or" or "union" of clusters
cc                           CLNAME1 and CLNAME2.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                         cluster CLNAME cl CLNAME1 .xnor. CLNAME2
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           that are either in both clusters or in neither
cc                           cluster CLNAME1 and CLNAME2.  This is the Boolean
cc                           "complement" of the "exclusive or" of clusters
cc                           CLNAME1 and CLNAME2.
cc                           Note:  "cl" is a synonym for "cluster".
cc
cc                         cluster CLNAME cl CLNAME1 .xor. CLNAME2
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           that are in cluster CLNAME1 but not in cluster
cc                           CLNAME2, or in cluster CLNAME2 but not in cluster
cc                           CLNAME1.  This is the Boolean "exclusive or" of
cc                           clusters CLNAME1 and CLNAME2.
cc                           Note:  "cl" is a synonym for "cluster".
cc
ccin                       cluster CLNAME brick BRNAME
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           inside of brick BRNAME.
cc
ccin                       cluster CLNAME tetrahedron TETNAME
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           inside of tetrahedron TETNAME.
cc
ccin                       cluster CLNAME zone ZNAME
cc
cc                           Create new cluster CLNAME, consisting of all points
cc                           inside of zone ZNAME.
cc
ccin                       cluster CLNAME mesh [all]
cc
cc                           Create new cluster CLNAME, consisting of all mesh
cc                           points.
cc
ccin                       cluster CLNAME nomesh
cc
cc                           Create new cluster CLNAME, consisting of all
cc                           non-mesh points.
cc
ccin                       cluster CLNAME mesh block
cc
cc                           Create new cluster CLNAME, consisting of all mesh
cc                           points in the mesh block.
cc
ccin                       cluster CLNAME mesh vacuum
cc
cc                           Create new cluster CLNAME, consisting of all mesh
cc                           points with x, y, z coordinates of -10^99 or less.
cc
cc                         Synonyms:  [.and., .int.], [.or., .un.],
cc                         [.xnor., .eqv.], [block, bl],
cc                         [brick, br], [cluster, cl], [help, h],
cc                         [tetrahedron, tet, tetra], [zone, z, zn].

cc    cluster            An option in commands "accelerate", "area", "cluster",
cc                         "copy", "delete", "distance", "invert", "last",
cc                         "list", "mesh", "move", "project", "reflect",
cc                         "rename", "repack", "rotate", "scale", "sort",
cc                         "twist" and "where", to perform the specified
cc                         operation on one or more clusters of points.
cc                         Synonyms:  [cluster, cl].

cg    cluster arrays     Clusters may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cc    cmd                A synonym for command in command "input".

cn    cmd                Indicates a command summary file.
cn                         The command summary file from GEOM is geom_cmd.

cg    cn                 In columns 1-2 of geom_base, indicates a file name.

cc    CNNAME             The name of a circular cone.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.  May not be the same as any other quadric
cc                         surface name QNAME.  Also referred to as CNNAME1,
cc                         CNNAME2, ..., CONE(1), CONE(2).  Specified with
cc                         commands "cone", "quadric" or "axisym".
cc                         No cone name may be "+", "-", "all", "h", "help",
cc                         "list", "point" or "thru", begin with "!" or contain
cc                         ";".
cc
cc                         Cones may appear in commands:
cc                         accelerate, arc, axisym, cone, copy, delete,
cc                         distance, extrema, help, invert, last, list, move,
cc                         operator, point, project, proximal, quadric, reflect,
cc                         rename, repack, rotate, scale, search, side, slice,
cc                         sort, track, triple, vector.

cg    code               The code name is GEOM.

cg    coefficient        To find the real coefficients of a polynomial equation
cg                         with from two to four specified real and/or complex
cg                         roots, use command "roots", option "=".
cg
cg                         To find the real coefficients of an Nth-order
cg                         polynomial equation with N specified real roots,
cg                         use command "root", option "=".
cg
cg                         To find the real and/or complex roots of a linear,
cg                         quadratic, cubic or quartic polynomial equation with
cg                         real coefficients, use command "roots".
cg
cg                         To search for real roots, extrema and inflection
cg                         points of a polynomial equation by Newtonian
cg                         iteration, use command "root".

cg    cofactor           If an integer N has an integer factor M, the cofactor
cg                         of M is N / M.
cg

cg    coil               A coil is a curve at a fixed distance from an axis,
cg                         with its angular coordinate around the axis a
cg                         linear function of its displacement parallel to the
cg                         axis.  To generate a family of points arrayed along a
cg                         coil, use command "point ... move" to create a family
cg                         of points along a straight line parallel to the coil
cg                         axis, then use command "cluster" to create a cluster
cg                         consisting of the points, and then operate on the
cg                         cluster with command "twist", option "axial".

cg    coincident         Two geometric objects are coincident if they have the
cg                         same size, shape, position and orientation, or the
cg                         difference between them is no more than the
cg                         estimated error in calculating that difference, based
cg                         on the tolerance limit TOL.
cg
cg                         To test points, lines, triangle, planes, spheres or
cg                         circular cylinders for coincidence, use command
cg                         "distance".
cg
cg                         To test two spheres for coincidence of their centers
cg                         or their surfaces, or to test two circular cylinders
cg                         for coincidence of their axes or their surfaces,
cg                         use command "distance".

cg    coincident         See "coincident planes".

cg    coincident planes
cg                       A pair of coincident planes may be represented by a
cg                         single quadric surface for which the standard
cg                         equation is x^2 = 0.
cg                         This factors into the two equations:
cg                           x = 0 (a simple plane),
cg                           x = 0 (a simple plane, coincident with the first).
cg
cg                         The general implicit quadric equation for coincident
cg                         planes is the square of the implicit quadric equation
cg                         for a simple plane:
cg                           (QC + QX * x + QY * y + QZ * z)^2 = 0.
cg
cg                         This is a limiting case of parallel planes or
cg                         intersecting planes.
cg                         See "non-simple planes".
cg
cg                         To create such a quadric surface, use command
cg                         "quadric", option "plane".

cc    col                An option in command "sudoku", to assign up to 9
cc                         digits to a Sudoku column.

cg    collision          See "mean free path".

cg    colon              The colon character, ":".

cg    column             In an input line, the character position counting from
cg                         the first (leftmost) character, including blank
cg                         characters.  Control characters displayed as two
cg                         characters count as only one column.  Tab characters
cg                         may be any number of columns.

cg    column             See "family of objects", "move", "scale".

cc    comb               An option in command "big", to find the number of
cc                         combinations of M things taken N at a time.
cc
cc                         big BIGNAME =  M comb N


cc    comb               An option in command "icalc", to find the number of
cc                         combinations of M things taken N at a time.
cc
cc                         icalc M comb N

cg    combinations       The number of combinations of M things taken N at a
cg                         time is C(M,N) = M! / (N! * (M - N)!).
cg                         Use command "icalc M comb N" or
cg                         "big BIGNAME = INT1 comb INT2".
cg                      .  See "permutations".

cc    comm               A synonym for commands.

cc    comma              The comma character, ",".

cc    COMMAND            In command "help COMMAND" or "COMMAND help", any
cc                         command name or the word "all".
cc                         See "commands", "help".

cc    COMMAND            In any command, the command key word.  See "commands".
cc
cc                         Forms include:
cc
ccin                       COMMAND
cc
cc                           For most commands, display forms and options.
cc                           Exceptions include commands "angles", "coordinate",
cc                           "input", "output", "time", "title", and "when",
cc                           which display other information, and
cc                           commands "delimiter", "endif", "indo", "redo",
cc                           "return" and "undo" which perform some operation.
cc
ccin                       help COMMAND
ccin                       COMMAND help
cc
cc                           For all commands, display forms and options.
cc
ccin                       COMMAND ...
cc
cc                           See the description for each command in geom_base.
cc
ccin                       COMMAND ... ! COMMENT
cc
cc                           Command COMMAND with appended comment COMMENT.
cc
ccin                       COMMAND1 ...;COMMAND2 ... ; COMMAND3 ...
cc
cc                           Multiple commands COMMAND1, COMAND2 and COMMAND3 on
cc                           the same line. The ";" needs no delimiter.
cc
ccin                       COMMAND1 ... ! COMMENT1 ; COMMAND2 ... ! COMMENT2
cc
cc                           Multiple commands COMMAND1 and COMMAND2, with
cc                           appended comments COMMENT1 and COMMENT2,
cc                           respectively.
cc
cc                         Synonyms:  [help, h].

cc    command            An option in command "input", used in place of a
cc                         file name, to indicate that one or more lines from
cc                         the input lines saved in memory are to be
cc                         executed or displayed.

cg    command            See "command form", "command list", "command summary".

cg    command form       Commands begin with a command word or one of its
cg                         synonyms, and may have additional arguments, and
cg                         must always be ended by typing the <return> key, or
cg                         with certain exceptions, by the command separator
cg                         character ";" if followed by another command on the
cg                         same input line, or by the comment separator
cg                         character "!" if followed by a comment on the same
cg                         input line.
cg                         The default field delimiter is a blank character,
cg                         but may be changed with command "delimiter".
cg                         Arguments are separated by the field delimiter.
cg                         Any argument beginning with "!" is the beginning of
cg                         an appended comment.
cg                         The <return> key must be typed at the end of every
cg                         input line typed at the user's terminal.
cg
cg                         Command arguments are displayed in lower case for
cg                         arguments to be typed literally as shown, and in
cg                         UPPER CASE for arguments for which the user
cg                         must provide an ASCII name (using any case), an
cg                         integer or floating point value, an integer or
cg                         floating point variable or a symbol.
cg
cg                         Any argument which allows a floating point value may
cg                         be replaced by the name of an integer or floating
cg                         point variable.
cg
cg                         Except for the argument LINE in command "indo" or
cg                         "redo", any argument which only allows an integer
cg                         value may be replaced by an integer variable.
cg
cg                         With certain exceptions (see "symbol"), any argument,
cg                         may be replaced by a symbolic word which, after
cg                         symbol replacement, is an acceptable value for that
cg                         argument.
cg
cg                         The first word of a command, and additional words if
cg                         desired, may be replaced by a previously defined
cg                         alias (see command "alias").
cg
cg                         See "commands", "key words", "arguments", "help".

cg    command list       See "input", "macro".
cg                         A list of commands from the current session is saved
cg                         in file geom_cmd, and may be displayed or executed
cg                         again with command "input geom_hsp [options]".
cg                         Another such list is saved internally, and may be
cg                         displayed or executed again with command
cg                         "input command [options]".

cg    command summary    See "geom_cmd", command "input command".

cg    command use        See the entries in this file for each object type.

cc    commands           Command to display an alphabetic list of command words,
cc                         including synonyms.
cc
cc                         Forms include:
cc
ccin                       help commands
ccin                       commands [help]
cc
cc                           Display an alphabetic list of command words,
cc                           including synonyms.
cc
cc                         Synonyms:  [commands, alph, comm], [help, h].

cc    commands           The currently implemented commands are listed below,
cc                         with their synonyms.  See the entry for each command,
cc                         for a description of required and optional arguments.
cc                         Also see "key words", "help", "command form",
cc                         "arguments".  Also see files geom_commands and
cc                         geom_input.
cc                         Use command "alias" to create your own command names,
cc                         with or without additional arguments.
cc                         Use command "alph" to display an alphabetic list of
cc                         all commands and their synonyms.
cc                         Use command "synonym" to find any synonym for a
cc                         given command or any command with a given synonym.
cc
cc                         .
cc                         accelerate    accel
cc                         alias         al      a
cc                         alph          commands              comm
cc                         angles        angle   ang
cc                         arc
cc                         area
cc                         axisym
cc                         base
cc                         big
cc                         bin
cc                         bisect
cc                         brick         br
cc                         circle        circ
cc                         cluster       cl
cc                         commands      comm     alph
cc                         cone
cc                         contfr        cf
cc                         coordinate    coord    cs           system    sys
cc                         copy          cp       duplicate    dup
cc                         cross         outer
cc                         crt
cc                         cut           slice
cc                         cute
cc                         cuts
cc                         cylinder      cyl
cc                         define        def
cc                         debug
cc                         delete        del      remove       rm
cc                         delimiter     delimit  delim
cc                         disk          dk
cc                         distance      dist     intersect    inter     int
cc                         do
cc                         dot           inner
cc                         duplicate     dup      copy         cp
cc                         ellipsoid     ellipse  ell          el
cc                         end           exit     split        quit
cc                         enddo
cc                         endif
cc                         environment   env      setup
cc                         error         err
cc                         exit          end      quit         split
cc                         extrema       extr
cc                         factor        totient
cc                         goto
cc                         help          h
cc                         hex
cc                         hyperb        saddle
cc                         icalc
cc                         id            title
cc                         if
cc                         increment     incr
cc                         indo
cc                         inner         dot
cc                         input         in       i            read      rd  r
cc                         input         call
cc                         intcirc
cc                         intersect     inter    int          distance  dist
cc                         invert        inv
cc                         iris
cc                         kiss
cc                         last
cc                         line          ln       l
cc                         list          print
cc                         lock          zzz
cc                         marker        mark     m
cc                         math
cc                         mcvol
cc                         mesh
cc                         misc
cc                         move          mv       translate    trans
cc                         object        obj
cc                         operator      oper     op
cc                         outer         cross
cc                         output        out      o            write     wr  w
cc                         page
cc                         parse
cc                         pdf
cc                         plane         pl
cc                         plot
cc                         point         pnt      pt           p
cc                         polygon       polyg    pg
cc                         polyhedron    polyh    ph
cc                         project       proj
cc                         prompt
cc                         proximal      prox
cc                         quadric       quad     q
cc                         quit          end      exit         split
cc                         random        ?
cc                         ratio         rat
cc                         read          rd       r            input     in  i
cc                         redo
cc                         reflect       refl
cc                         remove        rm       delete       del
cc                         rename        switch   sw
cc                         repack
cc                         return        rtn
cc                         root
cc                         rootf
cc                         roots
cc                         rotate        rot
cc                         saddle        hyperb
cc                         sample
cc                         search        find
cc                         scale
cc                         set           s        variable     var
cc                         setup         environment           env
cc                         side
cc                         sizes         tables
cc                         slice         cut
cc                         solve
cc                         sort
cc                         sphere        sph
cc                         spin
cc                         split         end      exit         quit
cc                         status        stat
cc                         steiner       stein
cc                         sudoku        su
cc                         switch        sw       rename
cc                         symbol        symb     sym
cc                         synonym       syn
cc                         system        sys      coordinate   coord     cs
cc                         tables        sizes
cc                         tetrahedron   tetra    tet
cc                         time          t
cc                         title         id
cc                         tol
cc                         trace
cc                         track         trk
cc                         translate     trans    move         mv
cc                         triangle      tri
cc                         trig
cc                         triple
cc                         twist         tw
cc                         undo          u
cc                         variable      var      set          s
cc                         vector        vect     v
cc                         vi
cc                         volume        vol
cc                         walk
cc                         when
cc                         where
cc                         write         wr       w            output    out o
cc                         zone          zn       z

cc    comment            A comment line is any input line with one of the
cc                         following forms, where COMMENT is any desired
cc                         comment, which may include leading blanks:
cc
ccin                       !COMMENT      (starting in column 1)
ccin                       #COMMENT      (starting in column 1)
ccin                       *COMMENT      (starting in column 1)
ccin                       /COMMENT      (starting in column 1)
ccin                       C COMMENT     (starting in any column)
ccin                       c COMMENT     (starting in any column)
ccin                       C++++ COMMENT (starting in any column)
ccin                       c++++ COMMENT (starting in any column)
ccin                       C---- COMMENT (starting in any column)
ccin                       c---- COMMENT (starting in any column)
ccin                       C.... COMMENT (starting in any column)
ccin                       c.... COMMENT (starting in any column)
ccin                       CCCOMMENT     (starting in column 1)
ccin                       ccCOMMENT     (starting in column 1)
cc
cc                         Comment lines are echoed in the output file.
cc
cc                         An appended comment is any character "!" following
cc                         a command and a field delimiter, and the rest of the
cc                         input line after the "!":
cc
cc                           COMMAND !COMMENT
cc
cc                         You may create your own comment corresponding to any
cc                         single character you desire, with command "marker".
cc                         See "definition", "alias".
cc
cc                         You may create your own comment corresponding to any
cc                         object name or other word you desire, with
cc                         command "alias".  See "definition", "marker".

cg    Comments           See "comment".

cg    Common             See "Common factors".

cg    common             See "common divisor", "common multiplier".

cg    common divisor     A common divisor divides two or more numbers without
cg                         remainder.  A greatest common divisor (gcd) is the
cg                         largest such number, and must contain only those
cg                         prime factors common to all of the numbers, with
cg                         each such prime factor to the least power of that
cg                         factor in any of the numbers.
cg                         If the gcd of a set of numbers is 1, the numbers are
cg                         all relatively prime.
cg                         See command "icalc", option "gcm".

cg    Common factors     In the display for command "factor", the number of
cg                         integers in the set (1, ..., N - 1), where N is the
cg                         integer being factored, that have common factors
cg                         with N.
cg                         The value "Totient function" is the number of
cg                         integers in the set (1, ..., N - 1) that are
cg                         relatively prime to N (have no common factor other
cg                         than 1).

cg    common multiplier
cg                       A common multiplier is divisible by two or more numbers
cg                         without remainder.  A least common multiplier (lcm)
cg                         is the smallest such number, and must contain all of
cg                         the prime factors found any any of the numbers, with
cg                         each such prime factor to the greatest power of that
cg                         factor in any of the numbers.
cg                         See commands "icalc" and "variable", option "lcm".

cc    comp.              An option in command 'icalc", to multiply an integer
cc                         by the Boolean complement of another integer.
cc                         M comp. N means M times the complement of N.
cc                         comp. N:  replace 0 with 1 and 1 with 0.
cc                         See "Boolean".

cc    comp.              An option in command "cluster", to create a cluster
cc                         consisting of all of the points not in a specified
cc                         cluster.  This is the Boolean "complement" of the
cc                         specified cluster.

cg    complement         See "Boolean".

cg    complex            Complex numbers are not allowed as input, except in
cg                         command "roots", options "=" and "?".
cg                         Complex roots displayed by command "roots" are the
cg                         only complex numbers displayed in the output data.
cg
cg                         A complex variable may be written in various forms:
cg                         z = x + i*y = r*exp(i*a) = r*cos(a) + i*r*sin(a),
cg                         where r = sqrt(x^2 + y^2), a = arctan (y / x),
cg                         x = r*cos(a), y = r*sin(a).
cg                         Complex roots of polynomial equations with real
cg                         coefficients only occur in the conjugate pairs
cg                         z = x + i * y and z' = x - i * y.
cg                         Alternate forms for z' are:
cg                         z' = x - i*y = r*exp(-i*a) = r*cos(a) - i*r*sin(a).
cg                         Also, z * z' = r^2, z + z' = 2*x, z - z' = 2*y*i.

cg    component          A vector has a component in each of the 3 coordinate
cg                         directions.  See "coordinate system", "vector".
cg                         See "coordinate conversion", "vector conversion".
cg
cg                         A tensor operator has nine components, in a 3 by 3
cg                         matrix.  See "operator".

cg    composite          See "composite number".

cg    composite number   A composite number is a positive non-prime integer.
cg                         Every known positive composite (non-prime) integer
cg                         N can be expressed as the product of a unique
cg                         collection of positive primes.  These are the prime
cg                         factors of N.  If N has the prime factors P1, P2,
cg                         ..., PN, then N = P1^a1 * P2^a2 * ... * PN^aN, where
cg                         a1, a2, ..., aN are positive integer exponents.

cg    compress           Output files may be compressed by editing them.
cg                         Some examples, using UNIX text editor VI, follow:
cg
cg                         :%s/  *$//
cg
cg                           Remove all trailing blanks.
cg
cg                         :%s/  */ /g
cg
cg                           Reduce all multiple blanks to single.
cg
cg                         :%s/00*E/E/g
cg
cg                           Remove all trailing zeros on floating point output.
cg
cg                         :%s/E+00/g
cg
cg                           Remove all zero exponents on floating point output.
cg
cg                         :%s/.......E/E/y
cg
cg                           Remove the final seven (for example) significant
cg                           figures after the decimal point of floating point
cg                           output.
cg
cg                         :g/^$/d
cg
cg                           Remove blank lines.

cc    conc               A synonym for concentric.

cc    concatenation      A new integer may be created by concatenating two
cc                         integers, e.g., 111 and 33333 may be concatenated to
cc                         form 11133333.  See command "big".
cc                         This is a convenient way to add digits at the
cc                         beginning or end of a big integer.

cc    concentric         An option in command "cylinder", to create a family of
cc                         concentric circular cylinders with a specified axis
cc                         point, axis direction, and equally spaced between two
cc                         radii.
cc                         See "move", "nest", "parallel", "rotate", "scale".
cc
cc                         This command option can be replaced, using a do loop.
cc                         See "do loop use", commands "do", "enddo".

cc    concentric         An option in command "sphere", to create a family of
cc                         concentric spheres with a specified center point,
cc                         inner radius, outer radius, and equally spaced
cc                         between two radii.
cc                         See "move", "nest", "parallel", "rotate", "scale".
cc
cc                         This command option can be replaced, using a do loop.
cc                         See "do loop use", commands "do", "enddo".

cg    concentric         Two spheres are concentric if they share the same
cg                         center.  Two circular cylinders are concentric if
cg                         they share the same axis.  To see if two spheres are
cg                         concentric, use command "distance".  To see if two
cg                         circular cylinders are concentric, use command
cg                         "distance".

cg    conditional        The execution of a block of input lines may be made
cg                         conditional by preceding the block with command
cg                         "if ...", and following the block with command
cg                         "endif ...".  Command "if ..." specifies the
cg                         condition for executing the block of input lines.

cg    cone               A cone is a surface in 3-D space described by an
cg                         implicit quadric equation, and is a limit case of a
cg                         hyperboloid of one or two sheets.  A conical surface
cg                         includes both sheets, on opposite sides of the vertex
cg                         point.  At any point in the surface, a straight line
cg                         drawn through that point and through the vertex point
cg                         lies entirely within the surface.  Along with planes,
cg                         cylinders, hyperbolic paraboloids and hyperboloids
cg                         of one sheet, a cone is a ruled surface.
cg
cg                         The standard forms of the implicit equations for
cg                         conical quadric surfaces are as follows (each
cg                         coefficient must have the preceding sign):
cg
cg                         Real elliptic cone:
cg                                              x^2 + QYY*y^2 - |QZZ|*z^2 = 0
cg                         Real circular cone:
cg                                                  x^2 + y^2 - |QZZ|*z^2 = 0
cg                                         or       x^2 + y^2 = K^2 * z^2
cg                         Imaginary elliptic cone:
cg                                                x^2 + QYY*y^2 + QZZ*z^2 = 0
cg                         Imaginary circular cone:
cg                                                    x^2 + y^2 + QZZ*z^2 = 0
cg                                         or         x^2 + y^2 = -K^2 * z^2
cg                         where K = tan (ALPHA), and ALPHA is the half-angle
cg                         at the vertex.  K = sqrt (QZZ/QXX).
cg
cg                         Given a cone on the z axis, through the points
cg                         (R1, Z1), (R2, Z2), where R1^2 = X1^2 + Y1^2, and
cg                         R2^2 = X2^2 + Y2^2, the equation of the cone is
cg                         (Z2-Z1)^2*(x^2+y^2) - (R2-R1)^2*(z-ZVER)^2 = 0,
cg                         where ZVER = (Z1 * R2 - Z2 * R1) / (R2 - R1).
cg
cg                         See "half-angle".  Also see "phi cone".
cg
cg                         The word "cone" in a command means a real circular
cg                         cone.  Otherwise, use "quadric".
cg
cg                         To find the distance from a point to a cone, use
cg                         command "distance", "proximal" or "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and a cone, use command "distance".
cg
cg                         The following commands relate to circular
cg                         cones:
cg                         accelerate, arc, axisym, cone, copy, debug, delete,
cg                         distance, extrema, help, invert, last, list, move,
cg                         operator, point, proximal, quadric, reflect, rename,
cg                         repack, rotate, scale, search, side, slice, sort,
cg                         symbol, synonym, tables, track, triple, vector, zone.
cg
cg                         A family of nested cones, with the same vertex and
cg                         axis, but with equally spaced vertex half-angles,
cg                         may be created with command "cone", option "nest".
cg                         See "parallel planes", "rotated planes",
cg                         "concentric".
cg
cg                         Also see "phi cone".

cc    cone               In a command, means a circular cone, except in
cc                         command "cone", option "scale", when the scaling
cc                         operator is linear or radial with an axis not
cc                         parallel to that of the base cone, elliptic cones
cc                         are created.

cc    cone               Command to display or create circular cones (except
cc                         that linear or radial scaling may create elliptic
cc                         cones).  Creating a cone replaces any existing
cc                         quadric surface having the same name.
cc                         The normal vectors will be pointed radially outward.
cc                         See "quadric" for other options.
cc                         See "do loop use".
cc
cc                         Command "cone" relates to objects:  axisym, cone,
cc                         line, operator, point, quadric, symbol, variable
vv                         vector.
cc
cc                         Forms include:
cc
ccin                       help cone
ccin                       cone [help]
cc
cc                           Display the command options.
cc
ccin                       cone [all,list]
cc
cc                           Display all circular cones (short display).
cc
ccin                       cone list CNNAME1 CNNAME2 CNNAME3 ...
cc
cc                           Display circular cones CNNAME1, CNNAME2, CNNAME3,
cc                           ..., with or without subscripts (short display).
cc
ccin                       cone CNNAME
cc
cc                           Display circular cone CNNAME (long display).
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         CNNAME or CONE(2) to form the name by incrementing or
cc                         decrementing the base name for circular cones.
cc                         See command "last", "increment names".
cc
ccin                       cone CNNAME line PAXIS VAXIS ALNAME
cc
cc                           Create circular cone CNNAME on the axis specified
cc                           by point PAXIS and vector VAXIS, and passing
cc                           through the end points of line ALNAME.  This is the
cc                           best way to specify the bounding surfaces of an
cc                           axially symmetric zone whose projected area is a
cc                           polygon.
cc
ccin                       cone CNNAME fit PVER VAXIS PNAME2
cc
cc                           Create circular cone CNNAME with vertex point PVER,
cc                           axis vector VAXIS, and passing through the point
cc                           PNAME2.
cc
ccin                       cone CNNAME angle PVER VAXIS ANGLE
cc
cc                           Create circular cone CNNAME with vertex point PVER,
cc                           axis vector VAXIS, and vertex half-angle ANGLE.
cc
ccin                       cone CNNAME nest NUMCONE INC PVER VAXIS ANG1 ANG2
cc
cc                           Create a family of NUMCONE nested circular cones
cc                           CNNAME, ..., with names incremented by INC
cc                           characters, with the same vertex point PVER and
cc                           axis vector VAXIS, and with vertex half-angles
cc                           equally spaced from ANG1 to ANG2.
cc                           See "increment names".
cc
cc                           This command can be replaced, by using commands
cc                           "do" and "enddo" to create a do loop over any of
cc                           the "cone" commands, using cones with subscripted
cc                           names, with more general specification of
cc                           properties.
cc
ccin                       cone CONE(2) move NUMCONE INC CONE(1) VMOVE
cc
cc                           Create a family of NUMCONE circular cones CONE(2),
cc                           ..., with names incremented by INC characters, and
cc                           spaced at intervals of vector VMOVE, starting from
cc                           circular cone CONE(1).  See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       cone CONE(2) rotate NUMCONE INC CONE(1) OPNAME PINV
cc
cc                           Create a family of NUMCONE circular cones CONE(2),
cc                           ..., with names incremented by INC characters, by
cc                           rotating the preceding circular cone with operator
cc                           OPNAME and invariant point PINV, starting from
cc                           circular cone CONE(1).  See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       cone CONE(2) scale NUMCONE INC CONE(1) OPNAME PINV
cc
cc                           Create a family of NUMCONE cones CONE(2), ..., with
cc                           names incremented by INC characters, by scaling the
cc                           preceding cone with operator OPNAME and invariant
cc                           point PINV, starting from circular cone CONE(1).
cc                           See "increment names".  WARNING:  a linear or
cc                           radial scaling axis not parallel to that of CONE(1)
cc                           will produce elliptic cones.
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
cc                         Synonyms:  [help, h], [line, l, ln],
cc                         [move, mv, trans, translate], [point, p, pnt, pt],
cc                         [rotate, rot].

cc    cone               An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more circular cones.

cg    cone               See "cone, circular", "cone, elliptic".

cg    cone arrays        Cones may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cc    CONE(1)            In command "cone", the name of a base cone, used to
cc                         create a family of cones.

cg    cone, circular     For the circular cone:
cg                         QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QXX = QYY > 0, QZZ < 0),
cg                         the center and vertex are at x = y = z = 0,
cg                         the main axis is the z axis, and
cg                         the half-angle from the z axis is
cg                         atan (sqrt (-QZZ/QXX)).

cg    cone, elliptic     For the elliptic cone:
cg                         QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QXX => QYY > 0, QZZ < 0),
cg                         the center and vertex are at x = y = z = 0,
cg                         the main axis is the z axis,
cg                         the eccentricity is sqrt (1 - QYY / QXX) < 1,
cg                         the half-angle from the z axis toward the x axis is
cg                         atan (sqrt (-QZZ/QXX)), and
cg                         the half-angle from the z axis toward the y axis is
cg                         atan (sqrt (-QZZ/QYY).

cg    cones              A family of circular cones may be created with
cg                         command "cone", options "concentric", "move",
cg                         "rotate" or "scale" (with restrictions) or as
cg                         follows:
cg
cg                         p pinv ...
cg
cg                           Create an invariant point, if needed.
cg
cg                         cone CONE(1) (options)
cg
cg                           Create a base cone.
cg
cg                         op opr (options)
cg
cg                           Create a tensor operator, to reflect, rotate,
cg                           invert or scale, if needed.
cg
cg                         v vmove (options)
cg
cg                           Create a vector for use as a translation operator,
cg                           if needed.
cg
cg                         cp CONE(1) CONE(2)
cg
cg                           Copy cone CONE(1) to the first cone.
cg
cg                         Repeat the following block of commands as many times
cg                         as needed to create the rest of the family of cones.
cg                         See "shortcuts", "saved input lines".
cg
cg                         move cone CONE(1) vmove
cg
cg                           Move the base cone by amount vmove, if needed.
cg
cg                         [invert,reflect,rotate,scale] cone CONE(1) opr [pinv]
cg
cg                           Invert, reflect, rotate or scale the base cone with
cg                           tensor operator opr, if needed.
cg
cg                         copy cone CONE(1) +
cg
cg                           Copy the base cone to the next cone.

cg    conflicts          Tests are made for conflicts between the names of
cg                         preset synonyms, symbols and aliases created by
cg                         the user.  If any are found, a warning message is
cg                         displayed, and the objects are displayed.

cg    congruent          Two geometric objects are congruent if they have the
cg                         same size and shape, and may be translated or rotated
cg                         to have the same position and orientation, such that
cg                         the difference between them is no more than the
cg                         estimated error in calculating that difference, based
cg                         on the tolerance limit TOL.
cg                         Objects that are mirror images, and can not be
cg                         superimposed, are not congruent.

cg    conic              See "conic sections".

cg    conic sections     The conic sections are the curves resulting from the
cg                         intersection of a plane with a circular cone.
cg                         The type of curve depends on the angle THETA of the
cg                         plane relative to the axis of the cone, and the
cg                         half-angle THHALF of the cone:
cg
cg                         hyperbola:  THETA = 0.
cg
cg                         parabola:   0 < THETA < THHALF.
cg
cg                         ellipse:    THHALF < THETA < 90 degrees.
cg
cg                         circle:     THETA = 90 degrees.

cg    conjugate          Conjugate complex numbers are pairs of complex numbers
cg                         of the form x + i*y and x - i*y, where i is the
cg                         square root of -1.
cg                         z  = x + i*y = r*exp( i*a) = r*cos(a) + i*r*sin(a),
cg                         z' = x - i*y = r*exp(-i*a) = r*cos(a) - i*r*sin(a).
cg                         where r = sqrt(x^2 + y^2), a = arctan(y / x),
cg                         x = r*cos(a), y = r*sin(a).
cg                         Note that the sum of the two conjugate complex
cg                         numbers is 2*x, while the product is x^2 + y^2.

cg    conjugate          In modular arithmetic, two numbers are conjugate if
cg                         they have the same value modulo the modulus.  That
cg                         is, they only differ by a multiple of the modulus.

cg    conjugate          On a hyperbola or hyperbolic cylinder, the conjugate
cg                         semiaxis is the line perpendicular to the transverse
cg                         semiaxis, from either of the two vertices to either
cg                         of the asymptotes of the hyperbola.  Its length is
cg                         half the distance between the asymptotes at the
cg                         vertex, in the direction of a tangent to the vertex.
cg                         See "transverse".
cg
cg                         For the hyperbola or hyperbolic cylinder,
cg                         QC + QXX * x^2 + QYY * y^2 = 0,
cg                         (QXX > 0, QYY < 0),
cg                         the length of the conjugate semiaxis is:
cg                         sqrt (QC / QXX), if QC > 0, and
cg                         sqrt (QC / QYY), if QC < 0.
cg
cg                         For the hyperboloid of one sheet,
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QC < 0, QXX => QYY > 0, QZZ < 0),
cg                         the length of the conjugate semiaxis is:
cg                         sqrt (QC / QZZ).
cg
cg                         For the hyperboloid of two sheets,
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QC > 0, QXX => QYY > 0, QZZ < 0),
cg                         the length of the conjugate semiaxis is:
cg                         sqrt (QC / QYY), in the plane x = 0, and
cg                         sqrt (QC / QXX), in the plane y = 0.

cg    constant           See "variable".

cg    constant           In data displayed for a quadric surface, the notation
cg                         "x const", "y const" or "z const" describes a plane
cg                         parallel to a major plane.  In data displayed for a
cg                         quadric curve, those notations describe a line
cg                         line perpendicular to a major axis.

cg    constants          See "internal" for GEOM internal numerical constants.

cg    construct          See "create".

cc    contfr             Command to find the integer coefficients of the
cc                         continued fraction for a specified value, and to
cc                         find the value of a continued fraction given the
cc                         coefficients (in any mode).
cc                         See "continued fraction".
cc
cc                         Command "contfr" relates to objects:  symbol,
cc                         variable.
cc
cc                         Forms include:
cc
ccin                       help contfr
ccin                       contfr [help]
cc
cc                           Display the command options.
cc
ccin                       contfr = VALUE
cc
cc                           Find the integer coefficients kc1, kc2, kc3, ...
cc                           of the continued fraction with value VALUE:
cc                           VALUE = kc1 + 1/(kc2 + 1/(kc3 + 1/(kc4 + ...))),
cc                           and the successive partial sums for VALUE, and
cc                           the integer numerators and denominators of the
cc                           corresponding rational fraction convergents.
cc                           If VALUE < 1, kc1 = 0.  Otherwise kc1 > 0.
cc                           All other coefficients > 0.
cc                           The coefficients of the continued fraction for 1/x
cc                           will differ from those of x only by a leading zero.
cc
cc                           The series will terminate when the relative
cc                           difference between successive partial sums is less
cc                           than TOL.  A very large final coefficient may
cc                           sometimes be ignored.
cc
ccin                       contfr C1 C2 C3 ... CN
cc
cc                           Find the value x of a continued fraction,
cc                           x = C1 + 1/(C2 + 1/(C3 + 1/(C4 + ...))), given
cc                           the coefficients C1, C2, C3, ... CN, which may be
cc                           integer or floating point, positive, zero or
cc                           negative.  Also find
cc                           the numerators and denominators of the rational
cc                           fraction convergents (when all coefficients are
cc                           integers) equivalent to the successive partial sums
cc                           of the continued fraction.
cc                           The method will fail if an infinity occurs.
cc                           Final coefficients of 0 will cause the values of
cc                           successive partial sums to oscillate, if they have
cc                           not already converged.
cc                           A final coefficient of 1 may be added to the
cc                           preceding coefficient.
cc
cc
cc                         See "continued fraction".
cc
cc                         Synonyms [contfr, cf], [help, h].

cg    continue           See "restart".

cg    continued          See "continued fraction".

cg    continued fraction
cg                       An expression of the form
cg                         x = C1 + 1/(C2 + 1/(C3 + 1/(C4 + ... 1/(CN))))...
cg                         where x > 0, and the coefficients are all positive
cg                         integers, except for C1, which is zero if x < 1.
cg                         If x is a rational number (an integer or a ratio of
cg                         two integers) the continued fraction terminates.
cg                         If x is an irrational number (a root of a polynomial
cg                         equation) the coefficients repeat in groups.
cg                         If x is a transcendental number (neither rational or
cg                         irrational, such as pi or e) the coefficients do not
cg                         repeat in groups, but may still form a predictable
cg                         pattern.
cg
cg                         To find the integer coefficients, start with the
cg                         integer part of x, find the reciprocal of the
cg                         remainder, and repeat.  See command "contfr".
cg
cg                         To evaluate a continued fraction, see "convergent",
cg                         "partial sum", command "contfr".
cg
cg                         If the first coefficient is zero, x is the reciprocal
cg                         of the x evaluated from the remaining coefficients.
cg
cg                         The final N coefficients may be replaced by a single
cg                         coefficient equal to the value x (not necessarily an
cg                         integer) of the continued fraction consisting of just
cg                         those N coefficients.
cg
cg                         Final coefficients of K, 1 may be replaced by K + 1.
cg
cg                         Golden Mean (n = 1) and Silver Means (n > 1):
cg                         the coefficients of the continued fraction with value
cg                         (n + sqrt (n^2 + 4)) / 2 are (n, n, n, n, ...)
cg                         for n = 1, 2, 3, ...
cg
cg                         A continued fraction solution for a root of the
cg                         quadratic equation P(x) = a + a*b * x - b * x^2 is
cg                           r1 = CF {a, b, a, b, a, b, ...}.
cg
cg                         See files cf.root and cf.size in ~edwards/geom/test

cg    contour            See "slice", "profile".

cg    control            See "control errors".

cg    control errors     To control typing errors, use command "undo", "redo"
cg                         or "indo", and see "input file".
cg                         To control numerical truncation errors, see "tol",
cg                         "error estimate".

cn    conv.mac           A macro file containing selected metric conversion
cn                         factors.  In directory ~edwards/work/geom/test .
cn                         To view or to read in conversion factors, use command
cn
cn                         input ~edwards/work/geom/test/conv.mac [options]
cn
cn                         See command "input" for options.

cg    conventions        See "command form", "default arguments", "key word",

cg    converge           See "convergence".

cg    convergence        Convergence of the iterative procedures used by
cg                         commands "side", "distance" and "proximal" is
cg                         determined by TOL.
cg                         The final relative change in distance must be no
cg                         greater than TOL, and the final angle between the
cg                         line from the specified point to the proximal point
cg                         and the normal vector of the quadric surface at the
cg                         proximal point must have a cosine within TOL of -1.0
cg                         or 1.0.

cg    convergent         The n'th convergent of a continued fraction is the
cg                         ratio of integers obtained by evaluating only the
cg                         first n coefficients of the continued fraction.
cg                         The value of the n'th convergent is the n'th
cg                         partial sum.
cg
cg                         To find the n'th convergent P(n) / Q(n), and the
cg                         n'th partial sum x(n), given the coefficients
cg                         C(n), n = 1, N:
cg                           P(0) = 1,            P(1) = C(1),
cg                           P(n) = C(n) * P(n-1) + P(n-2), n = 2, N,
cg                           Q(0) = 0,            Q(1) = 1,
cg                           Q(n) = C(n) * Q(n-1) + Q(n-2), n = 2, N,
cg                           x(n) = P(n) / Q(n), n = 2, N.
cg
cg                         These values are found and displayed when command
cg                         "contfr" is used.

cg    conversion         See "conversion factors", "coordinate conversion",
cg                         "convert output", "unit conversion",
cg                         "vector conversion".

cg    conversion factors
cg                       Selected metric conversion factors are in macro file
cg                         conv.mac in directory ~edwards/work/geom/test .

cg    convert            See "conversion factors", "coordinate conversion",
cg                         "convert output", "unit conversion",
cg                         "vector conversion".

cg    convert output     To convert the output file(s) (geom_hsp or the file(s)
cg                         specified by the user with command "output") to an
cg                         input file, see "input file".

cc    COORD              In command "plot", option "axis", the coordinate to be
cc                         plotted on the horizontal or vertical axis.  Must be
cc                         x, y, z, rcyl, rsph, theta or phi.  See commands
cc                         "coordinate", "angles".

cc    coord              A synonym for coordinate.

cg    coordinate         A value used to locate a mesh point in the mesh or
cg                         to locate a physical point in a physical space.
cg                         See "coordinate system", command "search".

cc    coordinate         An option in command "debug", to display GEOM internal
cc                         variables for the coordinate system and angles:
cc                         asys, acoordu, acoordv, acoordw, angunit.
cc                         Synonyms:  [coordinate, coord, cs, sys, system],

cc    coordinate         Command to display or specify the coordinate system of
cc                         the physical space.  See "coordinate conversion",
cc                         "coordinate system", "vector conversion", "angle".
cc                         Display of point coordinates and vector components
cc                         will be in the specified coordinate system, unless
cc                         specifically labeled otherwise, e.g., xyz.
cc                         Specification of point coordinates and vector
cc                         components must be in the specified coordinate
cc                         system.
cc                         Note:  variables are not affected by changes
cc                         in the coordinate system or angle units, so their
cc                         values and units remain as intended by the user when
cc                         originally specified.  Variables may be assigned a
cc                         descriptive tag to specify the units or to provide
cc                         other information.  See command "variable", option
cc                         "tag".
cc
cc                         Forms include:
cc
ccin                       help coordinate
ccin                       coordinate help
cc
cc                           Display the command options.
cc
ccin                       coordinate
cc
cc                           Display the coordinate system.
cc
ccin                       coordinate cartesian
cc
cc                           Set the coordinate system to rectangular (x, y, z).
cc
ccin                       coordinate cylindrical
cc
cc                           Set the coordinate system to cylindrical (rcyl,
cc                           theta, z).
cc
ccin                       coordinate spherical
cc
cc                           Set the coordinate system to spherical (rsph,
cc                           theta, phi).
cc
cc                         See command "angles" to specify the angle units.
cc
cc                         Synonyms:  [coordinate, coord, cs, sys, system],
cc                         [cartesian, Cartesian, cart, rectangular, rect, xyz],
cc                         [cylindrical, cyl], [help, h], [spherical, sph].

cg    coordinate         See "coordinate conversion", "coordinate surface",
cg                         "coordinate system", "surface of coord".

cg    coordinate conversion
cg                         To convert the coordinate system of existing points
cg                         and/or vectors, use commands "coordinate" and
cg                         "angles" to specify a new coordinate system, and
cg                         display the points and/or vectors with commands
cg                         "point" and "vector".  See "coordinate system",
cg                         "vector conversion".
cg                         Angles may be in degrees or radians.  See "angle".
cg
cg                         To convert the coordinate system of points and/or
cg                         vectors that appear in a data display, but are not
cg                         saved, see "(xyz)".
cg
cg                         rectangular (x, y, z):
cg                           x = rcyl * cos (theta),
cg                           x = rsph * cos (theta) * sin (phi),
cg                           y = rcyl * sin (theta),
cg                           y = rsph * sin (theta) * sin (phi),
cg                           z = rsph * cos (phi).
cg
cg                         cylindrical (rcyl, theta, z):
cg                           rcyl^2  = x^2 + y^2,
cg                           rcyl    = rsph * sin (phi),
cg                           theta   = arctan (y / x)        (azimuth angle),
cg                           sin (theta) = y / sqrt (x^2 + y^2),
cg                           cos (theta) = x / sqrt (x^2 + y^2),
cg                           tan (theta) = y / x.
cg
cg                         spherical (rsph, theta, phi):
cg                           rsph^2 = x^2 + y^2 + z^2,
cg                                   = rcyl^2 + z^2,
cg                           theta   = arctan (y / x)        (azimuth angle),
cg                           cos (theta) = x / sqrt (x^2 + y^2),
cg                           sin (theta) = y / sqrt (x^2 + y^2),
cg                           phi     = arccos (z / rsph)     (polar angle
cg                                   = arctan (rcyl / z)      from z axis),
cg                           sin (phi) = sqrt (x^2 + y^2) /
cg                                       sqrt (x^2 + y^2 + z^2),
cg                           sin (phi) = rcyl / sqrt (rcyl^2 + z^2),
cg                           cos (phi) = z / sqrt (x^2 + y^2 + z^2),
cg                           cos (phi) = z / sqrt (rcyl^2 + z^2),
cg                           tan (phi) = sqrt (x^2 + y^2) / z,
cg                           tan (phi) = rcyl / z.
cg
cg                         Note:  to avoid numerical error and insure that
cg                         all conversion are reversible, add 10^99 to x,
cg                         rcyl and rs, replace all sines and cosines less
cg                         than TOL with zero, and replace any results with
cg                         an estimated relative error less than TOL with zero.

cg    coordinate surface
cg                       A coordinate surface is a surface at a constant value
cg                         of one of the three coordinates of an orthogonal
cg                         coordinate system, such as rectangular (x, y z),
cg                         cylindrical (rcyl, theta, z) or spherical (rsph,
cg                         theta, phi).  To create a coordinate surface,
cg                         see "x plane", "y plane", "z plane", "rcyl cylinder",
cg                         "theta plane", "rsph sphere", "phi cone".
cg
cg                         To specify the coordinate system and the angle units,
cg                         use commands "coordinate" and "angles".
cg
cg                         To create a volume element bounded by coordinate
cg                         surfaces, use command "brick".
cg
cg                         To create a 3-D array of points on a specified family
cg                         of coordinate surfaces, use command "brick", followed
cg                         by command
cg                         "point PMESH brick BRNAME mesh block [volume]".
cg                         See "briquette".
cg
cg                         To create a family of variables whose values
cg                         are an arithmetic or geometric series, for use in
cg                         specifying coordinates, use command "variable",
cg                         option "series".

cg    coordinate system
cg                       The coordinate system used to specify the coordinates
cg                         of points is specified with command "coordinate",
cg                         and may be one of the following:
cg
cg                         Rectangular (x, y, z):
cg                           x = rcyl * cos (theta),
cg                           x = rsph * cos (theta) * sin (phi),
cg                           y = rcyl * sin (theta),
cg                           y = rsph * sin (theta) * sin (phi),
cg                           z = rsph * cos (phi).
cg
cg                         Cylindrical (rcyl, theta, z):
cg                           rcyl^2  = x^2 + y^2,
cg                           rcyl    = rsph * sin (phi),
cg                           theta   = arctan (y / x),
cg                             (azimuth angle around z axis, measured
cg                              counterclockwise from x axis).
cg
cg                         Spherical (rsph, theta, phi):
cg                           rsph^2 = x^2 + y^2 + z^2,
cg                                   = rcyl^2 + z^2,
cg                           theta   = arctan (y / x),
cg                           phi     = arccos (z / rsph),
cg                                   = arctan (rcyl / z),
cg                             (polar angle from positive z axis).
cg
cg                         For practical reasons, the coefficients of the
cg                         implicit equations for quadric surfaces are always
cg                         based on a rectangular (x, y, z) coordinate system.
cg
cg                         See "coordinate conversion", "vector conversion".

cg    coordinates        See "coordinate conversion", "coordinate",
cg                         "vector conversion".

cg    copy               See "copy objects".

cc    copy               Command to copy an alias, marker, variable, vector,
cc                         tensor operator, point, cluster, line, triangle,
cc                         regular polygon, annular disk, quadric surface
cc                         (including a plane, sphere, circular cylinder,
cc                         circular cone, hyperbolic paraboloid, ellipsoid,
cc                         axially symmetric quadric surface or general quadric
cc                         surface), regular polyhedron, tetrahedron, brick,
cc                         zone, probability bin, probability distribution
cc                         function (pdf), symbol or big integer.
cc                         You can not copy over an existing object.
cc                         The copy must not have the same name as the original.
cc                         This is an excellent way to create a backup before
cc                         modifying an object, in case of an error.
cc                         Note:  no replacement of symbols will be done
cc                         in the command "copy symbol ...".
cc
cc                         Command "copy" relates to objects:  alias, axisym
cc                         big, bin, brick, cluster, cone, cylinder, disk,
cc                         ellipsoid, hyperb, line, marker, operator, pdf,
cc                         plane, point, polygon, polyhedron, quadric, sphere,
cc                         symbol, tetrahedron, triangle, variable, vector,
cc                         zone.
cc
cc                         Forms include:
cc
ccin                       help copy
ccin                       copy [help]
cc
cc                           Display the command options.
cc
cc                         In the following command, use "+" or "-" instead of
cc                         OBJNAME2 to form the name by incrementing or
cc                         decrementing the base name for objects of type
cc                         OBJTYPE (which may not be OBJNAME).
cc                         See command "last", "increment names".
cc
ccin                       copy OBJTYPE OBJNAME OBJNAME2
cc
cc                           Copy object type OBJTYPE (alias, axisym, big, bin,
cc                           brick, cluster, cone, cylinder, disk, ellipsoid,
cc                           hyperb, line, marker, operator, pdf, plane, point,
cc                           polygon, polyhedron, quadric, sphere, symbol,
cc                           tetrahedron, triangle, variable, vector, zone)
cc                           with name OBJNAME to the same object type with name
cc                           OBJNAME2.
cc                           No copy will be made if OBJTYPE is symbol or alias.
cc
cc                         The command above may be placed between commands "do"
cc                         and "enddo", to create a do loop over subscripted
cc                         object names, to copy all of them to new names.
cc
cc                         Synonyms:  [alias, al, a,], [brick, br],
cc                         [cluster, cl], [copy, cp, dup, duplicate],
cc                         [cylinder, cyl], [disk, disc, dk],
cc                         [ellipsoid, el, ell, ellipse], [help, h],
cc                         [hyperb, saddle], [line, l, ln], [marker, mark, m],
cc                         [operator, oper, op], [plane, pl],
cc                         [point, pnt, pt, p], [polygon, polyg, pg],
cc                         [polyhedron, polyh, ph], [quadric, quad, q],
cc                         [sphere, sph], [symbol, sym, symb],
cc                         [tetrahedron, tetra, tet], [triangle, tri],
cc                         [variable, var, set, s], [vector, v, vect],
cc                         [zone, z, zn].

cg    copy objects       See command "copy".  See "cones", "cylinders",
cg                         "planes", "quadric families", "quadric surfaces",
cg                         "spheres".
cg                         To copy the output from a command for use in a new
cg                         command, see "cut and paste".

cc    cos                A synonym for cosine, in commands "project" and
cc                         "vector".

cc    cos                An option in command "project", when projecting onto
cc                         the surface of a circular cylinder, to make the axial
cc                         distance of each point from point PCEN on the axis
cc                         equal to the cosine of the original polar angle phi
cc                         of the point, as measured relative to point PCEN,
cc                         in the positive direction along the axis.

cc    cos                An option in command "variable", argument FUNCTION,
cc                         to indicate the trigonometric cosine function.
cc                         A cos B means A * cos (B).

cf    cos                The trigonometric cosine function.
cf                         Example:  y = cos (x) means y is the cosine of x.
cf                         x must be in radians.

cc    cosh               An option in command "variable", argument FUNCTION,
cc                         to indicate the hyperbolic cosine function.
cc                         A cosh B means A * cosh (B).

cf    cosh               The hyperbolic cosine function.  Example:  y = cosh (x)
cf                         means y = (1/2) * (e^x + e^(-x)).

cc    cosine             An option in command "vector", to create a vector by
cc                         randomly sampling from a cosine-power distribution
cc                         around a specified axis vector VAXIS, with a
cc                         specified power POWER.
cc
cc                         An option in command "project", to project onto the
cc                         surface of a circular cylinder by replacing the axial
cc                         distance of a point from the center of the cylinder
cc                         with the cosine of the polar angle phi of the
cc                         point, to preserve relative areas determined from
cc                         points projected from a spherical surface centered
cc                         at the center of the cylinder.  A special projection
cc                         used in cartography.
cc
cc                         Synonyms:  [cosine, cos].

cg    cosine             To find the cosine of the angle between two vectors,
cg                         use command "dot".
cg                         Also see "cosine-power", "Law of Cosines".

cg    cosine-power       A cosine-power distribution is one in which a power
cg                         of the cosine of the angle from a specified axis
cg                         is uniformly distributed from 0 to 1, and all angles
cg                         are within 90 degrees of the positive axial
cg                         direction.
cg
cg                         For an isotropic distribution the power must be zero.
cg
cg                         For a cosine distribution, the power must be 1.
cg
cg                         For distributions more strongly peaked toward the
cg                         axis, the power must be larger than 1.
cg
cg                         For a unit vector randomly sampled from a general
cg                         cosine-power distribution, with a power POWER from
cg                         zero to infinity, the relationship between POWER and
cg                         the expected value <ua> of the axial component of the
cg                         vector is as follows:
cg
cg                           <ua>  = 1 -  1 / (POWER + 2)
cg                           POWER = 1 / (1 - <ua>) - 2
cg
cg                           <ua>        POWER   angle (degrees)
cg                           0.5            0        60.00
cg                           2/3            1        48.19
cg                           0.8            3        36.87
cg                           0.9            8        25.84
cg                           0.99          98         8.11
cg                           0.999        998         2.56
cg                           0.9999      9998         0.81
cg                           0.99999    99998         0.26

cg    count              If you need to count the number of times a command is
cg                         repeated, initialize a variable to zero, then
cg                         increment it each time the command is repeated.
cg
cg                         variable ntot [=] 0
cg
cg                           Initialize ntot to zero.
cg
cg                         alias t = "variable ntot = ntot + 1 ; COMMAND"
cg
cg                           Create alias "t" to increment ntot and do COMMAND.
cg
cg                         t
cg
cg                           Increment ntot and do command COMMAND.
cg
cg                         Any block of commands such as those above may be
cg                         executed together in several ways.  See "alias",
cg                         "macro", "saved input lines", "shortcuts".

cc    cp                 A synonym for copy.

cg    cpar               In columns 1-4 of geom_base, indicates a parameter.

cg    CPU                Central processing unit.  In the time summary at the
cg                         end of the run, a component of machine time use,
cg                         separate from input/output (I/O) and system (SYS)
cg                         time.

cn    craps.mac          A macro file to create a pdf representing rolling two
cn                         dice, to demonstrate how to work with bins and pdfs.
cn                         In ~/work/geom/test .

cg    create             To describe an environmental, operational, geometric,
cg                         or Monte Carlo object, using a command.
cg                         See "create objects", "create environment",
cg                         "create operators".

cg    create             See "create environment", "create Monte Carlo",
cg                         "create objects", "create operators".

cg    create environment
cg                       The commands for modifying the default values of
cg                         certain environmental parameters are:
cg                         alias, marker, angles, coordinate, delimiter, icalc,
cg                         input, output, plot, tol.

cg    create Monte Carlo
cg                         The commands for creating Monte Carlo objects are:
cg                         bin (probability bins) and pdf (probability
cg                         distribution functions).
cg                         Also see "random", "sample", "spin".

cg    create objects     Geometric objects are created with commands:
cg                         point, cluster, line, vector, triangle, polygon,
cg                         disk, plane, sphere, cylinder, cone, hyperb,
cg                         ellipsoid, axisym, quadric, polyhedron, tetrahedron,
cg                         brick, zone.
cg
cg                         Other objects are created with commands "alias",
cg                         "bin", "marker", "operator", "pdf", "symbol",
cg                         "variable".
cg
cg                         The commands for creating objects have the general
cg                         forms:
cg
cg                         OBJTYPE [OBJNAME,+,-] [various specifications]
cg
cg                           Create one or more objects of type OBJTYPE with
cg                           initial name [OBJNAME, or the name incremented or
cg                           decremented from the base name for this object
cg                           type], with various specifications.
cg                           Names of objects after the first object will be
cg                           incremented in ASCII sequence.
cg
cg                         copy OBJTYPE OBJNAME [OBJNAME2,+,-]
cg
cg                           Copy the object of object type OBJTYPE with name
cg                           OBJNAME to object [OBJNAME2, or the name
cg                           incremented or decremented from the last name
cg                           specified for an object of this type].
cg
cg                         rename OBJTYPE OBJNAME [OBJNAME2,+,-]
cg
cg                           Rename the object of object type OBJTYPE with name
cg                           OBJNAME to object [OBJNAME2, or the name
cg                           incremented or decremented from the last name
cg                           specified for an object of this type].
cg
cg                         rename OBJTYPE OBJNAME OBJNAME2 array
cg
cg                           Rename the object of object type OBJTYPE by
cg                           replacing the subscripted stem name OBJNAME with
cc                           the subscripted stem name OBJNAME2.
cg
cg                         delete OBJTYPE [OBJNAME,all]
cg
cg                           Delete the object of object type OBJTYPE [with name
cg                           OBJNAME, or all such objects].
cg
cg                         See "OBJTYPE", "OBJNAME".

cg    create operators   Operators are created with commands:  variable, vector,
cg                         operator.

cg    cross              See "cross product", command "cross".

cc    cross              Command to find the cross (outer) product of two
cc                         vectors, which has a magnitude equal to the area of
cc                         the parallelogram specified by the two vectors, or
cc                         twice the area of the triangle specified by the two
cc                         vectors.
cc
cc                         Command "cross" relates to objects:  symbol, vector.
cc
cc                         Forms include:
cc
ccin                       help cross
ccin                       cross [help]
cc
cc                           Display the command options.
cc
ccin                       cross VNAME1 VNAME2
cc
cc                           Find the vector which is the cross (outer) product
cc                           of vectors VNAME1 and VNAME2.
cc
cc                         The cross product may be saved as another vector:
cc
cc                         vector VNAME cross VNAME1 VNAME2
cc
cc                           Create vector VNAME, the cross (outer) product of
cc                           vectors VNAME1 and VNAME2.
cc
cc                         Synonyms:  [cross, outer], [help, h].

cc    cross              An option in command "vector", to create a vector which
cc                         is the cross product of two vectors.

cg    cross product      The cross (or outer or vector) product of two vectors
cg                         A = (AX, AY, AZ) and B = (BX, BY, BZ) is the vector
cg                         C = A cross B = (CX, CY, CZ), where
cg                         C = (AY*BZ - AZ*BY, AZ*BX - AX*BZ, AX*BY - AY*BX).
cg                         |C| = |A| * |B| * sin (THETA), where THETA is the
cg                         angle between vectors A and B.
cg                         Note that B cross A = -(A cross B), and that the
cg                         cross product of two parallel vectors is zero.

cc    crt                Command to solve the Chinese Remainder Theorem for
cc                         a specified set of integer divisors and remainders,
cc                         or to find the remainders for a specified integer
cc                         value, using the first few prime numbers as divisors,
cc                         or with a specified set of relatively prime
cc                         divisors.  See command "factor".
cc
cc                         Command "crt" relates to objects:  symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help crt
ccin                       crt [help]
cc
cc                           Display the command options.
cc
ccin                       crt IDIV1 IREM1 IDIV2 IREM2 ... IDIVN IREMN
cc
cc                           Find the least integer value that has the
cc                           remainders IREM1, IREM2, ..., IREMN when divided
cc                           by the integer divisors IDIV1, IDIV2, ..., IDIVN,
cc                           respectively, where the integer divisors are 2 or
cc                           more, and have no common factor.  Also display the
cc                           next four higher integer values with the same
cc                           remainders, if less than 10^18.
cc
ccin                       crt = IVAL
cc
cc                           Find the remainders resulting from dividing the
cc                           integer IVAL by the first few prime numbers
cc                           (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...),
cc                           using the minimum set of prime numbers for which
cc                           IVAL is the least integer with the same remainders.
cc                           IVAL must be 2 or more.
cc
ccin                       crt = IVAL IDIV1 IDIV2 ... IDIVN
cc
cc                           Find the remainders resulting from dividing the
cc                           integer IVAL by the integers IDIV1, IDIV2, ...,
cc                           IDIVN, which must be 2 or more, and have no common
cc                           factors.  IVAL must be 2 or more.
cc
cc                         Synonyms:  [help, h].

cg    cryptography       See command "big".

cc    cs                 A synonym for coordinate.

cg    cube               A cube is a regular polyhedron with 8 vertices,
cg                         12 edges and 6 square faces.
cg                         The edge length, face area, volume, radius of
cg                         inscribed sphere, dihedral angle and central edge
cg                         angle, for a circumscribed sphere radius of 1.0 are:
cg
cg                         edge    = 2.0 / sqrt (3.0)        = 1.154700538379
cg                         area    = 4.0 / 3.0               = 1.333333333333
cg                         volume  = 8.0 * sqrt (3.0) / 9.0  = 1.539600717839
cg                         rinsc   = 1.0 / sqrt (3.0)        = 0.5773502691896
cg                         angdih  = acos (0.0)              = 90.00000000000
cg                         angcent = acos (1.0 / 3.0)        = 70.52877936551
cg
cg                         See "polyhedron", "brick".

cg    cubic              See command "roots".

cg    cubic              A cubic polynomial equation in z = x + i*y, where z is
cg                         real or complex, x and y are real, and i = sqrt (-1):
cg                           P(z) = a0 + a1*z + a2*z^2 + a3*z^3 = 0,
cg                         with real coefficients a0, a1, a2 and a3 = 1, has
cg                         three roots r1, r2, r3, of which one or three are
cg                         real, has two extrema (maxima and minima) e1, e2 of
cg                         which either zero or two are real, and has one real
cg                         inversion point v1.
cg                         The roots satisfy P(z) = 0.  The extrema satisfy
cg                         P'(z) = 0.  The inversion points satisfy P''(z) = 0.
cg                         P' and P'' are the first and second derivatives of
cg                         P with respect to z.
cg                         Some interesting relationships between the
cg                         coefficients and the properties of the polynomial
cg                         are as follows:
cg                          -a0     = r1*r2*r3
cg                           a1 / 3 = (1/3)*(r1*r2 + r2*r3 + r3*r1)
cg                                  = e1*e2
cg                          -a2 / 3 = (1/3)*(r1 + r2 + r3)
cg                                  = (1/2)*(e1 + e2)
cg                                  = v1
cg                           a3     = 1
cg
cg                         Note that the inversion point v1 is at the average z
cg                         value of the roots and the extrema, and that the
cg                         average value of the products of pairs of roots and
cg                         extrema are the same.
cg
cg                         See command "roots".

cg    curly bracket      The left or right curly bracket, "{" or "}".

cc    curt               An option in command "variable", argument FUNCTION,
cc                         to indicate the real cube root function.
cc                         A curt B means A * B^(1/3).

cf    curt               The cube root function.  Example:  y = curt (x) means
cf                         y = x^(1/3) (real value only).
cf
cf                         To find the real and complex cube roots of any
cf                         real nonzero number B, use command:
cf
cf                         roots B 0 0 -1

cg    curvature          The curvature of a curve is the inverse of its radius
cg                         of curvature.  See "radius of curvature",
cg                         "principal radius".
cg                         See commands "arc", "extrema".

cg    curve              See "quadric curve", "intersection curve", outline.

cc    cut                A synonym for slice.

cg    cut                See "cut and paste", "triangle cutting".

cg    cut and paste      If you have a mouse with cut and paste capability, you
cg                         can create variables, points, vectors, etc., by
cg                         cutting and pasting the output from previous commands
cg                         for use as the arguments of the new commands.
cg                         Note that the precision variable TOL should not be
cg                         less than the precision of command arguments.

cc    cute               Command to find all triangles formed by extending any
cc                         two sides of a specified triangle, to add a section
cc                         with the same perimeter and area as the specified
cc                         triangle.  See "cutting line", "triangle cutting",
cc                         and commands "cuts", "trig" and "triangle".
cc
cc                         Command "cute" relates to objects:  symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help cute
ccin                       cute [help]
cc
cc                           Display the command options.
cc
ccin                       cute VA VB VC
cc
cc                           Find each triangle that can be formed by extending
cc                           any two edges of the triangle with edges VA, VB and
cc                           VC, to add a section with the same perimeter and
cc                           area as the original triangle.  This is the inverse
cc                           of command "cuts".
cc
cc                         Synonyms:  [help, h].

cc    cuts               Command to find each straight line that cuts a triangle
cc                         into two parts with equal perimeters and areas.
cc                         See "cutting line", "triangle cutting", and commands
cc                         "cute", "trig" and "triangle".
cc
cc                         Command "cuts" relates to objects:  symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help cuts
ccin                       cuts [help]
cc
cc                           Display the command options.
cc
ccin                       cuts VA VB VC
cc
cc                           Find each straight line that cuts the triangle with
cc                           edges VA, VB and VC into two sections with equal
cc                           perimeters and areas, and display the edges cut,
cc                           the fractional and absolute distances of the ends
cc                           of the cutting line along the edges from their
cc                           common vertex, and the length of the cutting line.
cc                           There may be 1, 2 or 3 such cutting lines.
cc                           This is the inverse of command "cute".
cc
cc                         Synonyms:  [help, h].

cg    cutting            See "cutting line", "triangle cutting".

cg    cutting line       To partition a triangle into two parts of equal area
cg                         and equal perimeter with straight line cuts, see
cg                         "triangle cutting", commands "cute", "cuts",
cg                         "triangle", and "trig".
cg                         The cutting lines will be displayed whenever a
cg                         triangle is created or displayed.

cg    cx                 In columns 1-2 of geom_base, indicates an executable
cg                         program.

cc    cyl                A synonym for cylinder.

cc    cyl                A synonym for cylindrical in command "project".

cc    CYL(1)             In command "cylinder", the name of a base cylinder,
cc                         used to create a family of cylinders.

cg    cylinder           A cylinder is a surface in 3-D space described by an
cg                         implicit quadric equation, and is a limit case of a
cg                         hyperboloid of one sheet or an ellipsoid.
cg                         At any point in the surface, any line drawn parallel
cg                         to the cylindrical axis and perpendicular to the
cg                         normal vector, lies entirely within the surface.
cg                         Along with planes, cones, hyperbolic paraboloids and
cg                         hyperboloids of one sheet, a cylinder is a ruled
cg                         surface.
cg
cg                         The standard forms of the implicit equations for
cg                         cylinders are as follows (each coefficient must have
cg                         the preceding sign):
cg
cg                         Parabolic cylinder:                -QY*y + x^2 = 0
cg                         Hyperbolic cylinder:
cg                                                1 + QXX*x^2 - |QYY|*y^2 = 0
cg                         Real elliptic cylinder:
cg                                                - 1 + QXX*x^2 + QYY*y^2 = 0
cg                         Real circular cylinder:
cg                                                - 1 + QXX * (x^2 + y^2) = 0
cg                                              or        x^2 + y^2 =  rcyl^2
cg                         Imaginary elliptic cylinder:
cg                                                  1 + QXX*x^2 + QYY*y^2 = 0
cg                         Imaginary circular cylinder:
cg                                                  1 + QXX * (x^2 + y^2) = 0
cg                                               or       x^2 + y^2 = -rcyl^2
cg
cg                         The word "cylinder" in a command means a real
cg                         circular cylinder.  Otherwise, use "quadric".
cg
cg                         To find the distance from a cylinder to a point, use
cg                         command "distance", "proximal" or "side".
cg
cg                         To find the distance from a circular cylinder to a
cg                         point, use command "distance".
cg                         To find the distance and proximal point, and/or the
cg                         intersections of a circular cylinder and a line,
cg                         a triangle, plane, a sphere or another circular
cg                         cylinder, use command "distance".
cg
cg                         To find the distance or intersection between a plane
cg                         and a general cylinder, use command "distance".
cg
cg                         The following commands relate to circular
cg                         cylinders:
cg                         accelerate, arc, axisym, copy, cylinder, debug,
cg                         delete, distance, extrema, help, invert, last, list,
cg                         move, operator, point, project, proximal, quadric,
cg                         reflect, rename, repack, rotate, scale, search, side,
cg                         slice, sort, symbol, synonym, tables, track, triple,
cg                         vector, zone.
cg
cg                         Also see "rcyl cylinder".

cc    cylinder           In a command, means a circular cylinder, except in
cc                         command "cylinder", option "scale", when a linear or
cc                         radial scaling operator with an axis not parallel to
cc                         that of the base cylinder can create an elliptic
cc                         cylinder.

cc    cylinder           Command to display or create circular cylinders (except
cc                         that linear or radial scaling may create elliptic
cc                         cylinders).  Creating a cylinder replaces any
cc                         existing quadric surface having the same name.
cc                         The normal vectors will be pointed radially outward.
cc                         See "quadric" for other options.
cc                         See "do loop use".
cc
cc                         Command "cylinder" relates to objects:  axisym,
cc                         cylinder, operator, point, quadric, symbol, variable,
cc                         vector.
cc
cc                         Forms include:
cc
ccin                       help cylinder
ccin                       cylinder [help]
cc
cc                           Display the command options.
cc
ccin                       cylinder [all,list]
cc
cc                           Display all circular cylinders (short display).
cc
ccin                       cylinder list CYLNAME1 CYLNAME2 CYLNAME3 ...
cc
cc                           Display circular cylinders CYLNAME1, CYLNAME2,
cc                           CYLNAME3, ..., with or without subscripts (short
cc                           display).
cc
ccin                       cylinder CYLNAME
cc                           Display circular cylinder CYLNAME (long display.
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         CYLNAME or CYL(2) to form the name by incrementing or
cc                         decrementing the base name for circular cylinders.
cc                         See command "last", "increment names".
cc
ccin                       cylinder CYLNAME = PAXIS VAXIS RADIUS
cc
cc                           Create circular cylinder CYLNAME, with an axis
cc                           through point PAXIS in the direction of vector
cc                           VAXIS, and radius RADIUS.
cc
ccin                       cylinder CYLNAME fit VAXIS PNAME1 PNAME2 PNAME3
cc
cc                           Create circular cylinder CYLNAME with axis vector
cc                           VAXIS, through the three points PNAME1, PNAME2 and
cc                           PNAME3, which must not be coincident or collinear.
cc
ccin                       cyl CYLNAME conc NUMCYL INC PAXIS VAXIS RAD1 RAD2
cc
cc                           Create a family of NUMCYL concentric circular
cc                           cylinders CYLNAME, ..., with names incremented by
cc                           INC characters, with an axis through point PAXIS in
cc                           the direction of vector VAXIS, and with radii
cc                           equally spaced from radius RAD1 to radius RAD2.
cc                           See "increment names".
cc                           Note:  "cyl" is a synonym for "cylinder".
cc                           Note:  "conc" is a synonym for "concentric".
cc
cc                           This command can be replaced by using commands
cc                           "do" and "enddo" to create a do loop over any
cc                           of the "cyl" commands, using subscripted cylinder
cc                           names, with more general specifications.
cc
ccin                       cylinder CYL(2) move NUMCYL INC CYL(1) VMOVE
cc
cc                           Create a family of NUMCYL circular cylinders
cc                           CYL(2), ..., with names incremented by INC
cc                           characters, and spaced at intervals of vector
cc                           VMOVE, starting from circular cylinder CYL(1).
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
cc
ccin                       cylinder CYL(2) rotate NUMCYL INC CYL(1) OPNAME PINV
cc
cc                           Create a family of NUMCYL circular cylinders
cc                           CYL(2), ..., with names incremented by INC
cc                           characters, by rotating the preceding cylinder with
cc                           operator OPNAME and invariant point PINV, starting
cc                           from circular cylinder CYL(1).
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       cylinder CYL(2) scale NUMCYL INC CYL(1) OPNAME PINV
cc
cc                           Create a family of NUMCYL cylinders CYL(2), ...,
cc                           with names incremented by INC characters, by
cc                           scaling the preceding cylinder with operator OPNAME
cc                           and invariant point PINV, starting from circular
cc                           cylinder CYL(1).  WARNING:  a linear or radial
cc                           scaling axis not parallel to that of CYL(1) will
cc                           produce elliptic cylinders.  See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
cc                         Synonyms:  [help, h], [concentric, conc],
cc                         [cylinder, cyl], [move, mv, trans, translate],
cc                         [rotate, rot].

cc    cylinder           An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more circular
cc                         cylinders.
cc                         Synonyms:  [cylinder, cyl].

cc    cylinder           An option in command "project", to project onto the
cc                         surface of a circular cylinder.

cg    cylinder           See "cylinder, circular", "cylinder, elliptic",
cg                         "cylinder, hyperbolic", "cylinder, parabolic".

cg    cylinder arrays    Cylinders may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    cylinder, circular
cg                       For the circle or circular cylinder:
cg                         QC + QXX * x^2 + QYY * y^2 = 0, for all z,
cg                         (QC < 0, QXX = QYY > 0),
cg                         the center is at x = 0, y = 0,
cg                         the main axis is the z axis, and
cg                         the radius is RCYL = sqrt (-QC / QXX).
cg
cg                         The standard equation for the circle or circular
cg                         cylinder is x^2 + y^2 = RCYL^2.

cg    cylinder, elliptic
cg                       For the ellipse or elliptic cylinder:
cg                         QC + QXX * x^2 + QYY * y^2 = 0, for all z,
cg                         (QC < 0, QXX => QYY > 0),
cg                         the center is at x = 0, y = 0,
cg                         the main axis is the z axis,
cg                         the major semiaxis is A = sqrt (-QC / QYY),
cg                         the minor semiaxis is B = sqrt (-QC / QXX),
cg                         the eccentricity is sqrt (1 - QYY / QXX) < 1,
cg                         or sqrt (1 - (B / A)^2) < 1,
cg                         the two foci are at:
cg                         x = 0, y = (+/-)sqrt (QC / QXX - QC / QYY),
cg                         or y + (+/-)sqrt (B^2 - A^2),
cg                         the distance from the center to a focus is
cg                         sqrt (QC / QXX - QC / QYY), or sqrt (B^2 - A^2), and
cg                         the length of the latus rectum (width at focus) is
cg                         2 * sqrt (-QC * QYY) / QXX or 2 * B^2 / A.
cg
cg                         The standard equation for the ellipse or elliptic
cg                         cylinder is (x / B)^2 + (y / A)^2 = 1, for all z.

cg    cylinder, hyperbolic
cg                         For the hyperbola or hyperbolic cylinder:
cg                         QC + QXX * x^2 + QYY * y^2 = 0, for all z,
cg                         (QXX > 0, QYY < 0),
cg                         the center is at x = 0, y = 0,
cg                         the length of the transverse semiaxis is
cg                         A = sqrt (-QC / QYY), if QC > 0, and
cg                         A = sqrt (-QC / QXX), if QC < 0.
cg                         the length of the conjugate semiaxis is
cg                         B = sqrt (QC / QXX), if QC > 0, and
cg                         B = sqrt (QC / QYY), if QC < 0.
cg                         the main axis is the z axis,
cg                         the eccentricity is
cg                         sqrt (1 - QYY / QXX) > 1, if QC > 0, or
cg                         sqrt (1 - QXX / QYY) > 1, if QC < 0, or
cg                         sqrt (1 + (B / A)^2),
cg                         the two foci are at
cg                         x = 0, y = (+/-)sqrt (QC / QXX - QC / QYY),
cg                         or     y = (+/-)sqrt (A^2 + B^2), (QC > 0),
cg                         y = 0, x = (+/-)sqrt (QC / QYY - QC / QXX),
cg                         or     x = (+/-)sqrt (A^2 + B^2), (QC < 0),
cg                         the distance from the center to a focus is
cg                         sqrt (abs (QC / QXX - QC / QYY)), or
cg                         sqrt (A^2 + B^2), and
cg                         the length of the latus rectum (width at focus) is
cg                          2 * sqrt (-QC * QYY) / QXX, (QC > 0), or
cg                         -2 * sqrt (-QC * QXX) / QYY, (QC < 0), or
cg                         2 * B^2 / A.
cg                         The standard equation for the hyperbola or hyperbolic
cg                         cylinder is
cg                         (x / B)^2 - (y / A)^2 = 1, (QC < 0), or
cg                         (y / A)^2 - (x / B)^2 = 1, (QC > 0), for all z.

cg    cylinder arrays    Cylinders may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    cylinder, parabolic
cg                         For the parabola or parabolic cylinder:
cg                         QY * y + QXX * x^2 = 0, for all z,
cg                         the center is the vertex at x = 0, z = 0,
cg                         the focus is at x = 0, y = -0.25 * QY / QXX,
cg                         the distance from the vertex to the focus is
cg                         P = 0.25 * abs (QY / QXX), and
cg                         the length of the latus rectum (width at focus) is
cg                         abs (QY / QXX) = 4 * P.
cg
cg                         The standard equation for the parabola or parabolic
cg                         cylinder is
cg                         y = -x^2 / (4 * P), QY > 0, or
cg                         y =  x^2 / (4 * P), QY < 0.

cg    cylinders          A family of circular cylinders may be created with
cg                         command "cylinder", options "concentric", "move",
cg                         "rotate" or "scale" (with restrictions) or as
cg                         follows:
cg
cg                         p pinv (options)
cg
cg                           Create an invariant point, if needed.
cg
cg                         cyl CYL(1) (options)
cg
cg                           Create a base cylinder.
cg
cg                         op opr (options)
cg
cg                           Create a tensor operator, to reflect, rotate,
cg                           invert or scale, if needed.
cg
cg                         v vmove (options)
cg
cg                           Create a vector for use as a translation operator,
cg                           if needed.
cg
cg                         cp CYL(1) CYL(2)
cg
cg                           Copy cylinder CYL(1) to the first cylinder.
cg
cg                         Repeat the following block of commands as many times
cg                         as needed to create the rest of the cylinders.
cg                         See "shortcuts".
cg
cg                         mv cyl CYL(1) vmove
cg
cg                           Move the base cylinder by amount vmove, if needed.
cg
cg                         [invert,reflect,rotate,scale] cyl CYL(1) opr [pinv]
cg
cg                           Invert, reflect, rotate or scale the base cylinder
cg                           with tensor operator opr, if needed.
cg
cg                         cp cyl CYL(1) +
cg
cg                           Copy the base cylinder to the next cylinder.

cc    cylindrical        An option in command "coordinate", to use a 3-D
cc                         coordinate system with the radial, azimuthal and
cc                         axial coordinates (rcyl, theta, z).
cc                         Synonyms:  [cylindrical, cyl].

cg    cylindrical        To create volume elements bounded by surfaces of a
cg                         cylindrical coordinate system, use command "brick".
cg                         To create zones partially or completely bounded by
cg                         concentric cylindrical surfaces, use command
cg                         "cylinder", options "concentric", "scale", and
cg                         command "zone".
cg
cg                         To create arrays of points on surfaces of a
cg                         cylindrical coordinate system, see command
cg                         "point PMESH brick BRNAME mesh block [volume]".
cg
cg                         To find the volume of revolution of a polygonal area
cg                         rotated around an axis, use command "volume".

cc    CYLNAME            The name of a circular cylinder.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.  May not be the same as any other quadric
cc                         surface name QNAME.  Also referred to as CYLNAME1,
cc                         CYLNAME2, ..., CYL(1), CYL(2), ...
cc                         No cylinder name may be "+", "-", "all", "h", "help",
cc                         "list" or "thru", begin with "!" or contain ";".
cc                         Specified with command "cylinder".
cc
cc                         Circular cylinders may appear in commands:
cc                         accelerate, axisym, copy, cylinder, delete,
cc                         distance, invert, move, operator, point, proximal,
cc                         quadric, reflect, rename, rotate, scale, side, track,
cc                         vector.
cc                         Also see commands:  last, list, repack, search, sort.

cc    C____              A synonym for c____ in a comment line.

cc    c____              A synonym for c.

D-------------------------------------------------------------------------------

cc    D                  In command "delimiter", a character to be used as a
cc                         field delimiter, which must be one of the following:
cc                         (blank) (  )  ,  :  <  >  [  \  ]  ^  _  `  {  |  }

cc    D                  In command "solve", the vector of the constants on
cc                         the right-hand side of the three equations.

cc    DANGLE             In command "plane" with option "rotate",
cc                         the angular increment (counterclockwise, with the
cc                         axis pointed at the observer) around the axis vector
cc                         VAXIS, between the normal vectors of successive
cc                         planes.
cc                         May be integer, floating point or a variable.
cc                         Specified in the current angle units.
cc                         See "ANGLE", "angles".

cc    data               An optionin command "variable", to specify more than
cc                         one variable, with values specified in tabular form.
cc                         See "enddata".

cg    data               Each time a geometric or operational object is
cg                         specified or used, its name and some associated data
cg                         is displayed.  For some objects, more data is
cg                         displayed when the object is specified or when it
cg                         is listed by itself.  All data is labeled in a way
cg                         intended to be self-explanatory or to make it
cg                         possible to find a description in geom_base.

cg    data               See "data types", "output data".

cg    data types         Data entered into or displayed by GEOM can be any of
cg                         three types - character, integer or real:
cg
cg                         character - a delimited character string consisting
cg                                     of any of the allowed ASCII keyboard
cg                                     characters.  See "character set".
cg                                     All command names and command arguments
cg                                     shown in lower case are of type
cg                                     character.
cg
cg                         integer   - a delimited character string specifying a
cg                                     positive or negative whole number, in
cg                                     binary, octal, decimal or hexadecimal
cg                                     integer format.  See "integer".
cg                                     Any command argument shown in upper case
cg                                     which allows a numerical value may be of
cg                                     type integer.
cg
cg                         real      - a delimited character string specifying a
cg                                     positive or negative number, in decimal
cg                                     or hexadecimal floating point format.
cg                                     See "floating point".
cg                                     Any command argument shown in upper case
cg                                     which allows a non-integer numerical
cg                                     value may be of type real.

cc    DATAGRP            In command "debug", a data group for which GEOM
cc                         internal parameters and variables will be displayed.
cc                         Also referred to as DATAGRP1, DATAGRP2, ...
cc                         Defined in file geom_internal.  May be any of the
cc                         following:  angles, alias, arithmetic, bin, brick,
cc                         cluster, coordinates, disk, i/o, line, marker, mesh,
cc                         operator, parameter, parse, pdf, plot, point
cc                         polygon, polyhedron, proximal, quadric, symbol,
cc                         synonym, temp, tetrahedron, triangle, variable,
cc                         vector, zone.
cc                         Synonyms:  [alias, al, a], [angles, angle, ang],
cc                         [arithmetic, arith], [brick, br], [cluster, cl],
cc                         [coordinate, coord, cs, system, sys],
cc                         [disk, disc, dk], [environment, env, setup],
cc                         [help, h], [line, l, ln], [marker, mark, m],
cc                         [operator, oper, op], [parameter, param],
cc                         [point, pnt, pt, p], [polygon, polyg, pg],
cc                         [polyhedron, polyh, ph], [proximal, prox],
cc                         [quadric, quad, q], [symbol, sym, symb],
cc                         [synonym, syn], [tetrahedron, tetra, tet],
cc                         [triangle, tri], [variable, var, set, s],
cc                         [vector, vec, v], [zone, zn, z].

cg    date               Dates that may be displayed include the date the code
cg                         was made, and the date of the current run.
cg                         See command "when".
cg                         NOT CURRENTLY AVAILABLE ON THE OPEN YANA CLUSTER.

cg    dbl                See "dbl max", "dbl min".

cg    dbl max            In output from commands "root" and "roots", indicates
cg                         a root of order 2 of a polynomial equation, at a
cg                         maximum of the equation.

cg    dbl min            In output from commands "root" and "roots", indicates
cg                         a root of order 2 of a polynomial equation, at a
cg                         minimum of the equation.

cc    debug              Command to display the values of GEOM internal
cc                         parameters and variables defined in file store.h.
cc                         Most GEOM subroutines contain the command
cc                         "include 'store.h'".  These parameters and variables
cc                         are also listed and defined in file geom_internal,
cc                         in the same directory as the GEOM subroutines.
cc
cc                         Forms include:
cc
ccin                       help debug
ccin                       define [debug]
cc
cc                           Display the command options.
cc
ccin                       debug VARINT1 [VARINT2 ...]
cc
cc                           Display the current values of the GEOM internal
cc                           variables VARINT1 [, VARINT2, ...], which are
cc                           defined in file store.h, in the appropriate format
cc                           for their data type.  Do not use subscripts.
cc                           All values of arrays will be displayed.
cc
cc
ccin                       debug DATAGRP1 [DATAGRP2 ...]
cc
cc                           Display the current values of the GEOM internal
cc                           variables defined in file store.h, for data
cc                           group[s] DATAGRP1 [, DATAGRP2, ...], where DATAGRPn
cc                           may be any of the following:
cc                           angles, alias, arithmetic, big, bin, brick,
cc                           cluster, coordinates, disk, i/o, line, marker,
cc                           mesh, operator, parameter, parse, pdf, plot, point,
cc                           polygon, polyhedron, proximal, quadric, subscript,
cc                           symbol, synonym, temp, tetrahedron, triangle,
cc                           variable, vector, zone.
cc
cc                         Synonyms:  [alias, al, a], [angles, angle, ang],
cc                         [arithmetic, arith], [brick, br], [cluster, cl],
cc                         [coordinate, coord, cs, system, sys],
cc                         [disk, disc, dk], [environment, env, setup],
cc                         [help, h], [line, l, ln], [marker, mark, m],
cc                         [operator, oper, op], [parameter, param],
cc                         [point, pnt, pt, p], [polygon, polyg, pg],
cc                         [polyhedron, polyh, ph], [proximal, prox],
cc                         [quadric, quad, q], [symbol, sym, symb],
cc                         [synonym, syn], [tetrahedron, tetra, tet],
cc                         [triangle, tri], [variable, var, set, s],
cc                         [vector, vec, v], [zone, zn, z].

cc    dec                An option in command "icalc", to set the input mode for
cc                         integers in command "icalc" to decimal.

cg    decay              See "decay constant", "decay time".

cg    decay constant     In an exponential probability bin, the decay constant
cg                         is SIGMA in the relative differential probability
cg                         function exp (-SIGMA * v), where v is the random
cg                         variable, which ranges from VRANL to VRANR,
cg                         as specified in command "bin".  Note the minus
cg                         sign preceding SIGMA.

cg    decay time         To randomly sample decay times for an event with a
cg                         specified half-life T:
cg
cg                         bin BINAME 1 0 VRANR exp SIGMA
cg
cg                           Create a probability bin BINAME with an
cg                           exponentially decaying probability with decay
cg                           constant SIGMA = ln(2)/T, and VRANR = 100*T (which
cg                           has negligible probability).
cg
cg                         pdf PDFNAME = BINAME
cg
cg                           Create a probability distribution function (pdf)
cg                           PDFNAME, consisting only of the probability bin
cg                           BINAME.
cg
cg                         sample PDFNAME [NSAMP|1]
cg
cg                           Sample from pdf PDFNAME NSAMP times.

cg    decimal            The decimal form of an integer M, specified in the
cg                         current integer mode (see command "icalc"),
cg                         may be displayed with command:
cg
cg                         icalc M
cg
cg                           Display M in decimal, hexadecimal, octal and
cg                           (optionally) in binary mode.

cg    decrement          See "decrement names".

cg    decrement names    See "increment names", command "increment".

cg    decryption         Decryption is the process of making encrypted data or
cg                         text readable, using special knowlege intended to
cg                         be available only to certain recipients.
cg                         See command "big".

cc    def                A synonym for define.

cg    default            See "default arguments", "default values".

cc    default arguments
cc                       In commands, optional arguments are shown as follows:
cc
cc                         [argument]
cc
cc                           The literal argument is optional, and there is no
cc                           default value, unless specified in the description
cc                           of the command.
cc
cc                         [ARGUMENT|VALUE]
cc
cc                           The value of ARGUMENT is optional.  If omitted, the
cc                           argument defaults to VALUE.
cc
cc                         [OBJNAME,all]
cc
cc                           The object name OBJNAME or the literal "all" must
cc                           be specified.  No default is allowed.
cc
cc                         [arg1,arg2,arg3|arg1]
cc
cc                           The literal arguments arg1, arg2 or arg3 are
cc                           optional.  If omitted, the argument defaults to
cc                           arg1.
cc
cc                         arg1 [arg2|<]
cc
cc                           If arg2 is not specified, it defaults to arg1.
cc
cc                         Other defaults may be defined in the descriptions of
cc                         the commands.

cg    default values     A number of objects are assigned default values at the
cg                         beginning of each GEOM run.
cg
cg                         The input medium is the user's terminal
cg                           (do NOT use "<" or ">" on the GEOM execution line).
cg                           Change with command "input" or "return".
cg                           See "redirect input".
cg                         The output medium (in addition to the standard
cg                           output, normally to the user's terminal) is
cg                           file geom_hsp.  Change with command "output".
cg                           See "redirect output".
cg                         The field delimiter is a blank character.
cg                           Change with command "delimiter".
cg                         The numerical tolerance limit TOL = 10^(-11).
cg                           Change with command "tol".
cg                         The coordinate system is Cartesian (x, y, z).
cg                           Change with command "coordinate".
cg                         The unit for input and display of angles is degrees.
cg                           Change with command "angles".
cg                         The mode for integer input in command "icalc" is
cg                           decimal.  Change with command "icalc".
cg                         The mode for integer display in command "icalc" is
cg                           octal, decimal and hexadecimal.
cg                           Turn an added binary display on and off with
cg                           commands "icalc binary" and "icalc nobin".
cg
cg                         symbol pi      = 3.141592653589793
cg                         symbol deg/rad = 57.29577951308232
cg                         symbol rad/deg = 0.01745329251994330
cg                         symbol ebase   = 2.718281828459045
cg                         symbol rgold   = 1.618033988749894
cg                         symbol euler   = 0.577215664901533861
cg
cg                         Change or reinstate any of the five preceding
cg                         symbols with command "symbol".

cg    define             See "create", commands "define", "alias", "marker".

cc    define             Command to display an entry in file geom_base, which is
cc                         assumed to be in one of the locations:
cc                         geom_base                          (local)
cc                         ~edwards/work/geom/doc/geom_base   (YANA Cluster)
cc                         A list of all such entries is in file geom_define.
cc
cc                         Note:  no replacement of symbols will be done
cc                         in this command.
cc
cc                         See "character set" for the ASCII characters which
cc                         may be used in command "define".
cc
cc                         Forms include:
cc
ccin                       help define
ccin                       define [help]
cc
cc                           Display the command options.
cc
ccin                       define 'ENTRY'
cc
cc                           Display entry ENTRY of geom_base.  Case is ignored.
cc                           The first 19 characters of ENTRY must match columns
cc                           7-25 of the first line of entries in this file,
cc                           ignoring any trailing blanks.
cc                           If ENTRY is bracketed by single quotes, repeat any
cc                           internal single quote.  If ENTRY is bracketed by
cc                           double quotes, repeat any internal double quote.
cc
cc                         define 'X'
cc
cc                           Display the first line of all entries in geom.base
cc                           that begin with the letter "X".  Case is ignored.
cc
cc                         Synonyms:  [", ' (in pairs)], [define, def],
cc                         [help, h].
cc
cc                         See "help", "definition", "geom_define".

cg    definition         You can store and display your own comment, description
cg                         or definition of any object or word you desire,
cg                         OBJNAME, with command "alias", as follows:
cg
cg                         alias OBJNAME = "# DEFINITION"
cg
cg                           Assign alias OBJNAME to the comment line
cg                           "# DEFINITION", where DEFINITION is any character
cg                           string that will fit on the input line, describing
cg                           the object named OBJNAME.  OBJNAME should not be
cg                           the same as the first word of any command that you
cg                           might want to use.
cg
cg                         OBJNAME
cg
cg                           Display the character string for which OBJNAME is
cg                           the alias.
cg
cg                         Example:
cg
cg                         You type:
cg                         alias vnorm = "# The vector normal to plane plx."
cg
cg                         Any time later, you type:
cg                         vnorm
cg                         The code then displays:
cg                         # The vector normal to plane plx.
cg
cg                         You can store and display your own comment,
cg                         description or definition, to go with any single
cg                         character AMNAME you desire, with command "marker",
cg                         as follows:
cg
cg                         marker AMNAME = "TEXT"
cg
cg                           Assign to the 1-character marker AMNAME the text
cg                           "TEXT", where TEXT is any character string that
cg                           will fit on the input line.
cg
cg                         marker AMNAME
cg
cg                           Display the TEXT assigned to marker AMNAME.
cg
cg                         Example:
cg
cg                         You type:
cg                         marker a = "Note:  coordinates are in millimeters."
cg
cg                         Any time later, you type:
cg                         marker a
cg                         The code then displays:
cg                         mark a  Note:  coordinates are in millimeters.

cc    deg                A synonym for degrees in command "angles".

cn    deg.mac            A macro file for converting from degrees, minutes and
cn                         seconds to decimal degrees.  Type "in deg.mac ?" for
cn                         info.  In ~/work/geom/test .  See "dms.mac".

cc    deg/rad            An option in command "symbol", argument SYMBNAME,
cc                         to create symbol deg/rad with value
cc                         57.29577951308232087679815481410517033240547246658
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cg    deg/rad            Symbol, degrees per radian.  Generated as a
cg                         symbol at the beginning of the run.
cg                         deg/rad = 180.0 / pi.
cg                         deg/rad = 57.29577951308232 degrees per radian.
cg                         rad/deg = 0.01745329251994330 radians per degree.
cg                         pi      = 3.141592653589793... = acos (-1).
cg
cg                         symbol SYMBNAME deg/rad
cg
cg                           Set symbol SYMBNAME to
cg                           180 / pi = 57.29577951308232.
cg
cg                         symbol SYMBNAME rad/deg
cg
cg                           Set symbol SYMBNAME to
cg                           pi / 180 = 0.01745329251994330.

cg    degenerate         A quadric surface may be degenerate:
cg
cg                           F(x,y,z) =    QC          +
cg                           QX  * x     + QY  * y     + QZ  * z     +
cg                           QXY * x * y + QYZ * y * z + QZX * z * x +
cg                           QXX * x^2  + QYY * y^2  + QZZ * z^2  = 0
cg
cg                         Degenerate quadric surfaces (in standard form,
cg                         all coefficients positive):
cg
cg                         Imaginary intersecting planes:
cg                           (a straight line)              x^2 + QYY*y^2 = 0
cg                         Imaginary elliptic cone:
cg                           (a point)            x^2 + QYY*y^2 + QZZ*z^2 = 0
cg                         Imaginary circular cone:
cg                           (a point)                x^2 + y^2 + QZZ*z^2 = 0
cg
cg                         See "quadric surfaces".

cc    degree             A synonym for degrees in command "angles".

cc    DEGREES            In command "angles", the integer number of degrees of
cc                         an angle specified in degree, minutes and second
cc                         units.
cc                         May be integer, floating point or a variable.
cc                         The absolute value of DEGREES + MINUTES / 60
cc                         + SECONDS / 3600 must not exceed the largest machine
cc                         integer.

cc    degrees            An option in command "angles", to specify that angles
cc                         are to be input and displayed in degrees.

cg    degrees            A unit of measurement of angles.  A full circle has
cg                         360 degrees or 2 * pi radians.
cg                         One radian = 180/pi (~57.29577951308232) degrees.
cg                         One degree = pi/180 (~0.01745329251994330) radians.

cc    del                A synonym for delete.

cc    delete             An option in command "plot", to delete points from the
cc                         list of plot points.  This does not delete the points
cc                         from the list of geometric points.

cg    delete             See "delete objects", command "delete".

cc    delete             Command to delete aliases, bricks, probability bins,
cc                         big integers, clusters, annular disks, lines,
cc                         markers, tensor operators, planes, points,
cc                         regular polygons, quadric surfaces (including general
cc                         and axisymmetric, circular cones, circular cylinders,
cc                         ellipsoids, hyperbolic paraboloids and spheres),
cc                         regular polyhedrons, probability distribution
cc                         function (pdfs), tetrahedrons, triangles, vectors,
cc                         variables and zones.
cc
cc                         Command "delete" relates to objects:  alias, axisym,
cc                         big, bin, brick, cluster, cone, cylinder, disk,
cc                         ellipsoid, hyperb, line, marker, operator, pdf,
cc                         plane, point, polygon, polyhedron, quadric, sphere,
cc                         symbol, tetrahedron, triangle, variable, vector,
cc                         zone.
cc
cc                         Option "all" must be verified, if input is from the
cc                         user's terminal (do NOT use "<" on the GEOM execution
cc                         line).
cc
cc                         Fix errors with command "undo", "redo" or "indo",
cc                         or by cutting and pasting from the display.
cc
cc                         Note:  no replacement of symbols will be done
cc                         in the commands "delete symbol ...", "delete alias".
cc
cc                         Markers with dependencies (plot title, plot axis
cc                         labels or plot point titles) will not be deleted.
cc                         You must respecify the dependent object first.
cc
cc                         Points with dependencies (line, triangle, regular
cc                         polygon, annular disk, regular polyhedron or a
cc                         tetrahedron) will not be deleted.  You must delete
cc                         or respecify the dependent object first.  A warning
cc                         will be displayed if a deleted point is in a cluster,
cc                         unless it is the cluster that is being deleted.
cc
cc                         A warning will be displayed if a deleted bin is in a
cc                         probability distribution function (pdf), unless it is
cc                         the pdf that is being deleted.
cc
cc                         Do not use "+" or "-" for object names in these
cc                         commands.
cc
cc                         Forms include:
cc
ccin                       help delete
ccin                       delete [help]
cc
cc                           Display the command options.
cc
ccin                       delete OBJTYPE all
cc
cc                           Delete all objects of type OBJTYPE (alias, axisym,
cc                           big, bin, brick, cluster, cone, cylinder, disk,
cc                           ellipsoid, hyperb, line, marker, operator, pdf,
cc                           plane, point, polygon, polyhedron, quadric, sphere,
cc                           symbol, tetrahedron, triangle, variable, vector,
cc                           zone).
cc                           Requires verification by the user, if input is from
cc                           the user's terminal (do NOT use "<" on the GEOM
cc                           execution line).
cc
ccin                       delete OBJTYPE OBJNAME all
cc
cc                           Delete all objects of type OBJTYPE (see above) with
cc                           the names OBJNAME(NSUB1,NSUB2,...), where NSUBi is
cc                           any subscript or null.
cc
ccin                       delete OBJTYPE OBJNAME
cc
cc                           Delete the object of type OBJTYPE (see above) with
cc                           the name OBJNAME.
cc
ccin                       delete OBJTYPE OBJNAME1 thru OBJNAME2
cc
cc                           Delete any objects of type OBJTYPE (see above) with
cc                           names from OBJNAME1 through OBJNAME2.  WARNING:
cc                           the name comparison is with left-adjusted ASCII
cc                           names, except that subscripts are compared as
cc                           integers.
cc
cc                           NOTE!  No object may have the name "thru".
cc
ccin                       delete cluster CLNAME
cc
cc                           Delete cluster CLNAME.  The points in cluster
cc                           CLNAME are not deleted.
cc
ccin                       delete point cluster CLNAME
cc
cc                           Delete the points in cluster CLNAME.  Delete
cc                           cluster CLNAME if all of its points are deleted.
cc
ccin                       delete bin pdf PDFNAME
cc
cc                           Delete the probability bins in pdf PDFNAME.
cc                           Delete pdf PDFNAME if all of its bins are deleted.
cc
cc                         Synonyms:  [alias, al, a,], [brick, br],
cc                         [cluster, cl], [copy, cp, dup, duplicate],
cc                         [cylinder, cyl], [delete, del, remove, rm],
cc                         [disk, disc, dk],
cc                         [ellipsoid, ell, el, ellipse], [help, h],
cc                         [hyperb, saddle], [line, l, ln], [marker, mark, m],
cc                         [operator, oper, op], [plane, pl],
cc                         [point, pnt, pt, p], [polygon, polyg, pg],
cc                         [polyhedron, polyh, ph], [quadric, quad, q],
cc                         [sphere, sph], [symbol, sym, symb],
cc                         [tetrahedron, tetra, tet], [triangle, tri],
cc                         [variable, var, set, s], [vector, vect, v],
cc                         [zone, zn, z].

cc    delete             An option in command "mesh", to delete one or more
cc                         layers of the mesh.
cc                         Synonyms:  [delete, del, remove, rm].

cg    delete objects     See command "delete".

cc    delim              A synonym for delimiter.

cc    delimit            A synonym for delimiter.

cg    delimited          A delimited argument or character string is delimited
cg                         on the left by the beginning of the line or the
cg                         field delimiter, and on the right by the field
cg                         delimiter or the end of the line (up to 80
cg                         characters).

cg    delimiter          The field delimiter (FD) is the single character used
cg                         to delimit fields (arguments) in the input lines.
cg
cg                         The FD is initially a blank character, but may be
cg                         changed with command "delimiter", to one of the
cg                         following characters:
cg                         ,  :  <  >  [  \  ]  ^  _  `  {  |  }
cg                         Do not use a character as a delimiter in a command
cg                         that requires the same character as input.
cg
cg                         All commands shown in this file, and in the help
cg                         messages displayed by GEOM, are shown with a blank
cg                         character as the FD.  If the FD is not a blank
cg                         character, replace all blank characters or sequential
cg                         strings of blank characters shown in the examples
cg                         with the actual FD (except in the quoted strings of
cg                         commands "alias", "marker" and "define").
cg
cg                         If the FD is a blank character, any number of
cg                         adjacent blank characters may be used instead of a
cg                         single blank character, and blank characters used
cg                         before the first non-blank character and after the
cg                         last non-blank character will be ignored.
cg
cg                         If the FD is a blank character, do not include any
cg                         blank characters in any delimited argument or
cg                         character string, except in the quoted strings of
cg                         commands "alias", 'define" and "marker", or in the
cg                         variable tag specified in command "variable", option
cg                         "tag".
cg
cg                         If the FD is not a blank character, do not use it to
cg                         begin an input line, or an extra null field will be
cg                         produced.  Trailing null fields at the end of an
cg                         input line will be ignored, but any other null field
cg                         is an error.
cg
cg                         Sign prefixes on variable names do not use a field
cg                         delimiter.  See "sign".

cc    delimiter          Command to display or specify the field delimiter used
cc                         to delimit input line arguments.  The field delimiter
cc                         must be a blank character or one of the following
cc                         characters:
cc                         ,  :  <  >  [  \  ]  ^  _  `  {  |  }
cc
cc                         Forms include:
cc
ccin                       help delimiter
ccin                       delimiter help
cc
cc                           Display the command options, when the field
cc                           delimiter is a blank character.
cc
ccin                       helpCdelimiter
ccin                       delimiterChelp
cc
cc                           Display the command options, when the field
cc                           delimiter is any allowed character "C".
cc
ccin                       delimiter
cc
cc                           Change the field delimiter to a blank character,
cc                           when it is initially any character.
cc
ccin                       delimiter D
cc
cc                           Change the field delimiter to any allowed character
cc                           "D", when it is initially a blank character.
cc
ccin                       delimiterCD
cc
cc                           Change the field delimiter to any allowed character
cc                           "D", when it is initially the character "C".
cc
cc                         Synonyms:  [delimiter, delim, delimit], [help, h].

cg    dependent          Some geometric objects are dependent on points:
cg                         lines, triangles, regular polygons, annular disks,
cg                         regular polyhedrons, and tetrahedrons.  A change in
cg                         the coordinates of a point will change the geometry
cg                         of the dependent object.  A change in the name of a
cg                         point will have no effect on the dependent object,
cg                         except to change the name of the point in its
cg                         specification.  A point with dependent objects may
cg                         not be deleted.

cg    derivatives        When using command "roots" to find the real and/or
cg                         complex roots of a polynomial equation in z:
cg                         P(z) = a0 + a1*z + a2*z^2 + a3*z^3 + a4*z^4 = 0,
cg                         the real values of z and P(z) are displayed for which
cg                         each of the derivatives of P(z) are zero, up to the
cg                         highest derivative which is still a function of z.
cg                         These include the maximum and minimum points, where
cg                         the first derivative is zero:
cg                           P'(z) = a1 + 2*a2*z + 3*a3*z^2 + 4*a4*z^3 = 0,
cg                         the inflection points, where the second derivative
cg                         is zero:
cg                           P''(z) = 2*a2 + 6*a3*z + 12*a4*z^2 = 0,
cg                         and the point where the third derivative is zero:
cg                           P'''(z) = 6*a3 + 24*a4*z = 0.
cg
cg                         To search for real roots, extrema and inflection
cg                         points of a polynomial equation using Newtonian
cg                         iteration, use command "root".
cg
cg                         See "spatial derivatives".

cg    destroy            To destroy an object, use command "delete".

cg    det                An invariant of a quadric surface, for translations and
cg                         rotations.  See "invariants of quadric".

cg    determinant        The determinant of a tensor operator is displayed
cg                         whenever the tensor operator is displayed.
cg                         The determinant for a rotation operator is 1.
cg                         The determinant for an inversion operator is -1.
cg                         The determinant for a reflection operator is -1.
cg                         The determinant for a scaling operator is the
cg                         first, second or third power of the scale factor,
cg                         for linear, radial or 3-D (options "scale" or
cg                         uniform") scaling.

cg    determinant        To find the value of a 3 by 3 determinant:
cg
cg                         variable VARNAME triple VNAME1 VNAME2 VNAME3
cg
cg                           Create variable VARNAME with the value of the 3 by
cg                           3 determinant for which the three row or column
cg                           vectors are the vectors VNAME1, VNAME2 and VNAME3.

cc    DEV                In command "bin", the standard deviation.

cg    deviation          See "expected value".

cg    diatonic           See "diatonic scale".

cg    diatonic scale     In music, a diatonic scale in a major key consists
cg                         of eight tones, with intervals from the initial (key)
cg                         tone of two whole tones, a semitone, three whole
cg                         tones, and a semitone, ending on the octave of the
cg                         first tone.  For example, on a piano, the diatonic
cg                         scale for C major is entirely on the white keys,
cg                         consisting of C-D-E-F-G-A-B-C.  The diatonic scale
cg                         for a minor key, e.g. A minor, A-B-C-D-E-F-G-A, has
cg                         semitones for the second and fifth intervals.
cg
cg                         See "equally-tempered".

cn    dice.mac           A macro file for rolling two dice, to demonstrate the
cn                         use of bins and pdfs.  In ~/work/geom/test .

cg    digits             See "significant figures".

cg    dihedral           See "angle, dihedral".

cg    direction          Reverse the direction of a vector V1 with command
cg                         "vector V1 rel -1"

cg    direction          See "direction cosine", "direction vector".

cg    direction cosine   A direction cosine is a component of a unit direction
cg                         vector in the direction of one of the axes of the
cg                         coordinate system.

cg    direction vector   The direction vector of a moving object is the vector
cg                         in the direction of motion.  A unit vector parallel
cg                         to the direction vector has components in the
cg                         directions of the axes of the coordinate system.
cg                         These components are called direction cosines.
cg                         In linear motion, the direction cosines are constant
cs                         in a Cartesian (xyz) coordinate system, but change
cg                         continuously in a cylindrical or spherical coordinate
cg                         system.  See "coordinate conversion",
cg                         "vector conversion".

cg    directories        See "geom_summ", "geom_files".

cg    disallowed         See "disallowed names".

cg    disallowed names   Some names are disallowed for all or some object names,
cg                         because they make interpretation of a command
cg                         ambiguous.
cg                         No name may be "+", "-", "all", "h", "help",
cg                         "list" or "thru", begin with "!" or contain ";".
cg                         A point name must not be "cluster" or a synonym.
cg                         A cluster name must not be "point" or a synonym.
cg                         A quadric surface name must not be "types".

cc    disc               A synonym for disk.

cg    discrete           See "discrete event", "discrete object",
cg                         "discrete value".

cg    discrete event     A type of probability bin, which assigns a relative
cg                         total probability to a discrete event or object.
cg                         The discrete object may be any of the objects used
cg                         in GEOM, including a probability distribution
cg                         function (pdf).
cg                         See "discrete value", "uniform", "linear",
cg                         "power-law", "exponential".
cg                         See commands "bin", "pdf", "sample".

cg    discrete object    A type of probability bin, which assigns a relative
cg                         total probability to a discrete event or object.
cg                         The discrete object may be any of the objects used
cg                         in GEOM, including a probability distribution
cg                         function (pdf).
cg                         See "discrete value", "uniform", "linear",
cg                         "power-law", "exponential".
cg                         See commands "bin", "pdf", "sample".

cg    discrete value     A type of probability bin, which assigns a relative
cg                         total probability to a discrete value of a random
cg                         variable.
cg                         See "discrete event", "uniform", "linear",
cg                         "power-law", "exponential".
cg                         See commands "bin", "pdf", "sample".

cg    disk               An annular disk is a surface in 3-D space, lying in a
cg                         plane, and bounded by two circles having a common
cg                         center in the same plane.  For a simple disk, the
cg                         radius of the inner circle is zero.  The disk is
cg                         dependent on the point used to specify the center of
cg                         the circles, which may be changed.  An annular disk
cg                         may be used to create a plane, with command "plane".
cg                         The maximum number of disks is now 1000.
cg
cg                         The following commands relate to annular
cg                         disks:
cg                         copy, debug, delete, disk, help, last, list, plane,
cg                         point, rename, repack, search, sort, symbol, synonym,
cg                         tables.

cc    disk               An option in command "debug", to display GEOM internal
cc                         parameters and variables for annular disks:
cc                         ndiskm, ndisks, adisk, radkin, radkout, acendk,
cc                         xnormdk, ynormdk, znormdk, adisks, ladisks.
cc                         Synonyms:  [disk, disc, dk].

cc    disk               Command to display or create circular or annular
cc                         disks.  A disk may be replaced.  The display for a
cc                         disk includes the central point, the normal vector,
cc                         the inner and outer radii, the inner and outer
cc                         circumferences, and the areas of the central hole and
cc                         the annulus.
cc                         See command "circle" to find the circle through
cc                         three specified points.
cc                         See "do loop use".
cc
cc                         Command "disk" relates to objects:  disk, point,
cc                         variable, vector.
cc
cc                         Forms include:
cc
ccin                       help disk
ccin                       disk [help]
cc
cc                           Display the command options.
cc
ccin                       disk [all,list]
cc
cc                           Display all annular disks.
cc
ccin                       disk list DKNAME1 DKNAME2 DKNAME3 ...
cc
cc                           Display annular disks DKNAME1, DKNAME2, DKNAME3,
cc                           ..., with or without subscripts.
cc
ccin                       disk DKNAME
cc
cc                           Display annular disk DKNAME.
cc
ccin                       disk DKNAME fit PCEN VNORM RADKIN RADKOUT
cc
cc                           Create annular disk DKNAME, centered at point PCEN,
cc                           with normal vector VNORM, with inner radius RADKIN,
cc                           outer radius RADKOUT.  Any later change in the
cc                           coordinates or name of point PCEN will change the
cc                           definition of annular disk DKNAME.  Use "+" or "-"
cc                           instead of DKNAME to form the name by incrementing
cc                           or decrementing the base name for annular disks.
cc                           See command "last", "increment names".
cc
cc                         Synonyms:  [disk, disc, dk], [help, h].

cc    disk               An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more annular disks.
cc                         Synonyms:  [disk, disc, dk].

cg    disk arrays        Disks may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    displace           See "move".

cg    displacement       A displacement is the difference between the final and
cg                         initial position of an object.

cc    display            A synonym for list in command "list".

cg    display            The code displays data on the user's terminal
cg                         screen and in the output file (geom_hsp, or the
cg                         file(s) specified by the user with command "output"),
cg                         or specified on the GEOM execution line.
cg                         See "display objects", "redirect output", command
cg                         "list".

cg    display            See "display objects", command "list".

cg    display objects    The commands for displaying objects have the general
cg                         forms:
cg
cg                         OBJTYPE [all,list]
cg
cg                           Display all objects of type OBJTYPE.
cg                           Use the longest display.
cg
cg                         OBJTYPE OBJNAME
cg
cg                           Display object type OBJTYPE with name OBJNAME.
cg                           Use the longest display.
cg
cg                         OBJTYPE list OBJNAME1 OBJNAME2 OBJNAME3 ...
cg
cg                           Display objects of type OBJTYPE with names OBJNAME,
cg                           OBJNAME2, OBJNAME3, ..., with or without
cg                           subscripts.  Use the shortest display, unless there
cg                           is only one such object.
cg
cg                         list [OBJTYPE,all]
cg
cg                           Display only the names of all objects of [type
cg                           OBJTYPE, or all types].
cg                           Synonyms:  [list, display, print].
cg
cg                         last [OBJTYPE,all]
cg
cg                           Display only the names of the last object specified
cg                           of [type OBJTYPE, or all types].
cg
cg                         sort [OBJTYPE,all] [+,-,?|+]
cg
cg                           Display only the names of all objects of [type
cg                           OBJTYPE, or all types], after sorting the objects
cg                           into [increasing, decreasing, random] ASCII order,
cg                           with increasing order the default option.
cg
cg                         search OBJNAME1 OBJNAME2 OBJNAME3 ...
cg
cg                           Display all objects with object names the same as
cg                           OBJNAME1, OBJNAME2, OBJNAME3, ...  or if any of the
cg                           latter are a single character, with object names
cg
cg                           Display all marker texts, symbol and alias
cg                           replacements the same as OBJNAME1, OBJNAME2,
cg                           OBJNAME3, ...  or if any of the latter are a single
cg                           character, beginning with that character.
cg
cg                           Display all variables with numerical values equal
cg                           to OBJNAME1, OBJNAME2, OBJNAME3, ...
cg
cg                         search all
cg
cg                           Display all objects with object names.
cg
cg                         See "OBJTYPE", "OBJNAME", "create objects".

cg    dist               A synonym for distance in command "distance".

cg    dist               See "dist to collision".

cg    distal             The opposite of proximal.  The most distant.
cg                         See "distal point".

cg    distal point       For an external point P1, the distal point P2 (farthest
cg                         from point P1) on a quadric surface F(x,y,z) = 0,
cg                         with normal vector N = grad F = (NX, NY, NX),
cg                         satisfies the same equations as a proximal point:
cg
cg                         (X2 - X1) / NX =
cg                         (Y2 - Y1) / NY =
cg                         (Z2 - Z1) / NZ =
cg                         (P2 - P1) dot N / N^2 = distance / |N|.
cg
cg                         In general, this leads to a sixth order equation,
cg                         requiring iterative numerical methods of solution.
cg                         See commands "distance", "proximal", "side".

cg    distance           The straight-line distance between two points in 3-D
cg                         space, or the perpendicular distance from a point to
cg                         a line, plane or surface, or the perpendicular
cg                         distance between two parallel lines or planes.
cg                         See commands "accelerate", "distance", "proximal",
cg                         "side", "track".
cg
cg                         To find the distance between two points, use any of
cg                         the methods:
cg
cg                         distance point PNAME1 point PNAME2
cg
cg                           Display the distance between points PNAME1 and
cg                           PNAME2.
cg
cg                         line ALNAME = PNAME1 PNAME2
cg
cg                           Create line ALNAME1 from point PNAME1 to point
cg                           PNAME2, and display its direction and length.
cg
cg                         vector VNAME = PNAME1 PNAME2
cg
cg                           Create vector VNAME, directed from point PNAME1
cg                           toward point PNAME2, with length equal to the
cg                           distance from PNAME1 to PNAME2, and bound at
cg                           PNAME1, and display its direction and length.
cg
cg                         To find the distance between the two parallel planes
cg                         of a quadric surface of that form:
cg
cg                         quadric QNAME
cg
cg                           Display the coefficients of quadric surface QNAME
cg                           (before and after a principal axis transformation),
cg                           the intercepts on the axis, and the distance
cg                           between the parallel planes.
cg
cg                         To find the minimum distance between a point and a
cg                         quadric surface, use command "distance", "proximal"
cg                         or "side".
cg
cg                         To find the distances from a point to the six
cg                         surfaces bounding a brick, and to find if the point
cg                         is in the brick, use command "distance".
cg                         Also see command "where".
cg
cg                         To find the distances from a point to the four planes
cg                         bounding a tetrahedron, and to find if the point is
cg                         in the tetrahedron, use command "distance".
cg                         Also see command "where".
cg
cg                         To find the distances from a point to the quadric
cg                         surfaces bounding a zone, and to see if the point is
cg                         in the zone, use command "distance".
cg                         Also see command "where".
cg
cg                         To find the approximate distance or the intersection
cg                         between a line and a quadric surface, use command
cg                         "track".
cg
cg                         To find the distance between two circles in a major
cg                         plane, use command "intcirc".
cg
cg                         To find the distance and proximal points and/or
cg                         intersection of any combination of planes, spheres
cg                         and circular cylinders, use command "distance".
cg
cg                         To find the distance from a plane to any quadric
cg                         surface, use command "distance".
cg
cg                         To find the distance between a sphere and a quadric
cg                         surface, first find the distance from the point at
cg                         the center of the sphere to the quadric surface,
cg                         with command "distance", "proximal" or "side", then
cg                         subtract the radius of the sphere.  A negative result
cg                         indicates overlap.
cg
cg                         To find the distance between any two quadric
cg                         surfaces, use command "distance".
cg
cg                         To find the distance between a point and a toroid
cg                         generated by a quadric curve, see "toroid".

cc    distance           Command to find the distance between two geometric
cc                         objects, which may be points, lines, triangles or
cc                         planes, a point or plane and a quadric surface, or
cc                         a point and a brick, tetrahedron or zone, any
cc                         combination of spheres and circular cylinders, or
cc                         any two quadric surfaces.
cc                         When possible, find any symmetries, points of
cc                         intersection or tangency or curves of intersection.
cc                         In the case of clusters of points, also find the
cc                         points with the minimum and maximum distances.
cc                         In the case of two quadric surfaces, specify planes
cc                         before spheres before other quadric surfaces.
cc                         Also, see "toroid".
cc
cc                         Command "distance" relates to objects:  axisym,
cc                         brick, cluster, cone, cylinder, ellipsoid, hyperb,
cc                         line, plane, point, quadric, sphere, symbol,
cc                         tetrahedron, triangle, zone.
cc
cc                         The minimum distance found may be stored in variable
cc                         VARNAME with command:
cc                         variable VARNAME distance
cc
cc                         The proximal point found on the second object may be
cc                         stored in point PNAME with command
cc                         "point PNAME proximal".
cc
cc                         The vector between the proximal points may be stored
cc                         in vector VNAME with command
cc                         "vector VNAME proximal".
cc
cc                         Also see commands "accelerate", "intcirc",
cc                         "proximal", "side", "track", "variable".
cc
cc                         Forms include:
cc
ccin                       help distance
ccin                       distance [help]
cc
cc                           Display the command options.
cc
ccin                       distance point PNAME1 point PNAME2
cc
cc                           Find the distance from point PNAME1 to point
cc                           PNAME2.  Another way is to create a line from point
cc                           PNAME1 to point PNAME2.
cc
ccin                       distance point PNAME cluster CLNAME
cc
cc                           Find the distances from point PNAME1 to the points
cc                           in cluster CLNAME.
cc
ccin                       distance cluster CLNAME1 cluster CLNAME2
cc
cc                           Find the distances from the points in cluster
cc                           CLNAME1 to the points in cluster CLNAME2 (which
cc                           may be CLNAME1).
cc
ccin                       distance point PNAME line ALNAME
cc
cc                           Find the distance from point PNAME to line ALNAME,
cc                           and the nearest point on the line.
cc
ccin                       distance cluster CLNAME line ALNAME
cc
cc                           Find the distances from the points in cluster
cc                           CLNAME to line ALNAME, and the nearest points on
cc                           the line, for each point.
cc
ccin                       distance point PNAME triangle TRNAME
cc
cc                           Find the distance from point PNAME to triangle
cc                           TRNAME, and the nearest point on the triangle.
cc
ccin                       distance cluster CLNAME triangle TRNAME
cc
cc                           Find the distances from the points in cluster PNAME
cc                           to triangle TRNAME, and the nearest point on the
cc                           triangle for each point.
cc
ccin                       distance point PNAME QTYPE QNAME
cc
cc                           Find the distance from point PNAME to the quadric
cc                           surface of type QTYPE (plane, sphere, cylinder,
cc                           cone, hyperb, ellipsoid, axisym, quadric) with name
cc                           QNAME, and the nearest point on the quadric
cc                           surface.  Check with commands "proximal", "side".
cc                           See "toroid" for a method of finding the distance
cc                           from a point to a toroid generated by a quadric
cc                           curve.
cc
ccin                       distance cluster CLNAME QTYPE QNAME
cc
cc                           Find the distances from the points in cluster
cc                           CLNAME to the quadric surface of type QTYPE (plane,
cc                           sphere, cylinder, cone, hyperb, ellipsoid, axisym,
cc                           quadric) with name QNAME, and the nearest point on
cc                           the quadric surface, if possible.
cc                           See the preceding command form.
cc
ccin                       distance point PNAME brick BRNAME
cc
cc                           Find the distances from point PNAME to each of the
cc                           six surfaces bounding brick TETNAME, and find if
cc                           the point is inside the brick.  Also see command
cc                           "where".
cc
ccin                       distance point PNAME tetrahedron TETNAME
cc
cc                           Find the distances from point PNAME to each of the
cc                           four planes bounding tetrahedron TETNAME, and find
cc                           if the point is inside the tetrahedron.  Also see
cc                           command "where".
cc
ccin                       distance point PNAME zone ZNAME
cc
cc                           Find the distances from point PNAME to each of the
cc                           quadric surfaces bounding zone ZNAME, and find if
cc                           the point is inside the zone.  See the preceding
cc                           command forms.  Also see command "where".
cc
ccin                       distance line ALNAME1 line ALNAME2
cc
cc                           Find the distance from line ALNAME1 to line
cc                           ALNAME2, and the nearest points on each line.
cc
ccin                       distance line ALNAME triangle TRNAME
cc
cc                           Find the intersection of line ALNAME and triangle
cc                           TRNAME or find the distance, if they are parallel.
cc
ccin                       distance line ALNAME plane PLNAME
cc
cc                           Find the intersection of line ALNAME and plane
cc                           PLNAME or find the distance, if they are parallel.
cc
ccin                       distance triangle TRNAME1 triangle TRNAME2
cc
cc                           Find the line of intersection of the two triangles
cc                           TRNAME1 and TRNAME2 or find the distance, if they
cc                           are parallel.
cc
ccin                       distance triangle TRNAME plane PLNAME
cc
cc                           Find the line of intersection of triangle TRNAME
cc                           and plane PLNAME or find the distance, if they are
cc                           parallel.
cc
ccin                       distance plane PLNAME1 plane PLNAME2
cc
cc                           Find the distance and/or intersection of the two
cc                           planes PLNAME1 and PLNAME2.  For more information,
cc                           repeat with the second "plane" replaced by
cc                           "quadric".
cc
ccin                       distance plane PLNAME sphere SPHNAME
cc
cc                           Find the distance and/or intersection of plane
cc                           PLNAME and sphere SPHNAME.  See "separation".
cc
ccin                       distance plane PLNAME cylinder CYLNAME
cc
cc                           Find the distance and/or intersection of plane
cc                           PLNAME and circular cylinder CYLNAME.  See
cc                           "separation".  For more information, repeat with
cc                           "cylinder" replaced by "quadric".
cc
ccin                       distance plane PLNAME QTYPE QNAME
cc
cc                           Find the distance and/or intersection of plane
cc                           PLNAME and quadric type QTYPE (axisym, cone,
cc                           cylinder, ellipsoid, hyperb, plane, quadric or
cc                           sphere) with name QNAME.
cc                           See the preceding three commands.
cc                           Also see "quadric", "separation", "projection".
cc
ccin                       distance sphere SPHNAME1 sphere SPHNAME2
cc
cc                           Find the distance and/or intersection of the two
cc                           spheres SPHNAME1 and SPHNAME2.  See "separation".
cc                           If the spheres are concentric, there are an
cc                           infinite number of pairs of proximal points.
cc
ccin                       distance cylinder CYLNAME sphere SPHNAME
cc
cc                           Find the distance and/or intersection of sphere
cc                           SPHNAME and circular cylinder CYLNAME.  See
cc                           "separation".  If the sphere is on the axis of the
cc                           cylinder, there are an infinite number of proximal
cc                           points.
cc
ccin                       distance cylinder CYLNAME1 cylinder CYLNAME2
cc
cc                           Find the distance and/or intersection of the two
cc                           circular cylinders CYLNAME1 and CYLNAME2.  See
cc                           "separation".  If the cylinders are concentric,
cc                           there are an infinite number of pairs of proximal
cc                           points.
cc
ccin                       distance quadric QNAME1 quadric QNAME2
cc
cc                           Find the distance and proximal points, or find a
cc                           randomly sampled point on the intersection curve of
cc                           the two quadric surfaces QNAME1 and QNAME2.  May
cc                           fail if the surfaces are almost flat at their
cc                           proximal points, intersect at very small angles or
cc                           are tangent.
cc
cc                           Note:  the two surfaces QNAME3 and QNAME4 have the
cc                           same intersection curve as QNAME1 and QNAME2:
cc                             quadric QNAME3 sum QNAME1 QNAME2 N2 -N1
cc                             quadric QNAME4 sum QNAME1 QNAME2 N2  N1
cc                           and are orthogonal at point P, if N1 and N2 are the
cc                           magnitudes of the normal vectors of QNAME1 and
cc                           QNAME2 at point P.  See command "arc".
cc
cc                           Note:  either surface QNAME1 or QNAME2 may be
cc                           replaced by surface QNAME3 above, which will
cc                           have a zero value of one of the invariants
cc                           "trace", "dsum" or "det", if N1 and N2 are the
cc                           values of that invariant for QNAME1 and QNAME2.
cc
cc                         Synonyms:  [brick, br], [cluster, cl],
cc                         [cylinder, cyl],
cc                         [distance, dist, int, inter, intersect],
cc                         [ellipsoid, el, ell, ellipse], [help, h],
cc                         [hyperb, saddle], [line, l, ln], [plane, pl],
cc                         [point, p, pnt, pt], [proximal, prox],
cc                         [quadric, q, quad], [sphere, sph],
cc                         [tetrahedron, tet, tetra], [triangle, tri],
cc                         [zone, z, zn].

cc    distance           An option in command "variable", to use the value of
cc                         the last distance found with commands "distance",
cc                         "proximal", "side", "track" or "walk".
cc                         In the case of command "distance", the distance
cc                         between any two quadric surfaces, including planes,
cc                         is the distance to an external tangency, even if the
cc                         surfaces intersect.

cg    distance to collision
cg                       See "mean free path".

cg    distribution       See "cosine-power", "exponential", "isotropic",
cg                         "linear", "Maxwellian", "normal", "Planck spectrum",
cg                         "power-law", "uniform", "Wien spectrum".
cg                         See commands "bin", "pdf", "point", "sample",
cg                         "variable", "vector".

cg    divide             To divide real or integer numbers, see commands
cg                         "variable", "icalc", "big".

cg    division           The division or slash character, "/".

cg    divisor            See "common divisor", "gcd".

cc    dk                 A synonym for disk.

cc    DKNAME             The name of an annular disk.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.
cc                         No disk name may be "+", "-", "all", "h", "help",
cc                         "list", "point" or "thru", begin with "!" or contain
cc                         ";".
cc                         Also referred to as DKNAME1, DKNAME2, DKNAME3, ...
cc                         Specified with command "disk".
cc
cc                         Annular disks may appear in commands:
cc                         copy, disk, delete, plane, point, rename.
cc                         Also see commands:  last, list, repack, search, sort.

cc    DMAX               In command "walk", the maximum distance a point is to
cc                         be randomly walked from its initial position, if it
cc                         is not walked a total path length of PATHMAX first.
cc                         May be integer, floating point or a variable.

cc    DMEAN              In command "walk", the mean free path of a point being
cc                         displaced by a random walk process.
cc                         May be integer, floating point or a variable.

cc    dms                An option in command "angles" to convert angles from or
cc                         to degrees, minutes and seconds.

cn    dms.mac            A macro file for converting from degrees in decimal
cn                         form to degrees, minutes and seconds.
cn                         In ~/work/geom/test .
cn                         Type "in dms.mac ?" for info.  See "deg.mac".

cc    do                 A command to begin a do loop.  Must be followed, after
cc                         any desired block of commands, by command "enddo".
cc                         See "do while".
cc
cc                         Note:  do not use the statement separator ";" on the
cc                         same line as this command.
cc
cc                         Do loops may be nested, with up to 16 levels.
cc
cc                         Note:  command "goto" may not be used to jump into or
cc                         out of a do loop.  While in a do loop, any commands
cc                         read from another input file must end with a command
cc                         to return to the same do loop.
cc
cc                         Forms include:
cc
ccin                       help do
ccin                       do [help]
ccin                       help enddo
ccin                       enddo help
cc
cc                           Display the command options.
cc
ccin                       do VARNAME [=] N1 N2 [INC|1]
cc
cc                           Begin a do loop, with loop variable VARNAME ranging
cc                           from N1 to N2 with increment INC (default 1).
cc                           Name VARNAME has the same restrictions as any named
cc                           variable.  Arguments N1, N2 and INC must be
cc                           integers or integer variables.  Argument INC may be
cc                           negative.  The value of VARNAME starts at N1, and
cc                           is incremented by INC each time command "enddo" is
cc                           executed, until it falls outside the range from N1
cc                           to N2.  Until that happens, all the commands
cc                           between the initial command "do" of this do loop
cc                           and its corresonding command "enddo" are executed,
cc                           for each value of VARNAME.
cc
cc                           To loop over a range of floating point values,
cc                           e.g. from X1 to X2, with an increment of DX,
cc                           use the following commands:
cc
cc                           s XRANGE = X2 - X1
cc                           s N2 = XRANGE / DX
cc                           s N2 = N2 + 1
cc                           s N2 = 1 int N2
cc                           s X = X1
cc                           do N = 1 N2
cc                             ...
cc                             s X = X + DX
cc                           enddo
cc
ccin                       enddo
cc
cc                           Repeat the current "do-enddo" block with loop
cc                           variable VARNAME incremented from its current value
cc                           N to N + INC, unless N + INC is not in the range
cc                           from N1 to N2.  N1, N2 and INC are arguments in
cc                           command "do".
cc
cc                         Synonyms:  [help, h].

cg    do loop            A do loop consists of a block of commands beginning
cg                         with a "do VARNAME N1 N2 [INC|1]" command and ending
cg                         with an "enddo" command.  The loop variable VARNAME
cg                         will initially be set at value N1, and when command
cg                         "enddo" is reached, incremented by INC.  If the
cg                         value of VARNAME is still within the range from
cg                         N1 to N2, the loop will be repeated, starting at the
cg                         command after the command "do" that initiated this
cg                         do loop.
cg
cg                         Arguments N1, N2 and INC must be integers or integer
cg                         variables.  Argument INC may be negative, and if not
cg                         specified, defaults to 1.  If loop variable VARNAME
cg                         does not already exist, it will be created, as if
cg                         command "variable VARNAME = N1" had been used,
cg                         with the same restrictions on VARNAME.
cg
cg                         A do loop will be executed 1 + (N2 - N1) / INC
cg                         times (rounded to an integer).  This is the trip
cg                         count.
cg
cg                         Do loops may be nested, with up to 16 levels.
cg                         Each do loop must have a unique loop variable name.
cg                         Each do loop must be entirely contained within the
cg                         next outer do loop.
cg
cg                         Do loops may contain if-endif blocks, which may also
cg                         be nested.  Each such if-endif block must be
cg                         entirely within its do loop, and may be used to end a
cg                         do loop based on some other criterion than the values
cg                         of VARNAME, N1 and N2.  See "do while".
cg                         Jumps outside the do loop block are allowed only if
cg                         a jump back into the same do loop block follows.
cg                         Jumps into a do loop block from outside the block
cg                         should never be used.
cg
cg                         Do loops may not contain any of the following
cg                         commands or command synonyms:
cg                         ".", "call", "goto", "i", "in", "indo", "input",
cg                         "r", "rd", "read", "return", "rtn", "u", "undo".

cg    do loop use        Typically, use of do loops involves specifying an
cg                         initial set of objects with subscripted names, then
cg                         using a do loop over a range of values of the
cg                         suscripts to modify the values of the objects and use
cg                         the modified values to create additional objects with
cg                         or without subscripted names.  When the objects have
cg                         names with multiple subscripts representing, for
cg                         example, coordinate directions, two or more do loops
cg                         may be nested.
cg
cg                         Do loops may be used in place of certain commands,
cg                         including those with options for creating a group of
cg                         objects, including the options "brick", "concentric",
cg                         "family", "incr", "increment", "nest", "move",
cg                         "parallel", "rotate", "scale", "series", used in
cg                         one or more of the commands "cone", "cylinder",
cg                         "ellipsoid", "line", "plane", "quadric", "sphere",
cg                         "tetrahedron", "variable", "vector" and "zone".

cg    do until           See "do while".

cg    do while           The equivalent of a "do while" or "do until" loop may
cg                         be done by using an if-endif block within a do loop
cg                         executed with commands "do" and "enddo", placed
cg                         just before the "enddo" command, to reset the do loop
cg                         index outside its initial range.
cg
cg                         For example, a "do while x .ge. 1.e-12" loop or a
cg                         "do until x .lt. 1.e-12" loop might look like:
cg
cg                           do n 1 100
cg                             ...
cg                             if x .lt. 1.e-12
cg                               variable nlast = n
cg                               variable n = 101
cg                             endif
cg                           enddo
cg
cg                         Note the method of saving the last do loop index
cg                         used before exiting the loop.

cd    doc                Directory for documentation of GEOM.
cd                         Open YANA Cluster:  ~edwards/work/geom/doc
cd                         Archives:  /users/u47/edwards/geom/doc
cd                         See "make", "test".

cg    dodecahedron       A regular dodecahedron has 20 vertices, 30 edges, and
cg                         12 equilateral pentagonal faces.
cg                         The edge length, face area, volume, radius of
cg                         inscribed sphere, dihedral angle and central edge
cg                         angle, for a circumscribed sphere radius of 1.0 are:
cg
cg                         edge    = 4.0 / (sqrt (3.0) + sqrt (15.0))
cg                                 = 0.7136441795462
cg                         area    = 0.25 * sqrt (25.0 + 10.0 * sqrt (5.0)) *
cg                                   edge^2
cg                                 = 0.8762185201985
cg                         volume  = 0.25 * (15.0 + 7.0 * sqrt (5.0)) * edge^3
cg                                 = 2.785163863123
cg                         rinsc   = sqrt ((5.0 + 2.0 * sqrt (5.0)) / 15.0)
cg                                 = 0.7946544722918
cg                         angdih  = acos (-1.0 / sqrt (5.0))
cg                                 = 116.5650511771
cg                         angcent = acos (sqrt (5.0) / 3.0)
cg                                 = 41.81031489578
cg
cg                         See "polyhedron".

cg    dollar             The dollar character, "$".

cg    dot                The period or dot character, ".".  See command "dot".
cg                         See "dot product".

cc    dot                An option in command "variable", to find the dot
cc                         product of two vectors.  See "dot product".
cc                         Synonyms:  [dot, inner].

cc    dot                Command to find the dot product of two vectors, the
cc                         angle between them, and its cosine.
cc
cc                         Command "dot" relates to objects:  symbol, vector.
cc
cc                         Forms include:
cc
ccin                       help dot
ccin                       dot [help]
cc
cc                           Display the command options.
cc
ccin                       dot VNAME1 VNAME2
cc
cc                           Find the dot (inner) product of vectors VNAME1 and
cc                           VNAME2, the included angle, in the current angle
cc                           units, and its cosine.
cc
cc                         Synonyms:  [dot, inner], [help, h].

cg    dot product        The dot (or inner or scalar) product of two vectors
cg                         A = (AX, AY, AZ) and B = (BX, BY, BX) is the scalar
cg                         C = A dot B = AX*BX + AY*BY + AZ*BZ.
cg                         C = |A| * |B| * cos (THETA), where THETA is the angle
cg                         between vectors A and B.
cg                         Note that A dot B = B dot A, and that the dot product
cg                         of two perpendicular vectors is zero.

cg    double             In output from commands "root" and "roots", indicates
cg                         a root of order 2 of a polynomial equation.

cc    DPR                A uniform relative differential probability of a range
cc                         of values VRANL to VRANR of a random variable,
cc                         in command "bin".  Must not be negative.
cc                         See "PRTOT", "DPRL", "DPRR".

cc    DPRL               A relative differential probability of a value VRANL
cc                         of a random variable in command "bin".
cc                         Varies linearly from DPRL at VRANL to DPRR at
cc                         VRANR.  Must not be negative.
cc                         See "PRTOT", "DPR", "DPRR".

cc    DPRR               A relative differential probability of a value
cc                         VRANR of a random variable in command "bin".
cc                         Varies linearly from DPRL at VRANL to DPRR at
cc                         VRANR.  Must not be negative.
cc                         See "PRTOT", "DPR", "DPRL".

cg    dsum               An invariant of a quadric surface, for translations and
cg                         rotations.  See "invariants of quadric".

cc    DT                 In command "accelerate", the increment in time to be
cc                         used between the initial and final times for which
cc                         path length, velocity and position are to be
cc                         displayed.  If not specified, defaults to zero.
cc                         See "TMIN", "TMAX".

cc    DU                 In command "vector" or "plane", a vector component,
cc                         in the x, rcyl or rsph direction, for a rectangular,
cc                         cylindrical or spherical coordinate system,
cc                         respectively.
cc                         May be integer, floating point or a variable.

cg    du                 An increment or distance or vector component in the
cg                         u coordinate direction, which may be x, rcyl or rsph,
cg                         depending on the current coordinate system.

cg    du12               The u component of the distance between the centers of
cg                         the first and second circles, displayed after command
cg                         "intcirc".

cc    dup                A synonym for copy.

cc    duplicate          A synonym for copy.

cg    duplicate          To duplicate an object, use command "copy".

cc    DV                 In command "vector" or "plane", a vector component,
cc                         in the y, theta or theta direction, for a
cc                         rectangular, cylindrical or spherical coordinate
cc                         system, respectively.
cc                         May be integer, floating point or a variable.

cc    DV                 In command "slice", the increment between coordinate
cc                         values, for which points are to be found in a slice
cc                         of a quadric surface.  See "VMIN", "VMAX".

cg    dv                 An increment or distance or vector component in the
cg                         v coordinate direction, which may be y or theta,
cg                         depending on the current coordinate system.

cg    dv12               The v component of the distance between the centers of
cg                         the first and second circles, displayed after command
cg                         "intcirc".

cg    dval               An increment in the value of a random variable which
cg                         ranges from VRANL to VRANR.  Must be positive.

cc    DW                 In command "vector" or "plane", a vector component,
cc                         in the z, z or phi direction, for a rectangular,
cc                         cylindrical or spherical coordinate system,
cc                         respectively.
cc                         May be integer, floating point or a variable.

cg    dw                 An increment or distance or vector component in the
cg                         w coordinate direction, which may be z or phi,
cg                         depending on the current coordinate system.

cc    DX                 In command "root", the increment between successive
cc                         trial values of x, to search for roots, extrema
cc                         and inflection points of a polynomial equation using
cc                         Newtonian iteration.  No more than two roots,
cc                         extrema or inflection points can be found in each
cc                         increment DX.  Will be changed if either XMIN or
cc                         XMAX is changed to be within a calculated range of
cc                         x that contains all roots, extrema and inflection
cc                         points.
cc                         See "XMIN", "XMAX".

cg    dx                 An increment or distance or vector component in the
cg                         x coordinate direction.

cg    dy                 An increment or distance or vector component in the
cg                         y coordinate direction.

cg    dz                 An increment or distance or vector component in the
cg                         z coordinate direction.

E-------------------------------------------------------------------------------

cg    E                  See "E, e, E+, e+".

cg    e                  See "E, e, E+, e+".

cg    E+                 See "E, e, E+, e+".

cg    e+                 See "E, e, E+, e+".

cg    E, e, E+, e+       When preceded and followed by integers, indicates
cg                         multiplication by the power of ten indicated by the
cg                         final  integer.

cg    E-                 See "E-, e-".

cg    e-                 See "E-, e-".

cg    E-, e-             When preceded and followed by integers, indicates
cg                         division by the power of ten indicated by the final
cg                         integer.

cc    ebase              An option in command "symbol", argument SYMBNAME,
cc                         to create symbol ebase with value exp (1) =
cc                         2.71828182845904523536028747135266249775724709369995
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cg    ebase              The base of natural logarithms 2.718281828459045...
cg                         Generated as a symbol at the beginning of a
cg                         GEOM run.  ebase = exp (1).

cg    eccentricity       The eccentricity e of a hyperbola or a hyperbolic
cg                         cylinder is sqrt (1 + (B / A)^2), where
cg                         A is the transverse semiaxis, between the two
cg                         vertices, and B is the conjugate semiaxis, between
cg                         the two points where a line tangent to the vertices
cg                         intersects the asymptotes of the hyperbola.
cg
cg                         For the hyperbola or hyperbolic cylinder
cg                         QC + QXX * x^2 + QYY * y^2 = 0
cg                         (QXX > 0, QYY < 0)
cg                         the eccentricity is
cg                         sqrt (1 - QYY / QXX) > 1, if QC > 0 or
cg                         sqrt (1 - QXX / QYY) > 1, if QC < 0.
cg
cg                         The eccentricity e of an ellipse or elliptic cylinder
cg                         is sqrt (1 - (B / A)^2), where A is the major
cg                         semiaxis, and B is the minor semiaxis.
cg
cg                         For the ellipse or elliptic cylinder
cg                         QC + QXX * x^2 + QYY * y^2 = 0
cg                         (QC < 0, QXX => QYY > 0),
cg                         the eccentricity is sqrt (1 - QYY / QXX) < 1.
cg
cg                         For the ellipsoid
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0
cg                         (QC < 0, QXX => QYY => QZZ),
cg                         the eccentricities in planes of constant x, y and z
cg                         are sqrt (1 - QZZ / QYY), sqrt (1 - QZZ / QXX), and
cg                         sqrt (1 - QYY / QXX), respectively.

cg    echo               Each non-null input line is echoed in the output file,
cg                         preceded by the character ">", and if the input
cg                         medium is not the user's terminal, is echoed on the
cg                         user's terminal.

cc    edge               An option in command "trig", indicating that the next
cc                         argument is the positive length of a edge of a
cc                         triangle.  Synonyms:  [edge, side].

cg    edge               An edge of the mesh or the mesh block occurs wherever
cg                         two of the three mesh indices k, l and m have either
cg                         their minimum or maximum value, and the other index
cg                         ranges from its minimum to its maximum value.
cg                         The mesh and the mesh block each have 12 edges.
cg
cg                         To display the 12 edge lengths of a brick, use
cg                         command "brick".
cg                         The edges of a brick in the u, v and w directions
cg                         are labeled "min, min", "min, max", "max, min" and
cg                         "max, max", indicating the values of the other two
cg                         coordinates.  For example, in spherical coordinates,
cg                         edge phi (max, min) refers to the edge in the polar
cg                         angle phi direction, at maximum rsph, minimum theta.
cg
cg                         To display the edge length of a regular polygon,
cg                         use command "polygon".
cg
cg                         To display the average edge length of a general
cg                         polygon, use command "area".
cg
cg                         To display the edge length of a regular polyhedron,
cg                         use command "polyhedron".
cg
cg                         To find the plane containing the edge of a quadric
cg                         surface, as viewed from a point, use command "plane",
cg                         option "outline".

cg    edges              See "three edges", "triangle solution".

cg    edit               To display and/or edit a file, use the UNIX text
cg                         editor VI or GEOM command "vi".
cg                         For example, the saved input lines in file geom_cmd
cg                         could be edited to do a different problem, or to
cg                         correct errors before repeating a problem.

cd    edwards            Directory for Arthur L. Edwards, L-298, Ph. 422-4123
cd                         Open YANA Cluster:  ~edwards
cd                         Archives:  /users/u47/edwards

cc    el                 A synonym for ellipsoid.

cg    element            See "point element", "line element", "surface element",
cg                         "volume element", for elements of a mesh.

cg    element            See "brick", "tetrahedron", "zone".

cc    ell                A synonym for ellipsoid.

cc    ELL(1)             In command "ellipsoid", the name of a base ellipsoid,
cc                         used to create a family of ellipsoids.

cg    ellipse            An ellipse is a plane curve, for which the standard
cg                         equation in the x-y plane is:
cg                         - 1 + QXX * x^2 + QYY * y^2 = 0
cg                         (QXX => QYY).
cg                         The major semiaxis SAY = 1 / sqrt (QYY), and the
cg                         minor semiaxis SAX = 1 / sqrt (QXX).
cg                         The area is pi * SAX * SAY.
cg                         The perimeter requires an elliptic integral.

cc    ellipse            A synonym for ellipsoid.
cc                         Also see "cylinder, elliptic".

cg    ellipsoid          An ellipsoid is a quadric surface in 3-D space,
cg                         described by an implicit quadric equation.
cg                         An ellipsoid is a closed surface, with three axes.
cg                         An ellipsoid is axially symmetric if two of the three
cg                         axes are equal, and is a prolate spheroid if the
cg                         unequal axis is longer or an oblate spheroid if the
cg                         unequal axis is shorter.
cg
cg                         The standard forms of the implicit equations for
cg                         ellipsoids (with all coefficients positive) are:
cg
cg                         Real ellipsoid:
cg                                   - 1 + QXX*x^2 + QYY*y^2 + QZZ*z^2 = 0
cg                         Real sphere:
cg                                       - 1 + QXX * (x^2 + y^2 + z^2) = 0
cg                         Imaginary ellipsoid:
cg                                     1 + QXX*x^2 + QYY*y^2 + QZZ*z^2 = 0
cg                         Imaginary sphere:
cg                                         1 + QXX * (x^2 + y^2 + z^2) = 0
cg
cg                         The semiaxes in the x, y and z directions are
cg                         SX = 1/sqrt(QXX), SY = 1/sqrt(QYY), SZ = 1/sqrt(QZZ).
cg                         So, for a real ellipsoid, the standard equation is:
cg                              (x/SX)^2 + (y/SY)^2 + (z/SZ)^2 = 1
cg
cg                         For a real ellipsoid aligned with the x, y and z
cg                         axes, the eccentricities in the major planes are:
cg                         sqrt (1 - QZZ / QYY) or sqrt (1 - QYY / QZZ),
cg                         sqrt (1 - QZZ / QXX) or sqrt (1 - QXX / QZZ),
cg                         sqrt (1 - QYY / QXX) or sqrt (1 - QXX / QYY),
cg                         (use the positive argument of the square root).
cg
cg                         The distance D from the center to the surface in the
cg                         direction of any unit vector U = (UX, UY, UZ) is:
cg                           D = 1 / sqrt ((UX/SX)^2 + (UY/SY)^2 + (UZ/SZ)^2)
cg                         and the corresponding surface point P is:
cg                           P = (X, Y, Z) = (UX * D, UY * D, UZ * D).
cg
cg                         An ellipsoid centered on the z axis, at Z0, and
cg                         symmetric around the z axis (QXX = QYY), and passing
cg                         through the two points (R1, Z1) and (R2, Z2), where
cg                         R1^2 = X1^2 + Y1^2, and R2^2 = X2^2 + Y2^2,
cg                         has the equation:
cg                           QC + QXX * (x^2 + y^2) + QZZ = 0
cg                         where w = R2^2*(Z1-Z0)^2 - R1^2*(Z2-Z0)^2,
cg                         A1 = (R2^2 - R1^2) / w
cg                         A2 = [(Z1 - Z0)^2 - (Z2 - Z0)^2] / w
cg                         QC = -1 / sqrt(A2)
cg                         QZZ = A1 / A2
cg                         This requires that w not be zero, Z1 not be equal to
cg                         Z2, and that the point further from Z0 have the
cg                         smaller value of R.
cg
cg                         In a command, the word "ellipsoid" means a real
cg                         ellipsoid.   Otherwise, use "quadric".
cg
cg                         To find the distance from a point to an ellipsoid,
cg                         use command "distance", "proximal" or "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and an ellipsoid, use command "distance".
cg
cg                         The following commands relate to ellipsoids:
cg                         accelerate, arc, axisym, copy, debug, delete,
cg                         distance, ellipsoid, extrema, help, invert, last,
cg                         list, move, operator, point, proximal, quadric,
cg                         reflect, rename, repack, rotate, scale, search, side,
cg                         slice, sort, symbol, synonym, tables, track, triple,
cg                         vector, zone.

cc    ellipsoid          Command to display or create ellipsoids.
cc                         Creating an ellipsoid replaces any existing quadric
cc                         surface having the same name.
cc                         The normal vectors will be radially outward.
cc                         See "quadric" for other options.
cc                         See "do loop use".
cc
cc                         Command "ellipsoid" relates to objects:  axisym,
cc                         ellipsoid, operator, point, quadric, symbol,
cc                         variable, vector.
cc
cc                         Forms include:
cc
ccin                       help ellipsoid
ccin                       ellipsoid [help]
cc
cc                           Display the command options.
cc
ccin                       ellipsoid [all,list]
cc
cc                           Display all ellipsoids (short display).
cc
ccin                       ellipsoid list ELNAME1 ELNAME2 ELNAME3 ...
cc
cc                           Display ellipsoids ELNAME1, ELNAME2, ELNAME3, ...,
cc                           with or without subscripts (short display).
cc
ccin                       ellipsoid ELNAME
cc
cc                           Display ellipsoid ELNAME (long display).
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         ELNAME or ELL(2) to form the name by incrementing or
cc                         decrementing the base name for ellipsoids.
cc                         See command "last", "increment names".
cc
ccin                       ellipsoid ELNAME = PCEN SAX SAY SAZ
cc
cc                           Create ellipsoid ELNAME, centered at point PCEN,
cc                           with semiaxes SAX, SAY and SAX in the x, y and z
cc                           directions, respectively.
cc                           Note:  if any semiaxis is given a very large value,
cc                           such as 10^99, the resulting quadric surface will
cc                           be an elliptic cylinder, not an ellipsoid.
cc
ccin                       ellipsoid ELNAME fit PCEN VAXIS PNAME2 PNAME3
cc
cc                           Create circular ellipsoid ELNAME, centered at point
cc                           PCEN, with axis vector VAXIS, and passing through
cc                           the two points PNAME2 and PNAME3.  This is an
cc                           axially symmetric quadric surface, which may be an
cc                           oblate or prolate spheroid.
cc
ccin                       ellipsoid ELL(2) move NUMELL INC ELL(1) VMOVE
cc
cc                           Create a family of NUMELL ellipsoids ELL(2), ...,
cc                           with names incremented by INC characters, spaced at
cc                           intervals of vector VMOVE, starting from ellipsoid
cc                           ELL(1).  See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       ellipsoid ELL(2) rotate NUMELL INC ELL(1) OPNAME PINV
cc
cc                           Create a family of NUMELL ellipsoids ELL(2), ...,
cc                           with names incremented by INC characters, by
cc                           rotating the preceding ellipsoid with operator
cc                           OPNAME and invariant point PINV, starting from
cc                           ellipsoid ELL(1).  See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       ellipsoid ELL(2) scale NUMELL INC ELL(1) OPNAME PINV
cc
cc                           Create a family of NUMELL ellipsoids ELL(2), ...,
cc                           with names incremented by INC characters, by
cc                           scaling the preceding ellipsoid with operator
cc                           OPNAME and invariant point PINV, starting from
cc                           ellipsoid ELL(1).  See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
cc                         Synonyms:  [help, h], [ellipsoid, el, ell, ellipse],
cc                         [move, mv, trans, translate], [rotate, rot].

cc    ellipsoid          An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more ellipsoids.
cc                         Synonyms:  [ellipsoid, el, ell, ellipse].

cg    ellipsoid arrays   Ellipsoids may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    ellipsoids         A family of ellipsoids may be created with command
cg                         "ellipsoid", options "move", "rotate" or "scale"
cg                         (with restrictions) or as follows:
cg
cg                         p pinv (options)
cg
cg                           Create an invariant point, if needed.
cg
cg                         ell ELL(1) (options)
cg
cg                           Create a base ellipsoid.
cg
cg                         op opr (options)
cg
cg                           Create a tensor operator, to reflect, rotate,
cg                           invert or scale, if needed.
cg
cg                         v vmove (options)
cg
cg                           Create a vector for use as a translation operator,
cg                           if needed.
cg
cg                         cp ELL(1) ELL(2)
cg
cg                           Copy ellipsoid ELL(1) to the first ellipsoid.
cg
cg                         Repeat the following block of commands as many times
cg                         as needed to create the rest of the family of
cg                         ellipsoids.  See "shortcuts".
cg
cg                         mv ell ELL(1) vmove
cg
cg                           Move the base ellipsoid by amount move, if needed.
cg
cg                         [invert,reflect,rotate,scale] ell ELL(1) opr [pinv]
cg
cg                           Invert, reflect, rotate or scale the base ellipsoid
cg                           with tensor operator opr, if needed.
cg
cg                         cp ell ELL(1) +
cg
cg                           Copy the base ellipsoid to the next ellipsoid.

cg    elliptic           See "elliptic cone", "elliptic cylinder",
cg                         "elliptic paraboloid".

cg    elliptic cone      An elliptic cone is a quadric surface for which the
cg                         standard equation is:
cg                           x^2 + QYY*y^2 - |QZZ|*z^2 = 0, (QYY>0, QZZ<0).
cg                         An imaginary elliptic cone is a quadric surface for
cg                         which the standard equation is:
cg                           x^2 + QYY*y^2 + QZZ*z^2 = 0, (QYY>0, QZZ>0).
cg                         This last equation is also satisfied at the real
cg                         point (0,0,0).
cg
cg                         To find the distance from a point to an elliptic
cg                         cone, use command "distance", "proximal" or "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and an elliptic cone, use command "distance".

cg    elliptic cylinder
cg                       An elliptic cylinder is a quadric surface for which the
cg                         standard equation is:
cg                          - 1 + QXX*x^2 + QYY*y^2 = 0, (QXX>0, QYY>0).
cg                         An imaginary elliptic cylinder is a quadric surface
cg                         for which the standard equation is:
cg                            1 + QXX*x^2 + QYY*y^2 = 0.  (QXX>0, QYY>0).
cg
cg                         To create an elliptic cylinder aligned with the x,
cg                         y or z axis, use command:
cg
cg                         ellipsoid QNAME = PCEN SAX SAY SAZ
cg
cg                           Create elliptic cylinder QNAME, centered at point
cg                           PCEN, with semiaxes SAX, SAY and SAX (one of which
cg                           is very large, say 10^99) in the x, y and z
cg                           directions, respectively.  Any future reference to
cg                           this quadric surface must use option "quadric", not
cg                           "ellipsoid".
cg
cg                         To find the distance from a point to an elliptic
cg                         cylinder, use command "distance", "proximal" or
cg                         "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and an elliptic cylinder, use command "distance".

cg    elliptic paraboloid
cg                         An elliptic paraboloid is a quadric surface for which
cg                         the standard equation is:
cg                               -QZ*z + x^2 + QYY*y^2 = 0.
cg
cg                         To find the distance from a point to an elliptic
cg                         paraboloid, use command "distance", "proximal" or
cg                         "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and an elliptic paraboloid, use command "distance".

cc    ELNAME             The name of an ellipsoid.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.  May not be the same as any other quadric
cc                         surface name QNAME.  Also referred to as ELNAME1,
cc                         ELNAME2, ...  Specified with command "ellipsoid".
cc                         No ellipsoid name may be "+", "-", "all", "h",
cc                         "help", "list" or "thru", begin with "!" or contain
cc                         ";".
cc
cc                         Ellipsoids may appear in commands:
cc                         accelerate, arc, axisym, cone, copy, delete,
cc                         distance, ellipsoid, extrema, help, invert, last,
cc                         list, move, operator, point, project, proximal,
cc                         quadric, reflect, rename, repack, rotate, scale,
cc                         search, side, slice, sort, track, triple, vector.

cg    embedded           See "embedded lines".

cg    embedded lines     At any point on some quadric surfaces (all planes,
cg                         cylinders, cones, hyperbolic paraboloids, and
cg                         hyperboloids of one sheet) one or more straight lines
cg                         can be drawn, lying entirely on the surface.
cg                         These quadric surfaces are ruled surfaces.
cg                         See "ruled surface".
cg                         See commands "arc" and "hyperb".

cg    encryption         Encryption is the process of making data or a text
cg                         unreadable without special knowledge, intended to
cg                         be available only to certain recipients.
cg                         See command "big".

cg    end                See "end of command", "end of comment".

cc    end                Command to end the current run.
cc                         Synonyms:  [end, exit, quit, split].

cg    end of command     The unquoted character ";" ends a command or a comment,
cg                         when multiple commands and/or comments are on one
cg                         input line.  Ignored if between single or double
cg                         quotes (as in commands "alias", "define" and
cg                         "marker").  Does not need to be delimited.
cg                         If not between single or double quotes, will
cg                         terminate a comment, allowing a command to follow on
cg                         the same line.  Ignored in command "hex".
cg                         Can not be used after commands "indo", "input",
cg                         "read", "redo", "return" or"undo", because chaos
cg                         might ensue.

cg    end of comment     The unquoted character ";", if not between single or
cg                         double quotes, will terminate a comment, allowing a
cg                         command to follow on the same line.  Does not need to
cg                         be delimited.

cg    end-of-file        An end-of-file in the input file will have the same
cg                         effect as command "return" or "input #".

cc    enddata            An option in command "variable", indicating the end
cc                         of a table of variable values.  See "data".

cc    enddo              A command to repeat or end a do loop.  Must be preceded
cc                         by command "do".  Amy desired block of commands may
cc                         be used in between command "do" and command "enddo",
cc                         including other do loops.  See "do while".
cc
cc                         Note:  do not use the statement separator ";" on the
cc                         same line as this command.
cc
cc                         Note:  "do-enddo" blocks of commands may be nested.
cc
cc                         Forms include:
cc
ccin                       help enddo
ccin                       enddo help
cc
cc                           Display the command options.
cc
ccin                       enddo
cc
cc                           Repeat the current "do-enddo" block for loop
cc                           variable VARNAME incremented from its current value
cc                           N to N + INC, unless N + INC is not in the range
cc                           from N1 to N2.  N1, N2 and INC are arguments in
cc                           command "do".
cc
cc                         Synonyms:  [help, h].

cc    endif              The final command in an "if-endif" block of input
cc                         lines, which must begin with command "if ...".
cc
cc                         Command "endif" relates to objects:  symbol.
cc
cc                         Forms include:
cc
ccin                       help endif
ccin                       endif help
cc
cc                           Display the command options commands "if" and
cc                           "endif".  Note that the argument "help" is not
cc                           optional.
cc
ccin                       endif
cc
cc                           End the current "if-endif" block of input lines,
cc                           which begins with the command "if" defining the
cc                           current "if-endif" block.
cc
cc                         Synonyms:  [help, h].

cg    English            Commands and options in GEOM are based on English.
cg                         See "language".

cc    ENTRY              In command "define", the name of the entry in geom_base
cc                         (this file) to be displayed.  Must be bracketed by
cc                         single (') or double (") quotes.
cc                         If ENTRY is bracketed by single quotes, double any
cc                         internal single quotes.
cc                         If ENTRY is bracketed by double quotes, double any
cc                         internal double quotes.
cc                         The entry is found by matching columns 7-25 in
cc                         this file, geom_base, ignoring case, with ENTRY.
cc                         All lines are displayed until the next non-matching
cc                         entry.  Multiple entries with the same name will be
cc                         displayed.  ENTRY may have up to 19 characters.

cg    entry              In this file, a section delimited by blank lines, with
cg                         the name of the entry in columns 7-24 of the first
cg                         line.

cc    env                A synonym for environment.

cg    environment        See "environmental", command "environment".

cc    environment        An option in command "debug", to display GEOM internal
cc                         variables for the environment:
cc                         alodmach, aloddate, alodtime, aversion, amach,
cc                         adate, atime, atitle, aprompt, aprompts, lprompts.
cc                         Synonyms:  [environment, env, setup].

cc    environment        An option in command "help", to display a help message
cc                         for environment commands.

cc    environment        Command to display a help message for environment
cc                         commands, which include:
cc                         alias, angles, binary, commands, coordinate, dec,
cc                         define, degrees, delimiter, end, endif, help, hex,
cc                         icalc, if, indo, input, marker, nobin, oct, output,
cc                         page, plot, prompt, radians, redo, status, symbol,
cc                         synonym, tables, time, title, tol, tol, trace, undo,
cc                         variable, vi, when.
cc
cc                         Forms include:
cc
ccin                       help environment
ccin                       environment [help]
cc
cc                           Display a help message for environment commands.
cc
cc                         Synonyms:  [help, h], [environment, env, setup].

cg    environmental      To specify, create, display and use environmental
cg                         objects, see the entries and commands:  ".", ";",
cg                         alias, angles, binary, commands, coordinate, debug,
cg                         dec, define, degrees, delim, end, endif, help, hex,
cg                         icalc, if, indo, input, marker, nobin, oct, output,
cg                         page, plot, prompt, radians, redo, status, symbol,
cg                         synonym, tables, time, title, tol, tol, trace, undo,
cg                         variable, vi, when.

cg    equal              See "equal temperament", "equal zoning",
cg                         "equally-tempered".

cg    equal sign         The equal sign, "=".

cg    equal temperament
cg                       See "equally-tempered".

cg    equal zoning       Planes or other quadric surfaces may be created in a
cg                         family in which each surface is displaced by a
cg                         specified constant amount from the preceding surface,
cg                         in a specified direction.
cg                         Points may be created in a family in which each point
cg                         is displaced by a specified constant amount from the
cg                         preceding point, in a specified direction.
cg                         See option "move" of commands "point", "sphere",
cg                         "cylinder", "cone", "axisym", "quadric", and see
cg                         command "plane", option "parallel".
cg                         See "ratio zoning".
cg
cg                         Points may be created in a 3-D array in a brick,
cg                         in such a way that the volumes of each of the
cg                         smallest elements with 8 points at their vertices are
cg                         equal.  See command
cg                         "point PMESH brick BRNAME mesh bloxk [volume]".
cg                         See "briquette".
cg
cg                         See "do loops" and commands "do" and "enddo".

cg    equally-tempered   A musical scale in which adjacent semitones have a
cg                         fixed frequency ratio, 2^(1/12) = 1.059463094359.
cg                         For one octave, the results are as follows:
cg
cg                         Note  Equal-temp   Value  Just    Cents   Interval
cg                               ratio        cents  ratio   diff
cg
cg                          1  1.000000000000     0   1/1     0      unison
cg                          2  1.059463094359   100  16/15  -11.731  semitone
cg                          3  1.122462048309   200   9/8     3.910  whole tone
cg                          4  1.189207115003   300   6/5   -15.641  minor 3rd
cg                          5  1.259921049895   400   5/4    13.686  major 3rd
cg                          6  1.334839854170   500   4/3     1.955  perfect 4th
cg                          7  1.414213562373   600  45/32    9.776  tritone
cg                          8  1.498307076877   700   3/2    -1.955  perfect 5th
cg                          9  1.587401051968   800   8/5   -13.687  minor 6th
cg                         10  1.681792830507   900   5/3    15.641  major 6th
cg                         11  1.781797436281  1000  16/9     3.910  minor 7th
cg                         12  1.887748625363  1100  15/8    11.731  major 7th
cg                         13  2.000000000000  1200   2/1      0     octave
cg
cg                         Note  Equal-temp   Just ratio
cg                               ratio
cg
cg                         A   1.000000000000  1.00000
cg                         Bb  1.059463094359  1.06667
cg                         B   1.122462048309  1.12500
cg                         C   1.189207115003  1.20000
cg                         Db  1.259921049895  1.25000 or 1.26562 = 81/64
cg                         D   1.334839854170  1.33333
cg                         Eb  1.414213562373  1.40625 or 1.42383 = 729/512
cg                         E   1.498307076877  1.50000
cg                         F   1.587401051968  1.60000
cg                         Gb  1.681792830507  1.66667 or 1.68750 = 27/16
cg                         G   1.781797436281  1.77778
cg                         Ab  1.887748625363  1.82500 or 1.89844 = 243/128
cg                         A   2.000000000000  2.00000
cg
cg                         See "piano", "cents".

cg    equation           The real and/or complex roots of a quadratic
cg                         (a3 = a4 = 0), cubic (a4 = 0) or quartic polynomial
cg                         equation of the form:
cg                           a0 + a1 * z + a2 * z^2 + a3 * z^3 + a4 * z^4 = 0
cg                         may be found with command "roots".
cg
cg                         The real coefficients a0, a1, a2, a3 and a4 may be
cg                         found for any specified set of from two to four real
cg                         and/or complex roots with command "roots", option
cg                         "=".
cg
cg                         To search for real roots, extrema and inflection
cg                         points of a polynomial equation using Newtonian
cg                         iteration, use command "root".
cg
cg                         To find the coefficients of an Nth-order polynomial
cg                         equation with N specified real roots, use command
cg                         "root", option "=".

cg    equations          See "polygon equations", "polynomial", "quadric types",
cg                         "quadric curves", command "solve".

cg    equilateral        For every triangle, there are equilateral triangles
cg                         that have a parallel projection onto the triangle.
cg                         Whenever the data for a triangle are displayed, the
cg                         vertex coordinates and edge length of one such
cg                         triangle, such that one vertex is shared, are
cg                         displayed.

cc    err                A synonym for error.

cg    err                Estimated numerical truncation error in calculating
cg                         a value, based on TOL.  See "error estimate".

cg    error              Error messages are generated by the code whenever it
cg                         detects an error in an input line, which may include
cg                         an unrecognized command, the wrong number, mode or
cg                         sign of arguments, or the non-existence of an
cg                         operator, geometric object, file or key word referred
cg                         to in the command, or the non-existence of a
cg                         result satisfying the requirements of the command.
cg                         After an error message, check the command, or type
cg                         "help COMMAND" to display the options, or type
cg                         "define 'ENTRY'" to display the entry in geom_base
cg                         for ENTRY, and retry.
cg                         Use command "trace" to display the return path from
cg                         the current input file.
cg                         See command "error".

cc    error              Command to display a help message for error control
cc                         commands, which include: undo, indo, redo, input.
cc
cc                         Forms include:
cc
ccin                       help error
ccin                       error [help]
cc
cc                           Display a help message for error control commands.
cc
cc                         Synonyms:  [error, err], [help, h].

cc    error              An option in command "help", to display help messages
cc                         for error control commands.

cg    error              See "error estimate", "truncation error".

cg    error control      See commands "debug", "indo", "input", "parse", "redo",
cg                         "return", "symbol", "undo".
cg                         If you have ended the run, see "input file",
cg                         to find how to make a new input file to repeat
cg                         the run without the error(s).

cg    error correction   See "error control".

cg    error estimate     When calculating a value Z(A, B, C, ...), a function of
cg                         A, B, C, ..., a first-order estimate of the error
cg                         in Z, E(Z), is given by:
cg                         E(Z) = abs (Z'(A)) * E(A) + abs (Z'(B)) * E(B)
cg                                + abs (Z'(C)) * E(C) + ... ,
cg                         where Z'(A), Z'(B), Z'(C), ... are the partial
cg                         derivatives of Z with respect to A, B, C, ..., and
cg                         E(A), E(B), E(C), ... are the (positive) estimated
cg                         errors in A, B, C, ...
cg                         For an input value A, the estimated error E(A) is
cg                         TOL * abs (A).

cg    escape character   A character used to force the following character to be
cg                         interpreted literally, instead of having a special
cg                         meaning or function.  For example, adding an extra
cg                         quote mark to a quote mark inside of a quoted
cg                         character string or prefixing a character with a
cg                         backslash "\" or <control-V> in the text editor VI.

cg    estimate           To estimate the volume of a zone by a Monte Carlo
cg                         method, use command "mcvol".

cc    Euler              A synonym for euler.

cg    Euler              See "Euler's Totient", "totient function".

cc    euler              An option in command "symbol", argument VNAME,
cc                         to create symbol euler with value
cc                         0.57721566490153386061...
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cg    euler              Euler's constant.  Generated as a symbol at the
cg                         beginning of a GEOM run.
cg                         euler = the limit as z approaches infinity of
cg                                 1 + 1/2 + 1/3 + 1/4 + ... + 1/z - log z =
cg                         euler = 0.57721566490153386061...
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cg    Euler's            See "Euler's Totient", "totient function".

cg    Euler's Totient    See "totient function".

cg    eV                 Electron volts (an energy unit).
cg                         1 amu = 9.3149432E8 eV (+/- 0.3 ppm).
cg                         1 K = 8.617385E-5 eV (+/- 8.5 ppm).
cg                         1 Hz = 4.1356692E-15 eV (+/- 0.3 ppm).

cc    event              An option in command "bin", to create a probability
cc                         bin for a discrete event, identified by the name of
cc                         the bin.

cc    exch               A synonym for exchange.

cc    exchange           An option in command "mesh", to exchange the layers in
cc                         the directions of two different indices in a mesh
cc                         block, and to redefine the mesh indices of any mesh
cc                         points in the mesh block accordingly.  The mesh block
cc                         must have the same size in the two layer directions.
cc                         Along with option "invert" in command "mesh", this
cc                         allows all possible mesh reflections and rotations
cc                         to be done.
cc                         Synonyms:  [exchange, exch].

cg    EXCHANGED          In the display for a point, indicates a mesh point in
cg                         a mesh block for which mesh indices have been
cg                         exchanged.

cg    excluded           See "excluded names".

cg    excluded names     Some names are disallowed for all or some object names,
cg                         because they make interpretation of a command
cg                         ambiguous.
cg                         No name may be "+", "-", "all", "h", "help",
cg                         "list" or "thru", begin with "!" or contain ";".
cg                         A point name must not be "cluster".
cg                         A cluster name must not be "point".
cg                         A quadric surface name must not be "types".

cg    exclusive          See "exclusive or", "Boolean".

cg    exclusive or       See "Boolean".

cg    execution          See "GEOM execution".

cc    execution line     See "GEOM execution".

cc    exit               A synonym for end.

cc    exp                An option in command "bin", to create an exponential
cc                         probability distribution function.

cc    exp                An option in command "variable", argument FUNCTION,
cc                         to indicate the base e exponential function.
cc                         A exp B means A * exp (B).

cc    exp                An option in command "big", to find a base e
cc                         exponential function of a specified argument,
cc                         in the form of a big integer.

cf    exp                The exponential function.  Example:  y = exp (x)
cf                         means y = e^x, where e is the base of natural
cf                         logarithms.

cg    expected           See "expected value".

cg    expected value     The expected value of a random variable v(x)^n with
cg                         a probability distribution p(x) from x1 to x2 is
cg                         <v^n>, and is equal to the integral from x1 to x2
cg                         of v(x)^n * p(x) * dx, divided by the integral from
cg                         x1 to x2 of p(x) dx.
cg                         The variance of v is <v^2> - <v>^2, and the
cg                         standard deviation of v is the square root of the
cg                         variance.
cg                         See "random walk", commands "bin", "pdf", "sample",
cg                         "spin", "walk".

cg    exponent           A floating point number may be represented by a
cg                         mantissa, e.g. 1.23456789, followed by an exponent
cg                         indicating a power of ten, e.g, e-3, e22, meaning
cg                         10^(-3) or 0.001 and 10^22, respectively.
cg                         See "E, e, E+, e+", "E-, e-".

cg    exponential        An exponential probability distribution is a type of
cg                         probability bin or probability distribution function
cg                         (pdf) for which the relative differential probability
cg                         is an exponential function of the value of the random
cg                         variable, for a specified range of such values.
cg                         See "discrete event", "discrete object",
cg                         "discrete value", "uniform", "linear", "power-law".
cg                         See commands "bin", "pdf", "sample".

cg    extend             To extend two sides of a triangle, to make a new
cg                         triangle with twice the area, and a perimeter twice
cg                         that of the extended sides, see command "cute".

cg    exterior           An exterior mesh point is one that has exactly one
cg                         nearest neighbor in one or more of the three mesh
cg                         index directions k, l and m.
cg                         An exterior mesh block point is one that has exactly
cg                         one nearest neighbor in the mesh block, in one or
cg                         more of the three mesh directions k, l and m.

cc    extr               A synonym for extrema.

cg    extr               In display from commands "side" and "distance",
cg                         indicates an extremum on the quadric surface in a
cg                         specified vector direction.  An estimate of the
cg                         proximal point on the quadric surface, nearest point
cg                         PNAME.

cg    extract            See "extract entries".

cg    extract entries    To extract entries from geom_base that contain any
cg                         desired key words, using UNIX command "grep":
cg
cg                         For a single key word KEYWORD1:
cg
cg                         grep -p KEYWORD1 geom_base >! OUT_FILE
cg
cg                           For each occurrence in file geom_base of the key
cg                           word KEYWORD1, write the entry containing KEYWORD1
cg                           into file OUT_FILE.
cg
cg                         For multiple key words KEYWORD1, KEYWORD2, ...:
cg
cg                         grep -p -f KEYWORD_FILE geom_base >! OUT_FILE
cg
cg                           For each occurrence in file geom_base of any of the
cg                           key words in file KEYWORD_FILE, write the entry
cg                           containing the key word into file OUT_FILE,
cg
cg                         where KEYWORD_FILE is a file containing all of the
cg                         key words, each on a single line.

cg    extrema            The extrema of a curve are any points on the curve,
cg                         for which the normal vector to the curve is parallel
cg                         to a specified axis in the plane of the curve
cg                         An extremum may be a maximum, a minimum or an
cg                         inversion point.
cg
cg                         To find the extrema of a quadric curve in a major
cg                         plane, use command "quadric" or "slice".
cg
cg                         The extrema of a polynomial equation in z = x + i*y,
cg                         P(z) = a0 + a1*z + a2*z^2 + a3*z^3 + a4*z^4 + ...
cg                         are at the roots of the derivative of the equation,
cg                         P'(z) = a1 + 2*a2*z + 3*a3*z^2 + 4*a4*z^3 + ...,
cg                         where the slope of the polynomial is zero.
cg                         See "quadratic", "cubic", "quartic".
cg                         See commands "root" and "roots".
cg
cg                         The extrema of a surface are any points or curves on
cg                         the surface, for which the normal vector to the
cg                         surface is parallel to a specified axis.  An extremum
cg                         may be a maximum, minimum or saddle point.
cg
cg                         To find the extrema in any specified direction on a
cg                         quadric surface, use command "quadric" or "extrema".
cg
cg                         If a quadric surface is defined by a second order
cg                         implicit equation F(x,y,z) = 0, then the family of
cg                         quadric surfaces represented by the function
cg                         F(x,y,z) = C may have extrema at points on a line
cg                         or a curve.  In general, such points or curves are
cg                         not proximal or distal to the extrema on the quadric
cg                         surface for which F = 0.
cg
cg                         See "proximal".

cc    extrema            Command to find the extreme points on one or all
cc                         quadric surfaces in a specified direction, and the
cc                         nature of the curvature of the surface at any such
cc                         points.  See command "arc".
cc
cc                         Command "extrema" relates to objects:  axisym, cone,
cc                         cylinder, ellipsoid, hyperb, plane, quadric, sphere,
cc                         symbol, vector.
cc
cc                         Forms include:
cc
ccin                       help extrema
ccin                       extrema [help]
cc
cc                           Display the command options.
cc
ccin                       extrema QNAME VNAME
cc
cc                           Find the extrema of quadric surface QNAME in the
cc                           direction of vector VNAME, and the nature of the
cc                           curvature of the surface and any such extrema.
cc
ccin                       extrema all VNAME
cc
cc                           Find the extrema of all quadric surfaces in the
cc                           direction of vector VNAME, and the nature of the
cc                           curvature of the surface and any such extrema.
cc
cc                         Synonyms:  [extrema, extr], [help, h].

cg    extreme            To find the extreme points of a quadric surface in a
cg                         specified direction, see command "extrema".

F-------------------------------------------------------------------------------

cg    F                  The general equation of a quadric surface is:
cg                         F(x,y,z) = 0.  See "quadric surface".

cc    F1-FN              Arguments in command "variable", to be added (option
cc                         "+"), multiplied (option "*"), or their reciprocals
cc                         added (option "/"), or to be assigned to a set of
cc                         variables (option "data").

cg    face               A face of the mesh or the mesh block occurs wherever
cg                         one of the three mesh indices k, l and m has either
cg                         its minimum or maximum value, and each of the other
cg                         two indices ranges from its minimum value to its
cg                         maximum value.  The mesh and the mesh block each have
cg                         six faces.

cc    FACT               An argument in command "quadric" option "*" or "/",
cc                         which will multiply or divide, respectively, the
cc                         coefficients of the implicit equation for a quadric
cc                         surface.
cc                         May be integer, floating point or a variable.

cc    fact               An option in command "big", to find the factorial
cc                         function of a specified big integer.

cc    fact               An option in commands "icalc" and "variable", to find
cc                         the factorial function (to be done on the following
cc                         argument, which must be a non-negative integer).

cg    factf.mac          A macro file for finding the prime factors of factorial
cg                         functions up to 1012!.  In ~/work/geom/test .

cg    factor             To find all of the prime factors of any integer N from
cg                         1 to 10^18 use command "factor".
cg                         See command "big", option "//".

cg    factor             See "prime factors", "scale factor", "view factor",
cg                         command "factor".

cc    factor             Command to find all of the prime factors and their
cc                         cofactors, for any specified positive integers M from
cc                         1 to 10^18 and for each M, to find the Euler's
cc                         totient function for modulo M arithmetic, which is
cc                         the number of integers in the set (1, ..., M - 1)
cc                         that are relatively prime to M (have no common
cc                         factors other than 1), and to find the number of
cc                         integers that do have common factors with M.
cc                         See "totient function" and "modular arithmetic".
cc
cc                         Command "factor" relates to objects:  symbol,
cc                         variable.
cc
cc                         Forms include:
cc
ccin                       help factor
ccin                       factor [help]
cc
cc                           Display the command options.
cc
ccin                       factor M1 M2 [thru] M3 ...
cc
cc                           For each integer M = M1, M2 [, through] M3, ...,
cc                           find all of the prime factors and their cofactors,
cc                           and find the number of integers in the set
cc                           (1, ..., M - 1) that are relatively prime to M
cc                           (have no common factors other than 1), and the
cc                           number of integers that do have common factors
cc                           with M.  M must be from 1 to 10^18.
cc                           Negative values are changed to positive.
cc
cc                         Synonyms:  [factor, totient], [help, h].

cg    factorial          The factorial function, defined only for non-negative
cg                         integers:  n! = 1, n! = 1 * 2 * 3 * ... * n.
cg
cg                         The average derivative of n! with respect to n is:
cg                         from n - 1 to n    :  n! * (1 - 1/n)
cg                         from n     to n + 1:  n! *  n
cg                         from n - 1 to n + 1:  n! * (n + 1 - 1/n) / 2
cg                         and at n, approximatly for very large n:
cg                                               n! * (n*log(n) + 1/2) / n
cg
cg                         The options in commands "icalc" and "variable" are:
cg
cg                         icalc M fact N
cg
cg                           Display M * N!.  With M = 1, limited to 19!, with
cg                           18 digits:  121645100408832000.
cg
cg                         variable VARNAME = A fact N
cg
cg                           Create variable VARNAME, equal to A * N!.
cg                           With A = 1.0, limited to 170!, with 307 digits:
cg                           7.25741561530799896739672821112926311471699E+306.
cg
cgin                       big BIGNAME = BIGNAME2 fact
cg
cg                           Create big integer BIGNAME, equal to BIGNAME2!.
cg                           With the current limit of 1001 digits for big
cg                           integers, limited to 450!, with 1001 digits:
cg                           1.733368733112632659344713146104579399677811E+1000.
cg
cg                         See "prime products".

cg    factorial          See "factorial factors".

cg    factorial factors
cg                       The factorial function of n, n!, has the factors
cg                         1 * 2 * 3 * ... * (n-1) * n.  This may also be
cg                         expressed using only prime numbers as factors.
cg
cg                         Command "factor" may be used to find the prime
cg                         factors of a factorial.
cg
cg                         variable VARNAME = A fact N
cg
cg                           Create variable VARNAME, equal to A * N!.
cg                           With A = 1.0, limited to 170!.
cg
cg                         factor VARNAME
cg
cg                           Display the prime factors of VARNAME.
cg
cg                         In the case of factorials that are too big to store
cg                         as integers in a particular computer, the prime
cg                         factors of n! may be found as follows:
cg
cg                           n! = product {p(i)^m(i)}, for i = 1, imax,
cg
cg                         where the p(i) are all of the prime numbers less than
cg                         or equal to n, and the m(i) are the exponents
cg
cg                           m(i) = sum {int (p(imax) / p(i)^k)},
cg                           for k = 1, kmax > log (p(n)) / log (p(i)).
cg
cg                         where int (x) is the largest integer <= x, and kmax
cg                         is just sufficient to make the last term zero.
cg                         An upper bound on m(p(i)) is p(imax) / (p(i) - 1).
cg                         Note that in the sum, each term after the first is
cg                         equal to the preceding term divided by p(i).
cg
cg                         For example:
cg
cg                         13!  = 1*2*3*4*5*6*7*8*9*10*11*12*13 = 6227020800
cg
cg                         m(1) = int(13/1) + int(13/1) + ...
cg                              = infinity, but 1^infinity = 1, so no matter.
cg                         m(2) = int(13/2) + int(13/4) + int(13/8) + int(13/16)
cg                              = 6 + 3 + 1 + 0 = 10
cg                         m(3) = int(13/3) + int(13/9) + int(13/27)
cg                              = 4 + 1 + 0 = 5
cg                         m(5) = int (13/5) + int(13/25) = 2 + 0 = 2
cg                         m(7) = m(11) = m(13) = 1
cg
cg                         13!  = 2^10 * 3^5 * 5^2 * 7 * 11 * 13 = 6227020800
cg
cg                         In another example, 37! ~= 1.376375309122634E43,
cg                         the powers of 2, 3, 5, 7, 11, 13 and 17 are 34, 17,
cg                         8, 5, 3, 2, and 2, respectively, and the remaining
cg                         powers of 19, 23, 29, 31, and 37 are all 1.
cg
cg                         This method is implemented for factorials up to
cg                         1012! by using macro file factf.mac in directory
cg                         ~/work/geom/test .  See test problem factf.mac.test
cg                         and output file factf.mac.hsp.

cg    factors            Selected metric conversion factors volume are in macro
cg                         file conv.mac in directory ~edwards/work/geom/test .
cg                         See "prime factors".

cc    FADD               An additive constant used to create a family of
cc                         variables with command "variable", option "series".
cc                         The floating point value var(n) of each variable is
cc                         generated from the preceding value var(n-1) by the
cc                         recursion relation:
cc
cc                         var(n) = FADD + FMULT * var(n-1), n = 2, NUMVAR + 1,
cc
cc                         where var(1) is the value of variable VAR(1).

cc    fam                A synonym for family.

cc    family             An option in command "zone", to create a family of
cc                         zones bounded by from one to three specified families
cc                         of quadric surfaces.
cc                         Synonyms:  [family, fam].
cc
cc                         This command option can be replaced, using a do loop.
cc                         See "do loop use", commands "do", "enddo".

cg    family             A family is a group of objects of the same type,
cg                         sharing some characteristic, such as being parallel,
cg                         concentric, rotated, nested, coincident, congruent,
cg                         collinear, coplanar, etc.  See "family of objects".
cg
cg                         One particular type of family of quadric surfaces
cg                         has an implicit equation F(x,y,z) = C, where C is an
cg                         arbitrary constant, different for each member of the
cg                         family.  Sometimes the family member for which C = 0
cg                         is a different type of quadric surface than the other
cg                         members of the family, e.g., coincident planes
cg                         instead of parallel planes, intersecting planes
cg                         instead of hyperbolic cylinders, and cones instead of
cg                         hyperboloids.  Also, changing C may change a
cg                         hyperboloid of one sheet to a hyperboloid of two
cg                         sheets or vice versa.

cg    family             See "family of objects".

cg    family of objects
cg                       Some commands allow a family of objects to be created
cg                         with a single command.  The object names are
cg                         generated by incrementing the preceding object name
cg                         by a specified number of characters or digits,
cg                         starting with an initial object name.
cg                         The objects are created by transforming the preceding
cg                         object in a specified way, starting from an initial
cg                         object, which itself is not part of the family.
cg                         See "increment names".
cg
cg                         See the following commands and options:
cg
cg                         COMMAND      OPTIONS
cg
cg                         variable     series
cg                         vector       rotate, scale
cg
cg                         point        rotate, scale, move, polygon,
cg                                      polyhedron, brick
cg                         triangle     increment
cg
cg                         axisym       rotate, scale, move
cg                         cone         rotate, scale, move, nest
cg                         cylinder     rotate, scale, move, concentric
cg                         ellipsoid    rotate, scale, move
cg                         hyperb       rotate, scale, move
cg                         plane        rotate, scale, move, parallel
cg                         quadric      rotate, scale, move
cg                         sphere       rotate, scale, move, concentric
cg
cg                         brick        increment
cg                         tetrahedron  increment
cg                         zone         increment, family

cg    fan                To create a family of planes at equally spaced angles
cg                         around a common axis, use command "plane" with
cg                         option "rotate".

cg    farthest           See "maximum", "distal".

cg    fatal              See commands "undo", "redo", "indo".
cg                         See "error control", "fatal error".

cg    fatal error        See commands "undo", "redo", "indo".
cg                         See "error control".

cg    Fibonacci          See "Fibonacci series".

cg    Fibonacci series   To find the Fibonacci series for any two initial
cg                         integers, use input file fib.mac.  The series ends
cg                         when the ratio of the last two terms converges.

cg    field              See "argument", "delimited", "delimiter",
cg                         "field delimiter", "field width", "vector field".

cg    field delimiter    See "delimiter".

cg    field width        To find the minimum field width for output of floating
cg                         point numbers, see "format".

cg    figures            See "significant figures".

cg    filament           A filament is a single continuous curve in 3-D space,
cg                         either bounded, as in a brick with zero thickness in
cg                         two of the three coordinate directions, or unbounded,
cg                         as in a track.

cg    file               See "files", "i/o", "input file", "output file".

cg    files              The command to execute GEOM is as follows:
cg
cg                         geom [INFILE [OUTFILE]]
cg
cg                         If an input file INFILE is not specified on the
cg                         execution line, the initial input medium is the
cg                         user's terminal.
cg
cg                         You may change the input to a file, or back to the
cg                         user's terminal, at any time, with command "input"
cg                         or "return".  Options allow you to specify the line
cg                         numbers or the initial delimited character strings of
cg                         the first and last lines to be read, and to specify
cg                         that the contents of the input file only be
cg                         displayed, and not executed as commands.
cg
cg                         If an output file OUTFILE is not specified on the
cg                         execution line, the initial output file is
cg                         "geom_hsp".
cg
cg                         You may change the current output file at any time
cg                         with command "output".
cg                         All output sent to the current output file is also
cg                         sent to the user's terminal (usually too fast to
cg                         read, if input is from a file).
cg                         All non-null input lines typed by the user, or read
cg                         from the current input file, are printed in the
cg                         current output file, prefixed by the character ">"
cg                         or "<".  This allows the output file(s) to be easily
cg                         edited into a new input file. See "input".
cg
cg                         A command summary file, "geom.cmd" is printed fore
cg                         each run, and contains all input lines, with aliases
cg                         expanded, except for commands "call", "i", "in",
cg                         "indo", "input", "r", "rd", "read", "redo", "return",
cg                         "rtn", "u" or "undo, which are not needed to produce
cg                         the same output as the original run.
cg
cg                         The files and directories associated with the GEOM
cg                         code are summarized in the file "geom_summ", and
cg                         described in the file "geom_files".

cg    files              See file geom_files in ~edwards/work/geom/doc .

cg    final              See "value".

cg    final value        See "value".

cc    find               A synonym for search.

cg    find               Find means search for in a list, or calculate or
cg                         determine by some other means.
cg                         See commands "dot", "cross", "root", "roots",
cg                         "distance", "intersect", "intcirc", "proximal",
cg                         "track", "side", "where", "area", "volume", "slice",
cg                         "extrema".
cg                         Also see command "list", and all commands that have
cg                         "list" as an option.

cg    first              See "first character".

cg    first character    The first character of any variable name must be a
cg                         lower or upper case letter or "$", "%" or "@".
cg                         When a variable name is used in place of an integer
cg                         of floating point argument, the first character may
cg                         be "-" to indicate use of the negative of the actual
cg                         variable value, but the "-" is removed to obtain the
cg                         actual variable name.  See "arguments", "VARNAME".

cc    fit                An option in command "accelerate", to fit a parabolic
cc                         trajectory to the path of a uniformly accelerated
cc                         particle, given one, two or three points on the path
cc                         at one, two or three times.
cc                         If the initial position, the initial speed (but not
cc                         its vector components), the constant acceleration,
cc                         and a second point on the parabolic trajectory are
cc                         known, use command "accelerate", option "point",
cc                         with any initial velocity vector with the same
cc                         magnitude, to find the vector components of any
cc                         initial velocities that will make the path intersect
cc                         the second point.  None, 1 or 2 may be possible.

cc    fit                An option in commands "axisym", "cone", "cylinder",
cc                         "disk", "ellipsoid", "plane", "polygon",
cc                         "polyhedron", "quadric", "sphere" and "hyperb",
cc                         between the object name and the specification of the
cc                         object.  Usually means to fit the object to the
cc                         specified points and/or axis, etc.

cg    fit                A circle may be fitted to three points.  See "circle".
cg                         A sphere may be fitted to two points, with the
cg                         center on a specified axis.  See "sphere".
cg                         A sphere may be fitted to four points.  See "sphere".
cg                         A circular ellipsoid (an oblate or prolate spheroid)
cg                         may be fitted to two points, if the center and axial
cg                         direction are specified.
cg                         See "ellipsoid".
cg                         An axially symmetric quadric surface may be fitted
cg                         to two points, if the center and axial direction are
cg                         specified.
cg                         See "axisym".
cg                         A linear trajectory may be fitted to two points
cg                         at two times.  See command "accelerate".
cg                         A parabolic trajectory may be fitted to three points
cg                         at three times.  See command "accelerate".

cg    FLAG               Indicates type of entry in geom_base:
cg                         cc     Command
cg                         cd     Directory name.
cg                         cf     Function.
cg                         cg     General information.
cg                         cn     File name.
cg                         cpar   Code parameter.
cg                         cx     Executable file.

cg    flags              See commands "alias", "marker".

cg    flip               See "flip a coin".

cg    flip a coin        To randomly choose between two events:
cg
cg                         To spin the random number sequence first:
cg
cg                         spin random
cg
cg                           Set the random number seed to a new value based on
cg                           the date and clock.  This will make any results
cg                           based on random numbers not reproducible.
cg
cg                         spin N
cg
cg                           Use up the next N random numbers.
cg
cg                         icalc 1 ran 2
cg
cg                           Randomly sample one of the integers 1 and 2.

cg    floating           See "floating point".

cg    floating point     A decimal floating point argument is a delimited
cg                         character string consisting of a plus (optional) or
cg                         minus sign, followed by a mantissa consisting only
cg                         of one or more of the numerals 0-9, and (optionally)
cg                         a decimal point (in any order), followed (optionally)
cg                         by an exponent consisting (optionally) by D, d, E or
cg                         e, followed by a plus (optional) or minus sign,
cg                         followed by one or more of the numerals 0-9.  If
cg                         neither a decimal point or an exponent is present,
cg                         the string may also be interpreted as an integer.
cg                         Any number of meaningless zeroes may precede the
cg                         leftmost nonzero digit of the mantissa, to the left
cg                         of any decimal point or sign.
cg                         Any number of meaningless zeroes may precede the
cg                         leftmost nonzero digit of the exponent.
cg                         On the YANA machines, using 64 bit floating point
cg                         numbers, the mantissa may have a precision of about
cg                         1 part in 10^16 (53 bits).  Exponents may range from
cg                         about -308 to 308.
cg
cg                         Any decimal floating point argument in a command may
cg                         be replaced by an integer or floating point variable,
cg                         or by a symbolic word which, after symbol
cg                         replacement, is an integer or floating point value.
cg
cg                         Before any command is executed, all floating point
cg                         variables used for floating point arguments are
cg                         replaced by their floating point values.
cg
cg                         See "compress", "integer", "significant figures",
cg                         command "hex".

cc    FMULT              In command "variable", option "series", a multiplier
cc                         used to create a series of variables.
cc                         Each variable var(n) is generated from the preceding
cc                         variable var(n-1) by the recursion relation:
cc
cc                         var(n) = FADD + FMULT * var(n-1), n = 2, NUMVAR + 1,
cc
cc                         where var(1) is the value of variable VAR(1).
cc                         May be integer, floating point or an integer or
cc                         floating point variable.

cc    FMULT              In command "brick", option "x", "y", "z", "rcyl",
cc                         "theta", "rsph" or "phi", a multiplier used to create
cc                         a family of bricks.  The ratio of the thickness of
cc                         each brick in the family to that of the preceding
cc                         brick is FMULT, which must be positive.  May be
cc                         integer, floating point or a variable.

cc    foc                A synonym for focus.

cc    focal              A synonym for focus.

cg    focal              See "focus".

cg    focal point        See "focus".

cg    foci               See "focus".

cg    focus              A quadric surface may have one or more foci, or focal
cg                         points, if one of the types:
cg                         parabolic cylinder, hyperbolic cylinder,
cg                         elliptic cylinder, hyperbolic paraboloid,
cg                         elliptic paraboloid, circular paraboloid, or
cg                         ellipsoid.  See a math or geometry handbook for
cg                         definitions and equations.  For a circular cone,
cg                         the focal point is at the vertex.
cg
cg                         For the parabola or parabolic cylinder:
cg                         QY * y + QXX * x^2 = 0, for all z,
cg                         the focus is at x = 0, y = -0.25 * QY / QXX.
cg
cg                         For the parabola or parabolic cylinder:
cg                         QZ * z + QXX * x^2 = 0, for all y,
cg                         the focus is at x = 0, z = -0.25 * QZ / QXX.
cg
cg                         For the hyperbola or hyperbolic cylinder:
cg                         QC + QXX * x^2 + QYY * y^2 = 0, for all z,
cg                         (QXX > 0, QYY < 0), the two foci are at
cg                         x = 0, y = (+/-)sqrt (QC / QXX - QC / QYY), (QC > 0),
cg                         y = 0, x = (+/-)sqrt (QC / QYY - QC / QXX), (QC < 0).
cg
cg                         For the ellipse or elliptic cylinder:
cg                         QC + QXX * x^2 + QYY * y^2 = 0, for all z,
cg                         (QC < 0, QXX => QYY > 0), the two foci are at:
cg                         x = 0, y = (+/-)sqrt (QC / QXX - QC / QYY).
cg
cg                         For the circular or elliptic paraboloid:
cg                         QZ * z + QXX * x^2 + QYY * y^2 = 0,
cg                         (QXX => QYY => 0), the focus is at x = y = 0,
cg                         z = -0.25 * QZ / QYY, in plane x = 0, and
cg                         z = -0.25 * QZ / QXX, in plane y = 0.
cg
cg                         For the hyperboloid of one sheet:
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QC < 0, QXX => QYY > 0, QZZ < 0),
cg                         the two foci are at z = 0,
cg                         y = (+/-)sqrt (QC / QZZ - QC / QYY), in plane x = 0,
cg                         x = (+/-)sqrt (QC / QZZ - QC / QXX), in plane y = 0.
cg
cg                         For the hyperboloid of two sheets:
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QC > 0, QXX => QYY > 0, QZZ < 0),
cg                         the two foci are at x = y = 0,
cg                         z = (+/-)sqrt (QC / QYY - QC / QZZ), in plane x = 0,
cg                         z = (+/-)sqrt (QC / QXX - QC / QZZ), in plane y = 0.
cg
cg                         For the ellipsoid:
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QC < 0, QXX => QYY => QZZ => 0),
cg                         the two foci are at:
cg                         z = (+/-)sqrt (QC / QYY - QC / QZZ), in plane x = 0,
cg                         z = (+/-)sqrt (QC / QXX - QC / QZZ), in plane y = 0,
cg                         y = (+/-)sqrt (QC / QXX - QC / QYY), in plane z = 0.

cc    focus              An option in command "axisym", to fit an axially
cc                          symmetric quadric surface to a specified focal point
cc                          and axis, given two points in the surface.
cc                         Synonyms:  [focus, foc, focal].

cg    form               See "command form".

cg    format             The format to use for output of a floating point
cg                         number, to minimize the space required for a given
cg                         number of significant figures, is AB.C, which may be
cg                         found from:
cg
cg                         x = the value of the floating point number.
cg                         E = exponent of x = int (log10 (abs (x))).
cg                         N = the number of significant figures in output.
cg                         K = 0 for positive x, 1 for negative x.
cg
cg                         E            A     B               C
cg                         ---------    ---   -------------   ---------
cg                         < -3         1pe   N + 6 + K       N - 1
cg                         -3 to 0      f     N + 2 + K - E   N - 1 - E
cg                         0 to N - 1   f     N + 2 + K       N - 1 - E
cg                         N to N + 2   f         3 + K + E   0
cg                         > N + 2      1pe   N + 6 + K       N - 1
cg
cg                         After the output is produced, further reductions may
cg                         be obtained by editing the output file with UNIX text
cg                         editor VI as follows:
cg
cg                         command        purpose
cg                         ------------   --------------------------------------
cg                         :%s/00*E/E/g   eliminate trailing zeros in mantissa
cg                         :%s/E+00//g    eliminate zero exponent
cg                         :%s/E+0*/+/g   minimize characters in positive
cg                                        exponent
cg                         :%s/E-0*/-/g   minimize characters in negative
cg                                        exponent
cg                         :%s/\.\>//g    eliminate final decimal point of
cg                                        integer
cg                         :%s/  */ /g    eliminate multiple blank spaces
cg                         :%s.  *$//g    eliminate trailing blanks at end of
cg                                        line

cg    fraction           See "continued fraction", "fractional dist",
cg                         "rational fraction".

cg    fractional         See "fractional dist".

cg    fractional dist    A useful measure of the local coordinates of a point is
cg                         often the fractional distance from the beginning to
cg                         the end of a line, or the fractional distance from
cg                         one edge of a polygon to an opposite edge, or the
cg                         fractional distance from one face of a polyhedron to
cg                         an opposite face.
cg                         Used in GEOM for bricks, in the sense of dividing
cg                         the brick into fractional volumes.
cg                         See "local coordinates", "vertex weights".

cg    frequency          To randomly sample photon frequencies or energies from
cg                         a Planck spectrum or a Wien spectrum:
cg
cg                         bin BINAME [planck,wien] BTEMP
cg
cg                           Create probability bin BINAME with relative total
cg                           probability PRTOT, with a probability distribution
cg                           of a [Planck, Wien] spectrum at temperature BTEMP.
cg                           The random variable is a frequency XNU (same units
cg                           as BTEMP) from that spectrum, ranging from 0 to
cg                           infinity.  See "Planck spectrum", "Wien spectrum".
cg                           1 K (Kelvin) = 8.617385E-5 eV (+/- 8.5 ppm).
cg
cg                         pdf PDFNAME = BINAME
cg
cg                           Create new pdf PDFNAME, consisting only of the
cg                           probability bin BINAME.
cg
cg                         sample PDFNAME [NSAMP|1]
cg
cg                           Randomly sample from probability distribution
cg                           function (pdf) PDFNAME, to obtain NSAMP values of
cg                           XNU.  If specified, NSAMP must be a positive
cg                           integer or integer variable.  Only the first 26
cg                           samples will be displayed on the user's
cg                           terminal.  All will be displayed in the current
cg                           output file.  If two or more values are sampled,
cg                           the total, the minimum and maximum, the mean and
cg                           the standard deviation are displayed.

cg    frequency          See "frequency ratios".

cg    frequency ratios   For the frequency ratios of an equally-tempered musical
cg                         scale, see "equally-tempered", "scale.mac".

cg    FU                 See "FU, FV, FW".

cc    FU, FV, FW         Fractional distances (on a volume basis) of a point
cc                         between the minimum and maximum coordinates of a
cc                         brick, in the u, v and w coordinate directions,
cc                         which may be (x, y, z), (rcyl, theta, z) or
cc                         (rsph, theta, phi), based on the coordinate system
cc                         used to create the brick.  For example, the centroid
cc                         of a brick is at FU = 0.5, FV = 0.5, FW = 0.5.
cc                         Used in command "point'.

cc    FUNCTION           In command "icalc", indicates a type of mathematical
cc                         operation to be carried out, with two integer
cc                         arguments, in format "M FUNCTION N":.
cc
cc                         FUNCTION | value
cc                         ---------|--------------------------------------
cc                         +        | M + N
cc                         -        | M - N
cc                         *        | M * N
cc                         /        | M / N        (N nonzero)
cc                         ^        | M^N          (N > -1)
cc                         fact     | M * N!       (-1 < N < 20)
cc                         abs      | M * iabs(N)
cc                         ---------|--------------------------------------
cc                         max      | max0  (M, N)
cc                         min      | min0  (M, N)
cc                         mod      | mod   (M, N) (N nonzero)
cc                         sign     | isign (M, N)
cc                         ---------|--------------------------------------
cc                         comp.    | Boolean complement
cc                         .not.    | Boolean not
cc                         .and.    | Boolean intersection
cc                         .nand.   | Boolean nonintersection
cc                         .or.     | Boolean union
cc                         .nor.    | Boolean nor
cc                         .xor.    | Boolean exclusive or
cc                         .xnor.   | Boolean equivalence
cc                         ---------|--------------------------------------
cc                         perm     | M perm N     (permutations, P(M,N))
cc                         comb     | M comb N     (combinations, C(M,N))
cc                         ------------------------------------------------
cc
cc                         Synonyms:  [^, **], [.and., .int.], [.or., .un.]
cc                         [.xnor., .eqv.].

cc    FUNCTION           In command "variable", indicates a type of mathematical
cc                         operation to be carried out, with two numerical
cc                         arguments, which may be integer or floating point
cc                         values, variables, or symbolic words which, after
cc                         symbol replacement, are integer or floating point,
cc                         in format "A FUNCTION B":
cc
cc                         FUNCTION | value           || FUNCTION | value
cc                         ---------|-----------------||----------|-------------
cc                         +        | A + B           || *        | A * B
cc                         -        | A - B           || /        | A / B
cc                         ^        | A^B             || exp      | A*exp(B)
cc                         sqrt     | A*sqrt(B)       || curt     | A*B^(1/3)
cc                         log      | A*log(B)        || log10    | A*log10(B)
cc                         sin      | A*sin(B)        || asin     | A*asin(B)
cc                         cos      | A*cos(B)        || acos     | A*acos(B)
cc                         tangent  | A*tan(B)        || atan     | A*atan(B)
cc                         atan2    | atan2(A,B)      ||
cc                         sinh     | A*sinh(B)       || asinh    | A*atanh(B)
cc                         cosh     | A*cosh(B)       || acosh    | A*acosh(B)
cc                         tanh     | A*tanh(B)       || atanh    | A*atanh(B)
cc                         abs      | A*abs(B)        || sign     | sign(A,B)
cc                         int      | A*aint(B)       || nint     | A*nint(B)
cc                         max      | amax1(A,B)      || min      | amin1(A,B)
cc                         mod      | amod(A,B)       ||          |
cc                         ran      | a random number ||          |
cc                                  | between A and B ||          |
cc                         -----------------------------------------------------
cc
cc                         Synonyms:  [^, **], [tangent, tan].

cg    function           To do simple mathematics, including the use of in-line
cg                         FORTRAN functions, use command "icalc" or "variable".
cg                         See command "math".

cg    function           See "Totient function", "totient function",
cg                         "zeta function".

cg    FV                 See "FU, FV, FW".

cc    FVAR               The numerical value of a floating point variable
cc                         specified with command "variable".  If FVAR is within
cc                         fraction TOL of an integer, the variable will be an
cc                         integer variable.  FVAR itself may be numerical, the
cc                         name of an integer or floating point variable, or a
cc                         symbolic word which, after symbol replacement, is
cc                         integer or floating point.  See "AVAR", "IVAR".

cg    FW                 See "FU, FV, FW".

G-------------------------------------------------------------------------------

cg    games              Command files allowing several games to be played are
cg                        in directory ~/work/geom/test .  These files contain
cg                        instructions on how to use them, and include:
cg
cg                        card.mac       Deals from a deck of cards.
cg                        dice.mac       Rolls dice.
cg                        tictactoe.mac  Plays tic-tac-toe.

cg    GAMMA              The random variable in a relativistic Maxwellian
cg                         probability distribution function is BETA, the ratio
cg                         of the particle velocity, v, to the velocity of
cg                         light, c = 299,792,458 m / s.
cg                         A related variable is the relativistic function
cg                         GAMMA = 1 / sqrt (1 - BETA^2).
cg                         For RATIO << 1, where RATIO is the dimensionless
cg                         ratio of gas temperature to particle rest mass
cg                         energy, the expected value of BETA is approximately
cg                         1.60 * sqrt (RATIO), with a standard deviation of
cg                         approximately 0.67 * sqrt (RATIO), and the expected
cg                         value of BETA^2 is 3 * RATIO.
cg                         For RATIO >> 1, the expected value of BETA is
cg                         approximately 1 - 0.23 / RATIO^2, with a standard
cg                         deviation of approximately 1.26 / RATIO^2,
cg                         the expected value of BETA^2 is approximately
cg                         1 - 0.46 / RATIO^2, the expected value of GAMMA is
cg                         3 * RATIO, with a standard deviation of approximately
cg                         1.73 * RATIO, and the expected value of GAMMA^2 is
cg                         12 * RATIO^2.

cc    gcd                An option in commands "icalc" and "variable", to find
cc                         the greatest common divisor of a pair of numbers.
cc                         For more than two numbers, repeat the command with
cc                         the result and the next number.
cc                         See "common divisor".

cg    GEOM               The code name used in the output file.

cd    geom               Directory for GEOM files.
cd                         Open YANA Cluster:  ~edwards/geom
cd                         Subdirectories:  doc, make, test

cg    geom               See "GEOM execution".

cx    GEOM execution     Execution line command to run GEOM, the code,
cx                         application or executable program described in this
cx                         database.  Although printed as "GEOM" in this
cx                         database, use lower case "geom" to execute.
cx
cx                         In Open YANA Cluster:  ~edwards/work/bin/geom
cx
cx                         Forms include:
cx
cxin                       geom
cxin                       geom tty
cxin                       geom IN_FILE
cxin                       geom IN_FILE geom_hsp
cxin                       geom IN_FILE OUT_FILE
cxin                       geom IN_FILE OUT_FILE > STDOUT
cx
cx                           Execute the GEOM code.  If the input file tty or
cx                           IN_FILE is not specified, it defaults to tty,
cx                           indicating the user's terminal.
cx                           File IN_FILE must not be geom_hsp or geom_cmd, and
cx                           if not tty, it's first line should be command
cx                           "title".
cx
cx                           If IN_FILE is specified, and the output file
cx                           geom_hsp or OUT_FILE is not specified, it defaults
cx                           to geom_hsp.
cx                           File OUT_FILE must not be tty or geom_cmd.
cx
cx                           If the standard output STDOUT is not specified, it
cx                           defaults to the users terminal.  File STDOUT may be
cx                           specified to prevent large amounts of data from
cx                           being displayed on the user's terminal.
cx                           File STDOUT must not be tty, geom_hsp or geom_cmd.
cx
cx                           The various possibilities are described below.
cx
cx                         geom
cx                         geom tty
cx                         geom tty geom_hsp
cx
cx                           Execute the GEOM code, with input from the user's
cx                           terminal, and output to file geom_hsp and to the
cx                           user's terminal, and a command summary to file
cx                           geom_cmd.
cx
cx                           If executing GEOM for the first time, after
cx                           (optionally) specifying the problem title, type
cx                           command "help", then any other desired commands.
cx
cx                         geom IN_FILE
cx                         geom IN_FILE geom_hsp
cx
cx                           Execute the GEOM code, with input from file
cx                           IN_FILE, and output to file geom_hsp and to the
cx                           user's terminal, and a command summary to file
cx                           geom_cmd.
cx
cx                         geom IN_FILE OUT_FILE
cx
cx                           Execute the GEOM code, with input from file
cx                           IN_FILE, and output to file OUT_FILE and to the
cx                           user's terminal, and a command summary to file
cx                           geom_cmd.
cx
cx                         geom IN_FILE OUT_FILE > STDOUT
cx
cx                           Execute the GEOM code, with input from file
cx                           IN_FILE, and output to file OUT_FILE and to file
cx                           STDOUT, and a command summary to file geom_cmd.
cx
cx                         NOTE:  Do NOT use the UNIX execution line symbol
cx                         "<" to redirect the standard input.  You may use
cx                         the UNIX execution line symbol ">" to redirect the
cx                         standard output.  Use ">!" to write over an existing
cx                         file STDOUT.
cx
cx                         NOTE:  input will return to the user's terminal after
cx                         the last line of file IN_FILE, unless GEOM first
cx                         executes command "input", "return" or "end".
cx                         See commands "if", "endif".

cg    geometric          To display, create and use geometric objects, see the
cg                         entries and commands:  point, cluster, mesh, line,
cg                         triangle, polygon, disk, plane, sphere, cylinder,
cg                         cone, ellipsoid, axisym, quadric, polyhedron,
cg                         tetrahedron, brick, zone.

cg    geometric          See "geometric series".

cg    geometric optics   See "ray tracing".

cg    geometric series   A geometric series is one in which the value of each
cg                         term after the first term differs from the preceding
cg                         term by a constant ratio.  See command "ratio".
cg                         To create a family of variables whose values form a
cg                         geometric series:
cg
cg                         variable VAR(1) = VALUE
cg
cg                           Create variable VAR(1), with the value of the first
cg                           term.
cg
cg                         variable VAR(2) series NUMVAR INC VAR(1) 0 FMULT
cg
cg                           Create a series of NUMVAR variables VAR(2), ...,
cg                           with names incremented by INC characters, with each
cg                           value equal to the preceding value multiplied by
cg                           FADD, starting from the value of variable VAR(1).
cg                           See "increment names".
cg
cg                         To create a family of points, planes or quadric
cg                         surfaces in which the distance of each from an
cg                         invariant point, axis or plane forms a geometric
cg                         series, use option "scale" of command "point",
cg                         "plane", "sphere", "cylinder", "cone", "ellipsoid",
cg                         "axisym" or "quadric".

cn    geom_APT           A list of all APT subroutines, with use by the GEOM
cn                         code indicated.  See "calls", "map".

cn    geom_APT.not       A list of APT subroutines not used by GEOM.

cn    geom_base          This file.  Database of commands, key words, and all
cn                         related terminology in GEOM.
cn                         Long name:  ~edwards/work/geom/doc/geom_base
cn
cn                         In Open YANA Cluster:  ~edwards/work/geom/doc
cn
cn                         To display entries while executing GEOM, see
cn                         command "display".

cn    geom_cmd           A file containing all of the input commands needed to
cn                         reproduce the output produced by the current problem,
cn                         with aliases expanded, and with commands "call",
cn                         "goto", "i", "in", "indo", "input", "r", "rd",
cn                         "read", "redo", "return", "rtn", "u" and "undo
cn                         converted to comments, preceded by "! ".
cn                         The latter commands are preceded by the character
cn                         "<" in the current output file, initially geom_hsp.
cn
cn                         May not be specified as an input file on the
cn                         execution line, e.g. "geom geom_cmd" will not work.
cn
cn                         May be used as an input file after execution, using
cn                         command "input", but limited to the input commands
cn                         already written into it.

cn    geom_commands      A list of all GEOM commands, with a brief description
cn                         of their function.
cn                         Long name:  ~edwards/work/geom/doc/geom_commands

cn    geom_define        A list of all of the entries in this file, without
cn                         definitions.

cn    geom_diff.all      An executable script to compare old and new GEOM test
cn                         problem output files, after running geom_mv.hsp.all
cn                         and geom_exe.all.

cn    geom_exe.all       An executable script to run all GEOM test problems,
cn                         after running geom_mv.hsp.all, and before running
cn                         geom_diff.all.

cn    geom_files         A summary of all subroutines and files associated with
cn                         the GEOM code.
cn                         Long name:  ~edwards/work/geom/doc/geom_files

cn    geom_head          A summary of the header sections from all GEOM
cn                         subroutines
cn                         Long name:  ~edwards/work/geom/doc/geom_head

cn    geom_head.make     A file used to create file geom_head.
cn                         Long name:  ~edwards/work/geom/doc/geom_head.make

cn    geom_hsp           The initial output file of problem results.
cn                         See "output file", "geom_cmd".
cn                         Do not specify as the input file on the GEOM
cn                         execution line, e.g., "geom geom_hsp", or during a
cn                         GEOM run, unless the current output file is not
cn                         geom_hsp, and then only to display lines or execute
cn                         lines that have the form of commands.

cn    geom_input         A list of all GEOM code command lines, including all
cn                         options.
cn                         Long name:  ~edwards/work/geom/doc/geom_input

cn    geom_internal      A list of all GEOM internal parameters and variables in
cn                         file store.h, alphabetic and by data groups, with
cn                         definitions.  See command "debug".

cn    geom_log           A log of the development, modifications and maintenance
cn                         of the GEOM code.
cn                         Long name:  ~edwards/work/geom/doc/geom_log

cn    geom_make          A summary of procedures to maintain and update the
cn                         GEOM code, and add new subroutines.
cn                         Long name:  ~edwards/work/geom/make/geom_make

cn    geom_mv.hsp.all    An executable script to rename all GEOM test problem
cn                         output files, by appending the suffix ".old",
cn                         before running geom_exe.all and geom_diff.all.

cn    geom_summ          A summary of the names and locations of the GEOM code
cn                         files.
cn                         Long name:  ~edwards/work/geom/doc/geom_summ

cn    geom_tests         A summary of all command lines in GEOM test files.
cn                         Long name:  ~edwards/work/geom/doc/geom_tests

cg    getting started    See "starting up".

cc    go to              See command "goto".

cg    golden             See "golden ratio".

cg    golden ratio       The "golden ratio" 0.5 * (1.0 + sqrt (5.0)) or
cg                         1.618033988749894848204586834365638117720309189
cg                         Generated as a variable at the beginning of a
cg                         GEOM run.  The roots of x^2 - x - 1 = 0 are
cg                         rgold and 1 - rgold.  Also rgold = 1 + 1 / rgold.
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cc    goto               Command to jump to a new command in the current input
cc                         file, rather than the next command in sequence.
cc                         See command "input".
cc
cc                         Command "goto" relates to objects:  symbol, variable.
cc
cc                         Note:  do not use a "goto" command when input is from
cc                         the user's terminal.
cc
cc                         Note:  do not use a "goto" command to jump into or
cc                         out of a do loop.
cc
cc                         Note:  Do not use the statement separator ";" on the
cc                         same line as this command.
cc
cc                         Forms include:
cc
ccin                       help goto
ccin                       goto [help]
cc
cc                           Display the command options.
cc
ccin                       goto STRING1
cc
cc                           Read the current input file, from the first line
cc                           starting with STRING1 to the end of the file,
cc                           unless another command redirecting input is
cc                           executed before reaching the end of the file.
cc                           Equivalent to the command "input % STRING1 $".
cc
cc                           Comment lines may be used to provide the needed
cc                           character strings for this command.
cc
ccin                       goto LINE1
cc
cc                           Read the current input file, from line LINE1 to the
cc                           end of the file, unless another command
cc                           redirecting input is executed before reaching the
cc                           end of the file.
cc                           Equivalent to the command "input % LINE1 $".
cc
cc                           The line numbers needed for this command may be
cc                           obtained by displaying the input file with
cc                           command "input", option "?", before specifying it
cc                           as an input file, or by viewing it with an editor
cc                           such as VI that displays line numbers.

cc    grad               A synonym for grads, in command "angles".

cg    grads              A unit of measurement of angles.  A full circle has
cg                         400 grads.

cg    greater than       The "greater than" or right arrow character, ">".

cg    greatest           See "greatest common".

cg    greatest common    See "common divisor", "gcd", commands "icalc" and
cg                         "variable".

cx    grep               A UNIX command to search for patterns in files.
cx                         To find all entries containing the character string
cx                         STRING in this file, displaying the results in file
cx                         OUT_FILE:
cx
cx                         grep -p "STRING" geom_base > OUT_FILE
cx
cx                         Many other options are available.
cx                         See "extract entries".

cg    group              See "block", "cluster", "family of objects",
cg                         "group, modular".

cg    group, modular     In arithmetic modulo n, the multiplicative group
cg                         of n is the set of integers in the group
cg                         (1, ..., n -1) that have no common factors with n,
cg                         other than 1.  The size of this group is the
cg                         Euler's totient function for modulo n.
cg                         If all of the integers in the multiplicative group
cg                         for arithmetic modulo n are multiplied by any
cg                         integer in the group, the results modulo n are the
cg                         same set of integers, but possible in a different
cg                         order.

H-------------------------------------------------------------------------------

cc    h                  A synonym for help.  Do not use in command "delimiter".

cc    h                  An option im command "plot axis', to specify the
cc                         coordinate and label of the horizontal plot axis.
cc                         An option in command "plot limits", to specify the
cc                         minimum and maximum values SMIN and SMAX of the
cc                         horizontal axis coordinate.
cc                         An option in command "plot zoom", to specify a
cc                         multiplier ZOOMULT for the zoom factor for the
cc                         horizontal axis.

cg    half-angle         A quadric surface which is two intersecting planes,
cg                         a hyperbolic cylinder, a hyperbolic paraboloid, an
cg                         elliptic cone, a circular cone, a hyperboloid of one
cg                         sheet or a hyperboloid of two sheets, contains, or
cg                         is asymptotic to, straight lines passing through the
cg                         center of symmetry, and at a angle from one of the
cg                         symmetry axes of the surface.  In standard form, the
cg                         half-angles THHALF are given by:
cg
cg                         QZZ = 0, QXX * QYY < 0 (intersecting planes or a
cg                           hyperbolic cylinder):
cg                           From the x' axis toward the y' axis (labeled x'y'):
cg                           THHALF = atan (sqrt (-QXX/QYY))
cg
cg                           From the y' axis toward the x' axis (labeled y'x'):
cg                           THHALF = atan (sqrt (-QYY/QXX))
cg
cg                         QZZ < 0, QXX => QYY > 0:
cg                           The half-angle varies around the z' axis.
cg                           From the z' axis toward the x' axis (labeled z'x'):
cg                           THHALF = atan (sqrt (-QZZ/QXX))
cg
cg                           From the z' axis toward the y' axis (labeled z'y'):
cg                           THHALF = atan (sqrt (-QZZ/QYY))
cg
cg                         QZZ < 0, QXX = QYY > 0 (symmetric around z' axis):
cg                           From the z' axis:
cg                           THHALF = atan (sqrt (-QZZ/QXX))

cg    harmonic           The harmonics of a fundamental musical note are those
cg                         notes whose frequencies are that of the fundamental
cg                         multiplied or divided by an integer.

cg    harmony            Two musical notes are in harmony if their frequencies
cg                         are in the ratio of small whole numbers or differ
cg                         from such a ratio by an amount not easily detected
cg                         by most human ears.

cc    help               An option in commands, to display a help message.
cc                         For any command COMMAND, type "COMMAND help",
cc                         or "help COMMAND".
cc                         Synonyms:  [help, h] (except in command "delimiter").
cc
cc                         Do not name any object "help" or "h", but if you do,
cc                         use command "rename" to rename it, or command
cc                         "delete" to delete it.

cc    help               Command to get help messages.
cc                         See "commands", command "define".
cc
cc                         Forms include:
cc
ccin                       help
cc
cc                           Display the general help message.
cc
ccin                       help all
cc
cc                           Display the help messages for all commands.
cc                           WARNING:  this will display much output.
cc
cc                         help COMMAND
cc                         COMMAND help
cc
cc                           Display the help message for command COMMAND.
cc                           Case is not ignored.
cc
cc                           COMMAND = [accelerate, alias, all, angles, arc,
cc                           area, axisym, bin, bisect, brick, circle, cluster,
cc                           commands, cone, contfr, coord, copy, cross, cute,
cc                           cuts, cylinder, debug, define, delimiter, distance,
cc                           dot, ellipsoid, environment, error, extrema, hex,
cc                           hyperb, icalc, if, increment, indo, input, intcirc,
cc                           invert, iris, kiss, last, line, list, marker, math,
cc                           mcvol, mesh, misc, move, object, operator, output,
cc                           parse, pdf, plane, plot, point, polygon,
cc                           polyhedron, project, prompt, proximal, quadric,
cc                           random, ratio, redo, reflect, rename, repack,
cc                           return, root, rootf, roots, rotate, sample, scale,
cc                           side, slice, solve, search, sort, sphere, spin,
cc                           steiner, symbol, synonym, tables, tetrahedron,
cc                           time, title, tol, trace, track, triangle, trig,
cc                           triple, twist, undo, variable, vector, volume,
cc                           walk, when, where, zone].
cc
cc                         help commands
cc                         commands [help]
cc
cc                           Display an alphabetic list of command words,
cc                           including synonyms.
cc
cc                         help environment
cc                         environment [help]
cc
cc                           Display the help message for environment commands.
cc
cc                         help error
cc                         error [help]
cc
cc                           Display the help message for error control
cc                           commands.
cc
cc                         help math
cc                         math [help]
cc
cc                           Display the help message for mathematics
cc                           commands.
cc
cc                         help misc
cc                         misc [help]
cc
cc                           Display the help message for miscellaneous
cc                           commands.
cc
cc                         help object
cc                         object [help]
cc
cc                           Display the help message for commands that create
cc                           and display objects.
cc
cc                         help random
cc                         random [help]
cc
cc                           Display the help message for Monte Carlo commands.
cc
cc                         define 'ENTRY'
cc
cc                           Display entry ENTRY from geom_base.  Case is
cc                           ignored.  ENTRY may have up to 19 characters.
cc                           If ENTRY is bracketed by ', double any internal '
cc                           inside ENTRY.  If ENTRY is bracketed by ", double
cc                           any internal " inside ENTRY.
cc
cc                         Synonyms:  [", ' (in pairs)], [commands, alph, comm],
cc                         [define, def], [environment, env],
cc                         [error, err], [help, h], [object, obj].
cc                         Note:  do not use synonym "h" in command "delimiter".

cg    Hertz              A frequency unit, 1 / second.
cg                         1 Hz = 4.1356692E-15 eV (+/- 0.3 ppm).
cg                         The frequency of the musical note Concert A, or A4,
cg                         is 440 Hz.  Middle C (C4) is 523.2511306012 Hz.

cc    hex                An option in command "icalc", to set the input mode for
cc                         integers in command "icalc" to hexadecimal.

cc    hex                An option in command "big", to display a big integer
cc                         in hexadecimal.

cc    hex                Command to display the decimal form of one or more
cc                         character strings in hexadecimal floating point
cc                         format, or to display the hexadecimal machine words
cc                         and hexadecimal floating point forms, if any, of one
cc                         or more character strings, decimal constants or
cc                         decimal variables.
cc                         The special meanings of "!" and ";" are ignored.
cc                         See command "base".
cc
cc                         Command "hex" relates to objects:  symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help hex
ccin                       hex [help]
cc
cc                           Display the command options.
cc
ccin                       hex in  STRING1 STRING2 STRING3 ...
cc
cc                           Display any decimal floating point values for the
cc                           hexadecimal integer or hexadecimal floating point
cc                           values STRING1, STRING2, STRING3, ..., and their
cc                           ASCII and hexadecimal machine word formats.
cc
ccin                       hex out STRING1 STRING2 STRING3 ...
cc
cc                           Display any decimal integer, decimal floating point
cc                           and hexadecimal floating point values for the
cc                           character strings, decimal values or decimal
cc                           variables STRING1, STRING2, STRING3, ..., and their
cc                           ASCII and hexadecimal machine word formats.
cc
cc                         Synonyms:  [help, h].

cg    hexadecimal        The hexadecimal form of an integer M, specified in the
cg                         current integer mode (see command "icalc"), may be
cg                         displayed with command:
cg
cg                         icalc M
cg
cg                           Display M in decimal, hexadecimal and octal mode,
cg                           and if command "icalc bin" has been used, in
cg                           binary mode.
cg
cg                         The hexadecimal machine word for any character string
cg                         STRING may be displayed with command:
cg
cg                         hex out STRING
cg
cg                           Display the hexadecimal machine words for STRING1
cg                           and the hexadecimal machine words for its integer
cg                           and floating point equivalences, if any.
cg
cg                         The hexadecimal floating point equivalent for any
cg                         decimal value VALUE may be displayed with
cg                         command:
cg
cg                         hex out VALUE
cg
cg                           Display the hexadecimal machine words for VALUE,
cg                           the hexadecimal machine words for its integer
cg                           and decimal floating point equivalences, if any,
cg                           and its hexadecimal floating point equivalence.
cg
cg                         The decimal integer and/or floating point equivalent
cg                         of any hexadecimal value VALUE may be displayed with
cg                         command:
cg
cg                         hex in VALUE
cg
cg                           Display the decimal integer and/or floating point
cg                           value equivalents to the hexadecimal VALUE, and
cg                           and their machine word forms.

cg    hexahedron         See "cube".

cg    highest            See "maximum".

cg    history            To see a list of previously used commands in the
cg                         current session of GEOM, use command "input",
cg                         options "command" and "?".

cc    HPNAME             The name of a hyperbolic paraboloid.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.
cc                         No hyperbolic paraboloid name may be "+", "-", "all",
cc                         "h", "help", "list" or "thru", begin with "!" or
cc                         contain ";".
cc                         Specified with command "hyperb" or "quadric".
cc                         Also referred to as HPNAME1, HPNAME2, ...
cc
cc                         Hyperbolic paraboloids may appear in commands:
cc                         accelerate, arc, copy, delete, distance, extrema,
cc                         help, invert, last, list, move, operator, point,
cc                         project, proximal, quadric, reflect, rename, repack,
cc                         rotate, scale, search, side, slice, sort, track,
cc                         triple, vector.

cn    hsp                As a suffix, indicates a standard output file.
cn                         The initial output file from GEOM is geom_hsp.
cn                         In Open YANA Cluster:  ~edwards/work/geom/test
cn                         FILENAME.hsp is the output file from problem
cn                         FILENAME.test.

cc    hyperb             Command to display one or all hyperbolic paraboloids,
cc                         or to create the hyperbolic paraboloid containing the
cc                         vertices, edges and center of a specified nonplanar
cc                         quadrangle, or the plane containing a specified
cc                         planar quadrangle.
cc
cc                         Command "hyperb" relates to objects:  hyperb, point,
cc                         quadric, symbol,
cc
cc                         See commands "do" and "enddo", and "do loop use".
cc
cc                         Forms include:
cc
ccin                       help hyperb
ccin                       hyperb [help]
cc
cc                           Display the command options.
cc
ccin                       hyperb [all,list]
cc
cc                           Display all hyperbolic paraboloids (short display).
cc
ccin                       hyperb list HPNAME1 HPNAME2 HPNAME3 ...
cc
cc                           Display hyperbolic paraboloids HPNAME1, HPNAME2,
cc                           HPNAME3, ..., with or without subscripts (short
cc                           display).
cc
ccin                       hyperb HPNAME
cc
cc                           Display hyperbolic paraboloid HPNAME
cc                           (long display).
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         HPNAME to form the name by incrementing or
cc                         decrementing the base name for hyperbolic paraboloids
cc                         and hyperboloids of one sheet.
cc                         See command "last", "increment names".
cc
ccin                       hyperb HPNAME fit PNAME1 PNAME2 PNAME3 PNAME4
cc
cc                           Create the hyperbolic paraboloid that contains the
cc                           vertices, edges and center of the nonplanar
cc                           quadrangle with vertices at points PNAME1, PNAME2,
cc                           PNAME3 and PNAME4.  The center is at the
cc                           intersection of the lines joining the midpoints of
cc                           opposite edges.  If any two points coincide or if
cc                           all the points are coplanar, a plane through the
cc                           points will be created.
cc
cc                         Synonyms:  [help, h], [hyperb, saddle].

cg    hyperbolic         See "hyperbolic cylinder", "hyperbolic funct",
cg                         "hyperbolic paraboloid", "hyperboloid".

cg    hyperbolic cylinder
cg                         A hyperbolic cylinder is a quadric surface of two
cg                         sheets, for which the standard equation, with the
cg                         vertices on the y axis, is:
cg                           1 + QXX*x^2 - |QYY|*y^2 = 0  (QXX>0, QYY<0).
cg                         This is a limit case of a hyperboloid of two sheets.
cg                         To find the distance from a point to a hyperbolic
cg                         cylinder, use command "distance", "proximal" or
cg                         "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and a hyperbolic cylinder, use command "distance".

cc    hyperbolic funct   To do math using the FORTRAN hyperbolic functions
cc                         sinh, cosh, tanh, asinh, acosh and atanh, use
cc                         command "variable".

cg    hyperbolic paraboloid
cg                         A hyperbolic paraboloid is a quadric surface for
cg                         which the standard equation is:
cg                           x^2 - B^2 * y^2 + C * z = 0.
cg                         Every point on the surface of a hyperbolic paraboloid
cg                         is a saddle point.  This means that at every point on
cg                         the surface, there are two intersecting straight
cg                         lines that lie entirely within the surface, and are
cg                         perpendicular to the normal vector at that point.
cg                         The direction vectors of the two lines are:
cg                           V1 = ( B*C, C, 2*B*(B*y - x))
cg                           V2 = (-B*C, C, 2*B*(B*y + x))
cg                         Note that the x and y components of these vectors
cg                         are independent of x and y, so a view along the
cg                         z axis would show a grid of two sets of parallel
cg                         lines, intersecting at a constant angle theta,
cg                         where sin (theta) = (1 - B^2) / (1 + B^2).
cg
cg                         Planes, cylinders and cones are ruled surfaces,
cg                         while hyperboloids of one sheet and hyperbolic
cg                         paraboloids are doubly ruled surfaces.
cg                         To find the direction vector(s) of the embedded lines
cg                         for a particular point on a particular ruled surface,
cg                         use command "arc".
cg
cg                         To find the distance from a point to a hyperbolic
cg                         paraboloid, use command "distance", "proximal" or
cg                         "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and a hyperbolic paraboloid, use command "distance".
cg
cg                         To find the hyperbolic paraboloid that contains the
cg                         four edges of a nonplanar quadrangle, and the center
cg                         point (the average of the four vertices), use
cg                         command "hyperb".
cg
cg                         The following commands relate to hyperbolic
cg                         paraboloids:
cg                         accelerate, arc, copy, debug, delete, distance,
cg                         extrema, help, hyperb, invert, last, list, move,
cg                         operator, point, proximal, quadric, reflect, rename,
cg                         repack, rotate, scale, search, side, slice, sort,
cg                         symbol, synonym, tables, track, triple, vector, zone.

cg    hyperboloid        A hyperboloid is a quadric surface in 3-D space,
cg                         described by an implicit quadric equation.
cg                         Far away from its center, or the origin in standard
cg                         form, a hyperboloid asymptotically approaches the
cg                         shape of a wedge or cone, with zero curvature in at
cg                         least one direction, at a constant angle from an
cg                         axis.
cg
cg                         The standard forms of the implicit equations for
cg                         hyperboloids are as follows (each coefficient must
cg                         have the preceding sign):
cg
cg                         Real intersecting planes:    x^2 - |QYY|*y^2 = 0
cg                         Hyperbolic cylinder:
cg                                              1 + QXX*x^2 - |QYY|*y^2 = 0
cg                         Hyperbolic paraboloid (a saddle point surface):
cg                                           - |QZ|*z + x^2 - |QYY|*y^2 = 0
cg                         Hyperboloid of 1 sheet (a saddle point surface):
cg                                  - 1 + QXX*x^2 + QYY*y^2 - |QZZ|*z^2 = 0
cg                         Hyperboloid of 2 sheets:
cg                                    1 + QXX*x^2 + QYY*y^2 - |QZZ|*z^2 = 0
cg
cg                         Every point on a hyperbolic paraboloid or a
cg                         hyperboloid of one sheet is a saddle point.
cg                         This means that at every point on the surface there
cg                         are two intersecting straight lines that lie entirely
cg                         within the surface.
cg                         Along with planes, cylinders and cones, hyperbolic
cg                         paraboloids and hyperboloids of one sheet are ruled
cg                         surfaces.
cg
cg                         The equations for the direction vectors of these
cg                         lines may be found with command "arc" or as follows.
cg                         For a point P = (PX, PY, PZ) in the surface,
cg                         a straight line has the equation P' = P + f * V,
cg                         where the direction vector V = (VX, VY, VZ).
cg                         Substitute the following into the equation for a
cg                         ruled surface:
cg                         x = PX + f * VX, y = PY + f * VY, z = PZ + f * VZ.
cg                         Collect the coefficients of f and f^2, and set each
cg                         to zero (e.g., for a hyperboloid of one sheet):
cg                           QXX * PX * VX + QYY * PY * VY - |QZZ| * PZ * VZ = 0
cg                           QXX * VX * VX + QYY * VY * VY - |QZZ| * VZ * VZ = 0
cg                         Set VZ = 1, since the vector magnitude is arbitrary.
cg                         Solve the special cases PX = 0, PY = 0 and PZ = 0
cg                         first, to avoid indeterminate cases.  Otherwise,
cg                         solve the equations for VX and VY, by substitution,
cg                         to find the two solutions.  To make the values more
cg                         natural, then multiply each vector component by
cg                           sqrt ((QXX * PX^2 + QYY * PY^2) / |QZZ|).
cg                         To find the particular solution at a specified point
cg                         for a specified ruled surface, use command "arc".
cg
cg                         See "half-angle".
cg
cg                         To find the distance from a point to a hyperboloid,
cg                         use command "distance", "proximal" or "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and a hyperboloid, use command "distance".
cg
cg                         For the hyperboloid with the equation:
cg
cg                           QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg
cg                         the distance D from the center to the surface in the
cg                         direction of any unit vector U = (UX, UY, UZ) is:
cg                         D = sqrt (-QC / (QXX*UX^2 + QYY*UY^2 + QZZ*UZ^2))
cg                         (solutions exist only for non-negative arguments)
cg                         and the corresponding surface point P is:
cg                           P = (X, Y, Z) = (UX * D, UY * D, UZ * D).

cg    hyperboloid arrays
cg                         Hyperbolic paraboloids may be created as arrays with
cg                         subscripted names.  See "subscript",
cg                         "subscripted names", commands "do" and "enddo".

I-------------------------------------------------------------------------------

cc    i                  A synonym for input.

cg    i                  The character used in complex and imaginary numbers to
cg                         represent the square root of -1.  The only complex
cg                         numbers used in GEOM are in the input and output for
cg                         command "roots".

cg    I/O                Input/output.  A component of machine time use.
cg                         See "CPU", "SYS".  See commands "debug", "input",
cg                         "output".

cc    i/o                An option in command "debug", to display GEOM internal
cc                         parameters and variables for input and output files:
cc                         ainputex, ainput, iin, nfilem, nfiles, ainfile,
cc                         ainfbeg, ainfend, infbeg, infend, inline, incycle,
cc                         nflast, ninput, aoutputex, afilout, iout,
cc                         afilcmd, icmd, ncmdm, ncmds, ncharm, acmd.

cg    I1                 See "I1, I2, I3, ...".

cc    I1, I2, I3, ...    In command "indo", integer increments for arguments
cc                         1, 2, 3, ..., respectively, of the command being
cc                         redone.  A "0" means no increment, use original
cc                         argument.  The arguments being incremented may be
cc                         ASCII or integer.  See "increment names".
cc                         I1, I2 and I3 may be integers or integer variables.

cg    I2                 See "I1, I2, I3, ...".

cg    I3                 See "I1, I2, I3, ...".

cc    icalc              Command to do integer arithmetic.  Integers may be
cc                         specified in decimal, hexadecimal or octal mode, as
cc                         determined by the current mode, set by command
cc                         "icalc [dec,hex,oct].  Integer variables may not be
cc                         used.  Use no leading zeros, and use a minus sign for
cc                         negative integers.  See "integer" for limits.
cc                         Results will be displayed in decimal, hexadecimal and
cc                         octal, mode, and if command "icalc binary" was used,
cc                         in binary mode.  Use command "icalc nobin" to turn
cc                         off the binary mode display.
cc
cc                         Use command "variable" to do floating point or
cc                         integer arithmetic and save the results as a
cc                         variable.
cc
cc                         Use command "big" to do arithmetic with big integers
cc                         with up to 1001 digits.
cc
cc                         Command "icalc" relates to objects:  symbol.
cc
cc                         Forms include:
cc
ccin                       help icalc
ccin                       icalc [help]
cc
cc                           Display the command options and integer size
cc                           limits.  Display the input mode (hexadecimal,
cc                           decimal or octal).
cc
ccin                       icalc [octal,dec,hex]
cc
cc                           Set the mode of integer input to icalc to
cc                           [octal, decimal, hexadecimal].
cc                           In any mode, a negative integer is specified by
cc                           prefixing it with a minus sign.
cc
ccin                       icalc [binary,nobin]
cc
cc                           Turn [on, off] the display of the results of icalc
cc                           in binary mode.
cc
ccin                       icalc M
cc
cc                           Interpret integer M, which is specified in the
cc                           current integer input mode (not a variable).
cc
ccin                       icalc M FUNCTION N
cc
cc                           Interpret integers M and N, which are specified in
cc                           the current integer input mode (not variables),
cc                           and find the function of M and N specified by
cc                           FUNCTION, as follows:
cc
cc                           FUNCTION |  value
cc                           ---------|-----------------------------------------
cc                           +        |  M + N
cc                           -        |  M - N
cc                           *        |  M * N
cc                           /        |  M / N         (N nonzero)
cc                           ^        |  M^N           (N > -1, M^N not too big)
cc                           fact     |  M * N!        (-1 < N < 20)
cc                           ---------|-----------------------------------------
cc                           gcd      |  gcd (M, N)    (Greatest common divisor)
cc                           lcm      |  lcm (M, N)    (Least common multiple)
cc                           abs      |  M * iabs(N)
cc                           max      |  max0  (M, N)
cc                           min      |  min0  (M, N)
cc                           mod      |  mod   (M, N)  (N nonzero) (also order
cc                                    |                and inverse of M)
cc                           sign     |  isign (M, N)
cc                           ran      |  randomly from M to N
cc                           ---------|-----------------------------------------
cc                           comp.    |  M * comp. N   (Boolean complement)
cc                           .not.    |  M .not.   N   (Boolean not)
cc                           ---------------------------------------------------
cc                           .and.    |  M .and.   N   (Boolean intersection)
cc                           .nand.   |  M .nand.  N   (Boolean comp. .and.)
cc                           ---------------------------------------------------
cc                           .or.     |  M .or.    N   (Boolean union)
cc                           .nor.    |  M .nor.   N   (Boolean comp. .or.)
cc                           ---------------------------------------------------
cc                           .xor.    |  M .xor.   N   (Boolean exclusive or)
cc                           .xnor.   |  M .xnor.  N   (Boolean equivalence)
cc                           ---------|-----------------------------------------
cc                           perm     |  M perm    N   (permutations, P(M,N))
cc                           comb     |  M comb    N   (combinations, C(M,N))
cc                           ---------------------------------------------------
cc
cc                           Note:  see commands "variable", "base" and "hex".
cc
cc                         Synonyms:  [.and, .int.], [.or., .un],
cc                         [.xnor., .eqv.], [binary, bin], [help, h],
cc                         [octal, oct].

cg    icosahedron        A regular icosahedron has 12 vertices, 30 edges, and
cg                         20 equilateral triangular faces.
cg                         The edge length, face area, volume, radius of
cg                         inscribed sphere, dihedral angle and central edge
cg                         angle, for a circumscribed sphere radius of 1.0 are:
cg
cg                         edge    = 4.0 / sqrt (10.0 + 2.0 * sqrt (5.0))
cg                                 = 1.051462224238
cg                         area    = 0.25 * sqrt (3.0) * edge^2
cg                                 = 0.4787270691637
cg                         volume  = (5.0 / 12.0) * (3.0 + sqrt (5.0)) * edge^3
cg                                 = 2.536150710120
cg                         rinsc   = sqrt ((5.0 + 2.0 * sqrt (5.0)) / 15.0)
cg                                 = 0.7946544722918
cg                         angdih  = acos (-sqrt (5.0) / 3.0)
cg                                 = 138.1896851042
cg                         angcent = acos (1.0 / sqrt (5.0))
cg                                 = 63.43494882292
cg
cg                         See "polyhedron".

cc    id                 A synonym for title.

cg    IDIV1              See "IDIV1, IDIV2, ...".

cc    IDIV1, IDIV2, ...
cc                       In command "crt", a specified set of divisors, which
cc                         produce a specified set of remainders for an unknown
cc                         integer (to be found).

cg    IDIV2              See "IDIV1, IDIV2, ...".

cc    if                 The initial command in an "if-endif" block of input
cc                         lines, which must end with command "endif".
cc                         WARNING:  if any input error occurs in this command,
cc                         the following commands up to the next "endif" command
cc                         will not be executed.
cc
cc                         Command "if" relates to objects:  symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help if
ccin                       if [help]
cc
cc                           Display the options for commands "if" and "endif".
cc
ccin                       if VNAME1 <  VNAME2 [then]
ccin                       if VNAME1 <= VNAME2 [then]
ccin                       if VNAME1 =  VNAME2 [then]
ccin                       if VNAME1 /= VNAME2 [then]
ccin                       if VNAME1 >= VNAME2 [then]
ccin                       if VNAME1 >  VNAME2 [then]
cc
cc                           Execute any commands following this command,
cc                           through the corresponding command "endif", if
cc                           integer or floating point values or variables
cc                           VNAME1 and VNAME2 satisfy the condition VNAME1
cc                           [less than, less than or equal to, equal to,
cc                           not equal to, greater than or equal to,
cc                           greater than] VNAME2, respectively.
cc
cc                           If you want to test a problem parameter that is not
cc                           a variable, you must create VNAME1 variable with
cc                           a value that you can use in this command.
cc                           For example, the coordinate system could be
cc                           indicated by creating a variable ncoord = 1, 2, or
cc                           3 for xyz, cylindrical or spherical coordinates.
cc
ccin                       if VNAME1 null [then]
cc
cc                           Execute any commands following this command,
cc                           through the corresponding command "endif", if
cc                           integer or floating point variable VNAME1 has not
cc                           been created or does not currently exist.
cc                           Useful in a macro, to check input arguments.
cc
cc                         endif
cc
cc                           End the current "if-endif" block of input lines
cc                           (optional if the "if" condition was satisfied).
cc
cc                         Synonyms:  [/=, =/, .ne.], [<, .lt.], [<=, =<, .le.],
cc                         [=, .eq.], [=>, >=, .ge.], [>, .gt.], [help, h].

cg    if-endif           An "if-endif" block of input lines begins with command
cg                         "if" and ends with command "endif".  Command "if"
cg                         specifies a condition, based on the values of two
cg                         numerical constants or variables, for executing any
cg                         commands within the "if-endif" block.
cg
cg                         An number of "if-endif" blocks of input lines may be
cg                         nested, with each such block entirely contained
cg                         withing the next outer block.  This allows the
cg                         specification of multiple conditions for executing
cg                         any commands within the inner blocks.  If any command
cg                         "if" condition fails, no nested "if-endif" blocks
cg                         preceding the corresponding command "endif" will be
cg                         executed.
cg
cg                         Do loops may be used inside "if-endif" blocks, and
cg                         vice versa.  See "do loop".
cg
cg                         A single input file may contain many options,
cg                         each within an if-endif block executed if the
cg                         option flag, set before specifying the input file,
cg                         satisfies the "if" condition.

cg    imaginary          See "i", "complex", "roots", "imaginary quadrics".

cg    imaginary quadrics
cg                       A quadric surface may be imaginary or degenerate.
cg                         A general quadric surface has the equation:
cg                           F(x,y,z) =    QC          +
cg                           QX  * x     + QY  * y     + QZ  * z     +
cg                           QXY * x * y + QYZ * y * z + QZX * z * x +
cg                           QXX * x^2  + QYY * y^2  + QZZ * z^2  = 0
cg
cg                         The equations of all of the imaginary and degenerate
cg                         quadric surfaces are as follows (in standard form,
cg                         all coefficients positive):
cg
cg                         Imaginary parallel planes:        1 + QXX*x^2 = 0
cg                         Imaginary intersecting planes:
cg                           (a straight line)             x^2 + QYY*y^2 = 0
cg                         Imaginary elliptic cylinder:
cg                                                 1 + QXX*x^2 + QYY*y^2 = 0
cg                         Imaginary circular cylinder:
cg                                                 1 + QXX * (x^2 + y^2) = 0
cg                         Imaginary elliptic cone:
cg                           (a point)           x^2 + QYY*y^2 + QZZ*z^2 = 0
cg                         Imaginary circular cone:
cg                           (a point)               x^2 + y^2 + QZZ*z^2 = 0
cg                         Imaginary ellipsoid:
cg                                       1 + QXX*x^2 + QYY*y^2 + QZZ*z^2 = 0
cg                         Imaginary sphere:
cg                                           1 + QXX * (x^2 + y^2 + z^2) = 0
cg
cg                         See "quadric surfaces".

cc    in                 A synonym for input.

cc    in                 An option in command "hex", to display the decimal
cc                         form of one or more character strings in hexadecimal
cc                         floating point format.

cc    INC                In command "do", the increment in the loop variable
cc                         VARNAME for successive passes through the do loop.
cc                         If not specified, defaults to 1.  May be positive
cc                         or negative.  Must be an integer or integer variable.
cc                         Used to increment VARNAME each time command "enddo"
cc                         is executed, until the value of VARNAME is not in the
cc                         range from N1 to N2.

cc    INC                In commands of the form:
cc                         OBJTYPE OBJNAME OPTION NUMOBJ INC (...)
cc                         that create a family of NUMOBJ objects of type
cc                         OBJTYPE, INC is the number of characters to increment
cc                         (make INC negative to decrement) between successive
cc                         names of objects in the family, starting with name
cc                         OBJNAME.  INC must be a nonzero integer or integer
cc                         variable.  The total increment or decrement will be
cc                         INC * (NUMOBJ - 1).
cc
cc                         NUMOBJ may be NUMAX, NUMBR, NUMCONE, NUMCYL, NUMELL,
cc                         NUMPL, NUMPT, NUMQ, NUMSPH, NUMTET, NUMV, NUMVAR or
cc                         NUMZN.
cc
cc                         See "family of objects" for a summary of such
cc                         commands.  See "increment names".

cc    incidence          See "angle of incidence".

cc    inclusive          See "inclusive list".

cg    inclusive list     See "thru".

cc    INCR               In command "symbol", an integer or integer variable,
cc                         whose value is the increment (or decrement, if
cc                         negative) to be made in the ASCII replacement value
cc                         of the symbol or in the ASCII string STRING.
cc                         See "increment names", "subscript".

cc    incr               A synonym for increment.

cg    increment          See "increment args", "increment names",
cg                         command "increment".

cc    increment          An option in command "brick", to create a family of
cc                         bricks by incrementing the coordinates of the
cc                         bounding surfaces in a specified coordinate direction
cc                         from a base brick.
cc
cc                         An option in command "triangle", to create a family
cc                         of triangles by incrementing the names of the
cc                         vertex points of a base triangle.
cc
cc                         An option in command "tetrahedron", to create a
cc                         family of tetrahedrons by incrementing the names of
cc                         the vertex points of a base tetrahedron.
cc
cc                         An option in command "zone", to create a family of
cc                         zones by incrementing the names of the bounding
cc                         surfaces of a base zone.
cc
cc                         An option in command "symbol", to increment the
cc                         ASCII replacement value of a symbol.
cc
cc                         See "increment names".
cc
cc                         These commands can be replaced, using do loops.
cc                         See "do loop use", commands "do", "enddo".

cc    increment          Command to increment (or decrement) character strings
cc                         in the same way object names are incremented (or
cc                         decremented) by other commands.
cc                         This command may be used to check in advance that the
cc                         object names will be created as you wish.
cc                         See "increment names", "family of objects".
cc
cc                         Command "increment" relates to objects:  symbol,
cc                         variable.
cc
cc                         Forms include:
cc
ccin                       help increment
ccin                       increment [help]
cc
cc                           Display the command options.
cc
ccin                       increment STRING
cc
cc                           Display STRING and the result of incrementing
cc                           STRING by one character (or one digit, if STRING
cc                           is an integer or has one or more integer
cc                           subscripts).
cc                           STRING may have no more than 24 characters.
cc                           If STRING is subscripted, only the first subscript
cc                           will be incremented.  Use a do loop, with the
cc                           loop variable as the subscript, to increment
cc                           subscripts in any position.
cc
ccin                       increment STRING [INC|1] [[NUMSTR|2]]
cc
cc                           Display a family of NUMSTR character strings
cc                           created by incrementing (or decrementing, if INC
cc                           is negative) the preceding string INC times,
cc                           starting with string STRING.  INC may be negative
cc                           or positive, not zero.  The total number of
cc                           increments (or decrements) from STRING to the last
cc                           string will be abs (INC) * (NUMSTR - 1).
cc                           These strings are only displayed, not saved, and
cc                           have no effect on any other code objects.
cc                           STRING may have no more than 24 characters.
cc                           If STRING is subscripted, only the first subscript
cc                           will be incremented.  Use a do loop, with the
cc                           loop variable as the subscript, to increment
cc                           subscripts in any position.
cc
cc                         Synonyms:  [increment, incr], [help, h].

cg    increment args     See commands "indo", "redo".
cg
cg                         It may be useful to create a series of floating point
cg                         variables, with names differing be the same number
cg                         of characters, to use as arguments in a command which
cg                         will be repeated with modifications with command
cg                         "indo".  This allows floating point arguments to be
cg                         incremented by making integer increments in the
cg                         floating point variable names.

cg    increment names    Some commands generate one or a sequence of object
cg                         names, by incrementing (or decrementing) a specified
cg                         name a specified number of times.
cg                         See command "symbol", option "increment".
cg
cg                         If the name is an integer, then that integer is
cg                         incremented (or decremented).  If the number of
cg                         digits in the name changes, the length of the name
cg                         will change, but must not exceed 24.
cg
cg                         If the name has one or more integer subscripts NSUB1,
cg                         NSUB2, ..., where NSUBi is either an explicit integer
cg                         or a variable with an integer value, then only the
cg                         first integer is incremented (or decremented).
cg                         If the number of digits in the subscript changes,
cg                         the length of the name will change, but must not
cg                         exceed 24.
cg                         If STRING is subscripted, only the first subscript
cg                         will be incremented.  Use a do loop, with the
cg                         loop variable as the subscript, to increment
cg                         subscripts in any position.
cg
cg                         The following only applies if the name is not an
cg                         integer, and no integer subscript is present.
cg
cg                         A non-integer unsubscripted name is incremented
cg                         (or decremented) by the specified number of
cg                         characters, starting with the rightmost alphanumeric
cg                         character which is in one of the groups 0-9, A-Z or
cg                         a-z, and carrying, as needed, to the left.
cg                         Any other characters are ignored.
cg                         In each character position, the sequence for
cg                         incrementing is 0-9, carry, or A-Z, carry, or
cg                         a-z, carry, and the sequence for decrementing is
cg                         9-0, carry, or Z-A, carry, or z-a, carry.
cg                         Numerals remain numerals and case does not change.
cg                         The mode and length of the name does not change.
cg
cg                         In a non-integer unsubscripted name, an increment
cg                         (decrement) of 10 of a digit is the same as
cg                         incrementing (decrementing) the next character to
cg                         the left by 1, and an increment (decrement) of 26
cg                         of an alphabetic character is the same as
cg                         incrementing (decrementing) the next character to
cg                         the left by 1.  Therefore, to increment a non-integer
cg                         unsubscripted name by K characters, starting to the
cg                         left of the end of the name, the increment should be
cg                         K * 10^ND * 26^NA, where ND is the number of digits
cg                         and NA is the number of alphabetic characters to the
cg                         right of the character to be incremented.
cg
cg                         Unsubscripted names containing no alphanumeric
cg                         characters other than "z", "Z" or "9" can not be
cg                         incremented, and unsubscripted names containing no
cg                         alphanumeric characters other than "a", "A" or "0"
cg                         can not be decremented.
cg
cg                         If the required number of object names can not be
cg                         generated, because the base name can not be
cg                         incremented (or decremented) a sufficient number of
cg                         times, an error message is returned, and the command
cg                         is aborted.
cg
cg                         You must be sure that an incremented (or decremented)
cg                         name is not a disallowed name.  See the entry for the
cg                         particular name for examples.
cg
cg                         Examples:
cg
cg                         INCR
cg                         -1    abc(8)   def(99)   ghi(-11)  jkl(-100)  a9.8
cg                          0    abc(9)   def(100)  ghi(-10)  jkl(-99)   a9.9
cg                          1    abc(10)  def(101)  ghi(-9)   jkl(-98)   b0.0
cg
cg                         More examples:
cg
cg                         INCR
cg                         -100  --fails-  fails  #wd    fails  0N  s23  fails
cg                         -1    0.Z  2-Z  fails  #zy    fails  4J  t22  fails
cg                          0    1.A  3-A  +++    #zz    aaa    4K  t23  a_000
cg                          1    1.B  3-B  fails  fails  aab    4L  t24  a_001
cg                         26    2.A  4-A  fails  fails  aba    5K  t49  a_026
cg
cg                         More examples:
cg
cg                         INCR
cg                         -1    a122z   a001###8$   99   -1   -10  x(5)
cg                          0    a123a   a001###9$   100   0    -9  x(n), n = 6
cg                          1    a123b   a002###0$   101   1    -8  x(7)
cg
cg                         See "increment names".
cg                         See "+", "-", "last", "family of objects".

cg    index              See "mesh", "subscripted names".

cg    indices            See "mesh", "subscripted names".

cc    indo               Command to repeat an input line saved in memory,
cc                         optionally with one or more ASCII or integer
cc                         arguments incremented.  To display input lines saved
cc                         in memory, use command "input command ?".
cc                         This command is useful to correct an error in a
cc                         preceding command or to generate a series of
cc                         commands with only minor changes.  Also see command
cc                         "symbol" for replacing erroneous input data.
cc                         The modified command may not exceed 80 characters.
cc                         See "increment names", commands "undo", "redo".
cc                         To display all input lines saved in memory, use
cc                         command "input command ?".
cc
cc                         Command "indo" relates to objects:  symbol, variable.
cc
cc                         Commands of type "input", "return", "indo", "redo",
cc                         and "undo" are saved only as comments in the command
cc                         summary file geom_cmd, because of the difficulty of
cc                         avoiding ambiguous or infinite input chains.
cc
cc                         Do not use the statement separator ";" on the same
cc                         line as this command.
cc
cc                         Forms include:
cc
ccin                       help indo
ccin                       indo help
cc
cc                           Display the command options.
cc
ccin                       indo [$]
cc
cc                           Repeat the last input line saved in memory, with no
cc                           changes.  Same as command ".".
cc
ccin                       indo -N [I1 I2 I3 ...]
cc
cc                           Redo the Nth preceding input line saved in memory,
cc                           with arguments 1, 2, 3, ..., incremented by
cc                           I1, I2, I3, ...  characters, where I1, I2, I2, ...
cc                           may be integers or integer variables.
cc
ccin                       indo LINE [I1 I2 I3 ...]
cc
cc                           Redo line LINE in the input lines saved in memory,
cc                           with arguments 1, 2, 3, ..., incremented by
cc                           I1, I2, I3, ...  characters, where I1, I2, I2, ...
cc                           may be integers or integer variables.
cc                           LINE must be an integer, not an integer variable.
cc                           A negative value of LINE means the line preceding
cc                           the last input line saved in memory by that many
cc                           lines.
cc
ccin                       indo STRING [I1 I2 I3 ...]
cc
cc                           Redo the last input line saved in memory that
cc                           begins with STRING, with arguments 1, 2, 3, ...,
cc                           incremented by I1, I2, I3, ... characters, where
cc                           I1, I2, I2, ...  may be integers or integer
cc                           variables.
cc
cc                         Note:  for no increment, use "0" or "." for I1, I2,
cc                         ... (all trailing "0" or "." may be omitted).
cc
cc                         Synonyms:  [help, h], [0, .].

cg    inflection         See "inflection point".

cg    inflection point   The inflection points of a polynomial equation of
cg                         order 3 or higher in z = x + i*y,
cg                         P(z) = a0 + a1*z + a2*z^2 + a3*z^3 + a4*z^4 + ...
cg                         are at the roots of the second derivative of the
cg                         equation,
cg                         P''(z) = 1*2*a2 + 2*3*a3*z + 3*4*a4*z^2 + ... = 0.
cg                         See commands "root" and "roots".

cg    INij               A component of the inverse of a tensor operator,
cg                         with i = 1, 3, and j = 1, 3.

cc    inner              A synonym for "dot", in commands "dot" and "variable".
cc                         See "inner product",

cg    inner product      See "dot product".

cg    input              The input medium may be the user's terminal,
cg                         a file or the input lines saved in memory
cg                         from the current session (currently limited to the
cg                         last ncmdm input lines, where ncmdm may be displayed
cg                         with command "tables"), and may be changed as many
cg                         times as desired during a session with commands
cg                         "input" or "return".  The initial input file may be
cg                         specified on the execution line "geom IN_FILE".
cg
cg                         This is a convenient way to use "macros", or files of
cg                         commands to be used more than once, or to repeat a
cg                         block of commands used previously in the same
cg                         session, or when used in an "if-endif" block of input
cg                         lines, to carry out a recursive calculation with one
cg                         or more conditions for ending the calculation.
cg
cg                         If the input is from the user's terminal, it must
cg                         follow a prompt.  See command "prompt".
cg
cg                         Only the first 80 characters of each input line are
cg                         read.  If the input medium is not the user's
cg                         terminal, each non-null input line is echoed at
cg                         the user's terminal, preceded by its source and
cg                         line number.  Each non-null input line is also echoed
cg                         in the output file, preceded by its source and line
cg                         number, and on the same line, preceded by the
cg                         character ">".  This makes it easy to edit the output
cg                         file into a new input file, by removing all lines
cg                         that do not begin with ">", then deleting each
cg                         initial ">" and all trailing blanks.  Input summary
cg                         command file geom_cmd is already such a file.
cg
cg                         The following commands may affect or involve the
cg                         input file:  debug, geom, input, parser, return,
cg                         tables, trace.
cg
cg                         See "execution line", "macro", "redirect input",
cg                         "standard input".

cc    input              Command to display or specify previously used commands
cc                         or the input medium, and the line numbers or initial
cc                         character strings of the first and last lines to be
cc                         displayed (option "?"), or to be read (and executed
cc                         if commands).  Key word "%" indicates the current
cc                         input medium.  See command "goto" for a shorter way.
cc                         Key word "#" indicates the input medium preceding the
cc                         current one.  See command "return".
cc                         Use of "tty" for IN_FILE returns input to the user's
cc                         terminal.
cc
cc                         Command "input" relates to objects:  symbol,
cc                         variable.
cc
cc                         Do not use the statement separator ";" on the same
cc                         line as this command.
cc
cc                         The input medium is restored to the previous input
cc                         medium after executing command "input" with option
cc                         "?", or after the specified last line is reached,
cc                         or after an end-of-file is reached, or when command
cc                         "return" or "input #" is read and executed.
cc
cc                         In the commands below, LINE1 and LINE2 are integers
cc                         or integer variables, or the character "$" to
cc                         indicate the last input line saved in memory, the
cc                         last line of file geom_cmd, or the end-of-file of any
cc                         other file.  STRING1 and STRING2 are delimited
cc                         strings that are not integers or the names of integer
cc                         variables, and are not "$" or "?".
cc
cc                         Commands of type "input", "return", "indo", "redo",
cc                         and "undo" are saved only as comments in the command
cc                         summary file geom_cmd, because of the difficulty of
cc                         avoiding ambiguous or infinite input chains.
cc
cc                         Forms include:
cc
ccin                       help input
ccin                       input help
cc
cc                           Display the current input medium and position,
cc                           and command options.
cc
ccin                       input
cc
cc                           Display the input medium.
cc
cc                         return
cc
cc                           Read the previous (different) input medium.
cc                           If a file, start at the line following the line
cc                           previously read, and use the same specification
cc                           for the last line.  See "linked input", "trace".
cc                           An end-of-file has the same result.
cc                           Does not work if the execution line is
cc                           "geom < IN_FILE").
cc
ccin                       input #
cc
cc                           Same as "return".
cc
cc                         The following commands specify input from file
cc                         IN_FILE, which must not be the current output file,
cc                         but may be the character "%", indicating the current
cc                         input file (used to move to a different line in the
cc                         current input file), or may be file geom_cmd, the
cc                         command summary file (very useful to recover from
cc                         errors or to repeat a procedure with different
cc                         input), or may be any other external file:
cc
ccin                       input IN_FILE $ [?]
cc
cc                           Display the line count + 1 of IN_FILE, or if
cc                           IN_FILE is geom_cmd, read the last line of
cc                           geom_cmd [and display only].
cc
ccin                       input IN_FILE [?]
cc
cc                           Read file IN_FILE [and display only].
cc
ccin                       input IN_FILE LINE1 [LINE2|LINE1] [?]
cc
cc                           Read file IN_FILE, from line LINE1 to line LINE2
cc                           (default LINE1, read only one line) [and display
cc                           only].
cc
ccin                       input IN_FILE STRING1 [STRING2|STRING1] [?]
cc
cc                           Read file IN_FILE, from the first line starting
cc                           with STRING1 to the first line starting with
cc                           STRING2 (default STRING1, read only one line)
cc                           [and display only].  Do not begin STRING1 or
cc                           STRING2 with the character "!".
cc
ccin                       input IN_FILE LINE1 STRING2 [?]
cc
cc                           Read file IN_FILE, from line LINE1 to the first
cc                           line starting with STRING2 [and display only].
cc
ccin                       input IN_FILE STRING1 LINE2 [?]
cc
cc                           Read file IN_FILE, from the first line starting
cc                           with STRING1 to line LINE2 [and display only].
cc
cc                         The following commands specify input from the input
cc                         lines saved in memory from the current session, and
cc                         are the same as using "geom_cmd" for IN_FILE in the
cc                         commands above, except that no more than the last
cc                         ncmdm input lines are saved, where ncmdm may be
cc                         displayed using command "tables" (this is very useful
cc                         to recover from errors or to repeat a procedure with
cc                         different input:
cc
ccin                       input command $ [?]
cc
cc                           Read the last input line saved in memory [and
cc                           display only].
cc
ccin                       input command $
cc
cc                           Read the last input line saved in memory.
cc                           Same as command ".".
cc
ccin                       input command [?]
cc
cc                           Read all input lines saved in memory [and display
cc                           only].  See "input file".
cc
ccin                       input command LINE1 [LINE2|LINE1] [?]
cc
cc                           Read the input lines saved in memory, from line
cc                           LINE1 to line LINE2 (default LINE1, read only one
cc                           line) [and display only].
cc                           A negative value of LINE1 or LINE2 means the line
cc                           preceding the last input line saved in memory by
cc                           that many lines.  For example, to repeat the last
cc                           N + 1 commands, use command "input command -N $".
cc
ccin                       input command STRING1 [STRING2|STRING1] [?]
cc
cc                           Read the input lines saved in memory, from the
cc                           first line starting with STRING1 to the first line
cc                           starting with STRING2 (default STRING1, read only
cc                           one line) [and display only].
cc
ccin                       input command LINE1 STRING2 [?]
cc
cc                           Read the input lines saved in memory, from line
cc                           LINE1 to the first line starting with STRING2
cc                           [and display only].
cc                           A negative value of LINE1 means the line preceding
cc                           the last input line saved in memory by that many
cc                           lines.
cc
ccin                       input command STRING1 LINE2 [?]
cc
cc                           Read the input lines saved in memory, from the
cc                           first line starting with STRING1 to line LINE2
cc                           [and display only].
cc                           A negative value of LINE2 means the line preceding
cc                           the last input line saved in memory by that many
cc                           lines.
cc
cc                         Synonyms:  [command, c, cmd], [help, h],
cc                         [input, call, i, in, r, rd, read], [return, rtn].

cg    input              See "input file", "input line".

cg    input file         An input file for a GEOM run consists of a problem
cg                         title (optional) followed by comments and/or
cg                         commands, which may include specification of other
cg                         input files, and may be created with a text editor,
cg                         either from scratch, or by modifying the command
cg                         summary file geom_cmd produced by a previous GEOM run
cg                         and changing its name, or by modifying an output file
cg                         (e.g. geom_hsp) produced by a previous run.  In the
cg                         later type of file, all commands needed to reproduce
cg                         the run are prifixed with ">" in the first column.
cg
cg                         See "redirect input", "standard input".

cg    input line         An input line is either a blank line, a comment line,
cg                         a command or a response to a query concerning a
cg                         specified output file that already exists.
cg                         Any command may have an appended comment, beginning
cg                         with a field delimiter and the character "!".
cg                         Blank input lines will be ignored.
cg                         No input line may exceed 80 characters.

cg    ins                A synonym for insert.

cg    inscribed          See "inscribed circle", "inscribed sphere".

cg    inscribed circle   When a triangle is created or displayed, the center
cg                         and radius of the inscribed circle is displayed.
cg                         The center is at the intersection of the bisectors of
cg                         the vertex angles.
cg                         For a triangle with vertices A, B and C, edge vectors
cg                         AB, BC and CA, and edge lengths ab, bc and ca,
cg                         the inscribed circle center is
cg                         CI = (bc * A + ca * B + ab * C) / s, where
cg                         s = bc + ca + ab.
cg                         The radius RI = |AB x BC| / s.
cg                         See "circumscribed circle", "centroid",
cg                         "orthocenter", "trig".
cg                         The inscribed circle tangent externally to each of
cg                         three mutually tangent circles may be found with
cg                         command "kiss".

cg    inscribed sphere   A sphere may be inscribed within a regular polyhedron,
cg                         such as a tetrahedron, a cube, an octahedron,
cg                         a dodecahedron or an icosahedron, so that it is
cg                         tangent to every face of the regular polyhedron.
cg                         See "circumscribed sphere".
cg                         Any circumscribed sphere, tangent externally to each
cg                         of four mutually tangent spheres may be found with
cg                         command "kiss".

cc    insert             An option in command "plot", to insert k, l or m
cc                         layers into a mesh.
cc                         Synonyms:  [insert, ins].

cg    inside             To find if a point is inside a volume element (brick,
cg                         tetrahedron or zone), use command "distance" or
cg                         "where".
cg
cg                         To find if a point PNAME is assigned mesh indices,
cg                         use command "point PNAME".
cg
cg                         To find if the points in a cluster CLNAME are
cg                         assigned mesh indices, use command "cluster CLNAME".
cg
cg                         To put all points inside a volume element (brick,
cg                         tetrahedron or zone) into a cluster, use command
cg                         "cluster".
cg
cg                         To put all mesh points into cluster CLNAME, use
cg                         command "cluster CLNAME mesh [all]".
cg
cg                         To put all non-mesh points into cluster CLNAME, use
cg                         command "cluster CLNAME nomesh".
cg
cg                         To put all mesh points in the mesh block into cluster
cg                         CLNAME, use command "cluster CLNAME mesh [all]".

cc    int                A synonym for distance in command "distance".

cc    int                A synonym for intersect in command "quadric".

cc    int                An option in command "variable", argument FUNCTION,
cc                         to indicate the integer function.
cc                         A int B means A * int (B).

cf    int                The integer function.  Example:  y = int (x) means
cf                         that y is an integer, whose magnitude is no larger
cf                         than x, and with the same sign as x.

cc    INT1               See INT1, INT2, ...

cc    INT1, INT2, ...    In command "big", integers, integer variables, or
cc                         ASCII strings containing only the digits 0-9, to
cc                         be put into a big integer.  With certain command
cc                         options, INT1 and INT2 may be the names of big
cc                         integers.

cc    INT2               See INT1, INT2, ...

cc    INT3               See INT1, INT2, ...

cc    intcirc            Command to find any point(s) of intersection of two
cc                         circles in a major plane, the axis and distance
cc                         between the centers, the separation between the
cc                         circles, and the circumference and area of each
cc                         circle.  For plotting purposes, the circles are
cc                         assumed to be in the x-y plane.  See commands
cc                         "circle", "kiss", "plot".
cc
cc                         To make two circles tangent, move one of the circles
cc                         toward (positive) or away from (negative) the other,
cc                         in the direction of the axis, by the amount of the
cc                         separation.  See "separation", commands "circle",
cc                         "kiss".
cc
cc                         Command "intcirc" relates to objects:  symbol,
cc                         variable.
cc
cc                         Forms include:
cc
ccin                       help intcirc
ccin                       intcirc [help]
cc
cc                           Display the command options.
cc
ccin                       intcirc U1 V1 RADIUS1 U2 V2 RADIUS2
cc
cc                           Find any tangency, intersections or coincidence of
cc                           the circle at point (U1, V1) with radius RADIUS1,
cc                           and the circle at point (U2, V2) with radius
cc                           RADIUS2, in the (u, v) plane in any rectangular
cc                           coordinate system.  Also finds the axis and
cc                           distances of separation between the circles, and
cc                           finds the circumference and area of each circle.
cc
cc                         Synonyms:  [help, h].

cg    integer            An integer argument is a delimited character string
cg                         consisting of a plus (optional) or minus sign,
cg                         followed only by the numerals 0 through 7 (octal),
cg                         0 through 9 (decimal), or the characters 0 through 9,
cg                         a through f and A through F (hexadecimal), and
cg                         without a decimal point or exponent.
cg                         Any number of meaningless zeroes may precede the
cg                         leftmost nonzero digit of the integer.
cg
cg                         Integer arithmetic may be done on integers input in
cg                         decimal, hexadecimal or octal mode, using command
cg                         "icalc".  The integer results are not named or saved.
cg                         Maximum positive integer values are as follows:
cg
cg                         YANA machines (64 bits):
cg                         7FFFFFFFFFFFFFFF      hex   (16 hexadecimal digits)
cg                         999999999999999999    dec   (18 decimal digits)
cg                         777777777777777777777 oct   (21 octal digits)

cg    integer            See "integer data", "integer input", "integer names",
cg                         "integer variable", "big integer", command "big".

cg    integer data       Integer data is data which has been assigned type
cg                         Integer in GEOM.  On execution, all such data is
cg                         filled with zeros.  See "data types", command
cg                         "big", "big integer".

cg    integer input      Any integer argument in a command may be replaced by an
cg                         integer variable or a symbolic word which, after
cg                         symbol replacement, is an integer.
cg                         See "value", command "symbol".
cg
cg                         An integer argument is a delimited character string
cg                         that contains only digits and possibly an initial
cg                         "+" or "-".  The first digit may be preceded by any
cg                         number of meaningless zeroes.
cg
cg                         See "floating point".  See "big integer".

cg    integer names      Most objects, excluding variables, may be given
cg                         integer names.  Such names may be sorted,
cg                         incremented or decremented as integers.
cg                         See "increment names".
cg                         Names of big integers are upper case "A" to "Z".
cg                         See "big integer", command "big".

cg    integer variable   An integer variable is a variable of type integer,
cg                         created with command "variable", with a magnitude
cg                         no greater than 10^16.
cg                         Any integer or floating point argument (except for
cg                         LINE in commands "indo" and "redo") may be replaced
cg                         by the name of an integer variable.
cg                         See "big integer", command "big".
cg
cg                         Before a command is executed, all integer variable
cg                         names used for integer or floating point arguments
cg                         are replaced by their integer values.  See "value".
cg
cg                         A floating point variable will be rounded off to an
cg                         integer variable at the time it is created, if the
cg                         fractional amount by which it differs from an integer
cg                         is less than TOL, and it is not bigger than the
cg                         largest machine integer, and does not exceed 10^16.

cg    Inter              In the output display for a quadric surface, indicates
cg                         that the quadric surface is a pair of intersecting
cg                         planes.

cc    inter              A synonym for distance in command "distance".

cc    inter              A synonym for intersect in command "quadric".

cg    intercept          A quadric surface may have intercepts on the x, y and z
cg                         axes.  Any such intercepts are displayed with other
cg                         quadric surface data, for both the original
cg                         (x, y and z) and standard (x', y' and z') forms of
cg                         the quadric surface.
cg
cg                         x axis intercepts are found by setting y = z = 0,
cg                         and solving:  QC + QX * x + QXX * x^2 = 0
cg                         y axis intercepts are found by setting z = x = 0,
cg                         and solving:  QC + QY * y + QYY * y^2 = 0
cg                         z axis intercepts are found by setting x = y = 0,
cg                         and solving:  QC + QZ * z + QZZ * z^2 = 0

cg    internal           Refers to GEOM internal parameters, numerical constants
cg                         and variables.  These are specified in file store.h,
cg                         and listed and defined in file geom_internal, in the
cg                         same directory as the GEOM subroutines.
cg                         Their values may be displayed with command "debug".

cg    internet files     The GEOM and APT files are on web site
cg                         http://nuclear.llnl.gov/CNP/apt/.

cg    interpolation      Linear interpolation of a function G of three variables
cg                         k, l and m, which may be indices or coordinates, may
cg                         be done using a simple general rule.  If G is known
cg                         at the eight vertices of a hexahedron with faces at
cg                         k = k1 and k2, l = l1 and l2, and m = m1 and m2, and
cg                         fk is the fractional distance of k between k1 and k2,
cg                         fl is the fractional distance of l between l1 and l2,
cg                         fm is the fractional distance of m between m1 and m2,
cg
cg                           fk = (k - k1) / (k2 - k1)  (use zero if k2 = k1),
cg                           fl = (l - l1) / (l2 - l1)  (use zero if l2 = l1),
cg                           fm = (m - m1) / (m2 - m1)  (use zero if m2 = m1),
cg
cg                         then the interpolation formula for G(k,l,m) is:
cg
cg                           G(k,l,m) =
cg
cg                           (1 - fk) * (1 - fl) * (1 - fm) * G(k1,l1,m1) +
cg                                fk  * (1 - fl) * (1 - fm) * G(k2,l1,m1) +
cg                           (1 - fk) *      fl  * (1 - fm) * G(k1,l2,m1) +
cg                                fk  *      fl  * (1 - fm) * G(k2,l2,m1) +
cg                           (1 - fk) * (1 - fl) *      fm  * G(k1,l1,m2) +
cg                                fk  * (1 - fl) *      fm  * G(k2,l1,m2) +
cg                           (1 - fk) *      fl  *      fm  * G(k1,l2,m2) +
cg                                fk  *      fl  *      fm  * G(k2,l2,m2)
cg
cg                         For a function G(k,l) of two variables, use fm = 0.
cg                         For a function G(k) of one variable, use fl = fm = 0.
cg                         The formula may be extended to four or more variables
cg                         in an obvious way.

cg    interrupted        See "interrupted run".

cg    interrupted run    An interrupted run may be repeated up to the point of
cg                         interruption if the output file (initially geom_hsp)
cg                         or the command summary file geom_cmd is saved.
cg                         See "input file" to see how to make a new input file.
cg                         End the new input file with command "return" to
cg                         return control of input to your terminal.

cc    intersect          A synonym for distance in command "distance".
cc                         Also see "intersecting planes", "intersection",
cc                         "accelerate".

cc    intersect          An option in command "quadric", to create a cylindrical
cc                         quadric surface perpendicular to a specified plane,
cc                         and through the intersection of that plane with
cc                         another specified quadric surface.
cc                         See command "distance".

cg    intersect          See "intersecting planes", "intersection curve".

cg    intersecting       See "intersecting planes", "intersection curve".

cg    intersecting planes
cg                         A pair of intersecting planes may be represented by a
cg                         single quadric surface for which the standard
cg                         equation is:
cg                           x^2 - |QYY| * y^2 = 0 (QYY < 0),
cg                         which may be factored into the two equations:
cg                           x - sqrt (-QYY) * y = 0 (a simple plane),
cg                           x + sqrt (-QYY) * y = 0 (a simple plane).
cg                         The angle between the two planes is given by:
cg                           theta = arc cos ((1 + QYY) / (1 - QYY)).
cg                         The planes intersect at the origin, and along the
cg                         z axis.
cg
cg                         The general implicit quadric equation for
cg                         intersecting planes is the product of the implicit
cg                         quadric equations for two simple planes:
cg                           (QC1 + QX1 * x + QY1 * y + QZ1 * z) *
cg                           (QC2 + QX2 * x + QY2 * y + QZ2 * z) = 0.
cg
cg                         This is a limit case of a hyperbolic cylinder.
cg                         A pair of imaginary intersecting planes is a quadric
cg                         surface for which the standard equation is:
cg                           x^2 + QYY * y^2 = 0,    (QYY > 0).
cg
cg                         To create such a quadric surface, use command
cg                         "quadric", option "plane".
cg
cg                         A family of intersecting planes may be created by
cg                         rotating a base plane around a specified axis by a
cg                         specified angular increment.  See command "plane".
cg                         See "non-simple planes".

cg    intersection       See "Boolean".

cg    intersection       To find any intersection between two lines in the same
cg                         plane, between a line and the plane of a triangle,
cg                         between a line and a plane, between the plane of a
cg                         triangle and a plane, or between two planes, use
cg                         command "distance".  If the two objects are parallel,
cg                         the distance between them will be displayed.
cg
cg                         To find any intersection between a linear track and
cg                         a point, line, triangle, plane, quadric surface or
cg                         the bounding surfaces of a zone, use command "track".
cg
cg                         To find the intersection of the parabolic trajectory
cg                         of a uniformly accelerated particle with a plane or
cg                         a general quadric surface, use command "accelerate".
cg
cg                         To find any intersection of two circles in a plane,
cg                         use command "intcirc".
cg
cg                         To find the distance and proximal points and/or
cg                         intersection of any combination of planes, spheres
cg                         and circular cylinders, use command "distance".
cg
cg                         To find the distance or intersection between a plane
cg                         and any quadric surface, use commands "distance" and
cg                         "quadric", option "intersect".
cg
cg                         To find any lines of intersection perpendicular to a
cg                         specified axis, in a quadric surface describing
cg                         intersecting planes, use commands "quadric",
cg                         "extrema".
cg
cg                         To find the intersection of a major plane with a
cg                         quadric surface or a zone, use command "slice".
cg
cg                         To find a point on any intersection curve of two
cg                         quadric surfaces, use command "distance"
cg
cg                         To find any 3-way intersection between three planes,
cg                         use command "triple".
cg
cg                         To find 3-way intersections between three quadric
cg                         surfaces, use command "triple".

cg    intersection curve
cg                       The intersection between two surfaces can be imaginary,
cg                         a point of tangency or a curve in 3-D space.
cg                         Any point of the intersection must satisfy the
cg                         equations of both surfaces.
cg                         Any intersection curve must lie in both surfaces
cg                         simultaneously, and must therefore be perpendicular
cg                         to the normal vectors of both surfaces at each
cg                         point on the curve, or equivalently, parallel to the
cg                         cross product of the two normal vectors at the point.
cg                         The differential equation of the intersection curve
cg                         is given by dx / CX = dy / CY = dz / CZ = ds / |C|,
cg                         where vector C = (CX, CY, CZ) is the cross product of
cg                         the two normal vectors, dx, dy and dz are the
cg                         differential components of x, y and z along the
cg                         curve, and ds is the differential component of
cg                         distance along the curve.  The equation of the
cg                         straight line tangent to the intersection curve at
cg                         a point P1 = (X1, Y1, Z1) is obtained by substituting
cg                         (x - X1) for dx, (y - Y1) for dy, (z - Z1) for dz,
cg                         and evaluating CX, CY, and CZ at point P1, in the
cg                         equation above.
cg
cg                         To find the intersection curve between a plane and
cg                         a quadric surface, use commands "distance" and
cg                         "quadric", option "intersect".
cg
cg                         A line can be coincident with another line, can
cg                         intersect a point, line, triangle or plane at a
cg                         point or can intersect a nonplanar quadric surface at
cg                         one or two points.
cg
cg                         A triangle or plane can be coincident with another
cg                         triangle or plane, can intersect a point or line at
cg                         a point, can intersect a triangle or plane on a
cg                         straight line, or can intersect a nonplanar quadric
cg                         surface at a point, on a simple straight line, on two
cg                         coincident, parallel or intersecting straight lines,
cg                         or on a parabola, hyperbola, ellipse or circle.
cg                         A sphere can be coincident with another sphere or
cg                         can intersect a sphere at a point or a circle.
cg
cg                         A general quadric surface can be coincident with
cg                         another general quadric surface, or intersect it at
cg                         one or two points, on one or two straight lines,
cg                         on a circle, ellipse, parabola or hyperbola or on
cg                         higher order nonplanar curves.  However, such an
cg                         intersection curve can not intersect any plane in
cg                         more than four points.

cg    IntPlane           In the output display for a quadric surface, indicates
cg                         that the quadric surface is a pair of intersecting
cg                         planes.

cc    inv                A synonym for invert.

cg    invariant          See "invariant component", "invariant point",
cg                         "invariants of quadric".

cg    invariant component
cg                         The components of a vector are invariant to any
cg                         sequence of translations or rotations which change
cg                         only one coordinate of the coordinate system at a
cg                         time.
cg                         In a rectangular coordinate system, this includes
cg                         all translations.
cg                         In a cylindrical coordinate system, this includes
cg                         translations in the radial or axial direction, and
cg                         rotations around the z axis.
cg                         In a spherical coordinate system, this includes
cg                         translation in the radial direction, rotations around
cg                         the z axis, and rotations that change only the angle
cg                         phi (around an axis perpendicular to the z axis and
cg                         to the position vector of the bound point of the
cg                         vector).

cg    invariant point    In commands "invert", "reflect", "rotate" and "scale",
cg                         an invariant point must be specified.  This is a
cg                         point whose coordinates do not change when the
cg                         operation is done.  For inversion, this is the point
cg                         of inversion.  For reflection, this is any point in
cg                         the reflection plane.  For rotation, this is any
cg                         point on the equivalent single axis of rotation.
cg                         For uniform scaling, this is the center of scaling.
cg                         For radial scaling relative to an axis, this is any
cg                         point on the axis.  For linear scaling relative to a
cg                         plane, this is any point in that plane.

cg    invariants         See "invariants of quadric".

cg    invariants of quadric
cg                         The implicit equation of a quadric surface:
cg
cg                           F(x,y,z) =    QC          +
cg                           QX  * x     + QY  * y     + QZ  * z     +
cg                           QXY * x * y + QYZ * y * z + QZX * z * x +
cg                           QXX * x^2   + QYY * y^2   + QZZ * z^2   = 0
cg
cg                         has the invariants trace, dsum and det, as follows:
cg
cg                           trace = div dot (grad F) = 2.0 * (QXX + QYY + QZZ)
cg                           dsum  = 4.0 * (QXX * QYY + QYY * QZZ + QZZ * QXX) -
cg                                         (QXY^2     + QYZ^2     + QZX^2)
cg                           det   = 2.0 * (4.0 * QXX * QYY * QZZ +
cg                                                QXY * QYZ * QZX -
cg                                   (QXX * QYZ^2 + QYY * QZX^2 + QZZ * QXY^2))
cg
cg                         These invariants do not change as a result of
cg                         translation or rotation of the quadric surface,
cg                         but may be changed by multiplying function F by
cg                         a constant.
cg
cg                         If the quadric surface is aligned with the major
cg                         axes:
cg
cg                           trace = 2.0 * (QXX + QYY + QZZ)
cg                           dsum  = 4.0 * (QXX * QYY + QYY * QZZ + QZZ * QXX)
cg                           det   = 8.0 *  QXX * QYY * QZZ
cg
cg                         Invariant trace is zero for simple planes, and
cg                         nonzero for ellipsoids, including spheres.
cg
cg                         Invariant dsum is zero for simple planes and for
cg                         parabolic cylinders, and nonzero for all ellipsoids,
cg                         including spheres.
cg
cg                         Invariant det is zero for all real quadric surfaces
cg                         except hyperboloids and ellipsoids, including
cg                         spheres.

cg    inverse            See "inverse functions", "inverse, modular".

cg    inverse functions
cg                       See "acos", "acosh", "asin", "asinh", "atan", "atanh",
cg                         and command "variable".

cg    inverse, modular   In arithmetic modulo n,  an integer i has a
cg                         multiplicative inverse ir if i is nonzero, and has no
cg                         common factors with n, and i * ir = 1 mod n.  If the
cg                         order m of i in modulo n is known, then the
cg                         multiplicative inverse ir of i is i^(m-1) mod n.
cg                         If it exists, it is displayed by commands "icalc" and
cg                         "variable", when option "mod" is used.
cg
cg                         In arithmetic modulo n, the multiplicative group
cg                         of n is the set of integers in the group
cg                         (1, ..., n -1) that have no common factors with n,
cg                         other than 1.

cg    inversion          See "inversion point".

cg    inversion point    The inversion points of a polynomial equation in
cg                         z = x + i*y,
cg                         P(z) = a0 + a1*z + a2*z^2 + a3*z^3 + a4*z^4 + ...
cg                         are at the roots of the second derivative of the
cg                         equation,
cg                         P''(z) = 1*2*a2 + 2*3*a3*z + 3*4*a4*z^2 + ...
cg                         See "quadratic", "cubic", "quartic".
cg                         See commands "root" and "roots".

cg    invert             Inversion through the origin replaces each point
cg                         (x, y, z) with (-x, -y, -z).  Inversion through
cg                         invariant point (X0, Y0, Z0) replaces each point
cg                         (x, y, z) with (2*X0 - x, 2*Y0 - y, 2*Z0 - z).

cc    invert             Command to invert points, clusters, vectors or
cc                         quadric surfaces (including planes, spheres,
cc                         circular cylinders, circular cones, hyperbolic
cc                         paraboloids, ellipsoids, axially symmetric quadric
cc                         surfaces and general quadric surfaces), using an
cc                         inversion operator previously created with command
cc                         "operator".  The inversion is through the invariant
cc                         point PINV, which defaults to the origin if not
cc                         specified.  See commands "undo", "redo", "indo".
cc
cc                         Command "invert" relates to objects:  axisym,
cc                         cluster, cone, cylinder, ellipsoid, hyperb, operator,
cc                         plane, point, sphere, symbol, vector.
cc
cc                         Forms include:
cc
ccin                       help invert
ccin                       invert [help]
cc
cc                           Display the command options.
cc
ccin                       invert point PNAME OPNAME [PINV]
cc
cc                           Invert point PNAME with operator OPNAME, through
cc                           the invariant point PINV.
cc
ccin                       invert point all OPNAME [PINV]
cc
cc                           Invert all points with operator OPNAME, through
cc                           the invariant point PINV.
cc
ccin                       invert cluster CLNAME OPNAME [PINV]
cc
cc                           Invert cluster CLNAME with operator OPNAME, through
cc                           the invariant point PINV.
cc
ccin                       invert cluster all OPNAME [PINV]
cc
cc                           Invert all clusters with operator OPNAME, through
cc                           the invariant point PINV.
cc
ccin                       invert vector VNAME OPNAME [PINV]
cc
cc                           Invert vector VNAME with operator OPNAME, through
cc                           the invariant point PINV.
cc
ccin                       invert vector all OPNAME [PINV]
cc
cc                           Invert all vectors with operator OPNAME, through
cc                           the invariant point PINV.
cc
ccin                       invert QTYPE QNAME OPNAME [PINV]
cc
cc                           Invert the quadric surface of type QTYPE (plane,
cc                           sphere, cylinder, cone, hyperb, ellipsoid, axisym,
cc                           quadric) with name QNAME, with operator OPNAME,
cc                           through the invariant point PINV.
cc
ccin                       invert QTYPE all OPNAME [PINV]
cc
cc                           Invert all quadric surfaces of type QTYPE (plane,
cc                           sphere, cylinder, cone, hyperb, ellipsoid, axisym,
cc                           quadric) with operator OPNAME, through the
cc                           invariant point PINV.
cc
cc                         Synonyms:  [cylinder, cyl],
cc                         [ellipsoid, el, ell, ellipse], [help, h],
cc                         [hyperb, saddle], [invert, inv], [plane, pl],
cc                         [point, p, pnt, pt], [quadric, q, quad],
cc                         [sphere, sph], [vector, v, vect].

cc    invert             An option in command "mesh", to invert the k, l or
cc                         m-layers in the mesh block, and to redefine the mesh
cc                         indices of any mesh points in the mesh block
cc                         accordingly.
cc
cc                         Inverting two mesh indices is equivalent to rotating
cc                         the logical mesh 180 degrees around the other index.
cc
cc                         Along with option "exchange" in command "mesh", this
cc                         allows all possible mesh reflections and rotations to
cc                         be done.

cc    invert             An option in command "operator", to create a tensor
cc                         operator to invert all points through the origin.
cc                         Same as a "uniform" or "scale" operator with
cc                         RATIO = -1.
cc                         To invert perpendicularly through an axis, use
cc                         radial scaling with RATIO = -1.
cc                         To invert perpendicularly through a plane, use
cc                         linear scaling with RATIO = -1.

cg    invert             Points, clusters, vectors or quadric surfaces
cg                         (including planes, spheres, circular cylinders,
cg                         circular cones, hyperbolic paraboloids, ellipsoids,
cg                         axially symmetric quadric surfaces and general
cg                         quadric surfaces) may be inverted with command
cg                          "invert".
cg                         Lines, triangles, regular polygons, regular
cg                         polyhedrons and tetrahedrons may be inverted by
cg                         inverting the points used to create them.  This may
cg                         be easier if the points are put into clusters.
cg                         Zones may be inverted by inverting the quadric
cg                         surfaces used to create them.  Annular disks may be
cg                         inverted by redefining them, after inverting the
cg                         center point and normal vector.

cg    INVERTED           In the display for a point, indicates a mesh point in
cg                         a mesh block that has been inverted.

cc    IN_FILE            An input file specified on the GEOM execution line,
cc                         to take the place of the user's terminal, until and
cc                         unless changed later with command "input".

cc    IN_FILE            In command "input", the name of an input file, from
cc                         which to read commands until command "return" or
cc                         command "input #" is read or until an end-of-file
cc                         is reached.
cc                         May have as many characters as will fit on the
cc                         input line (limited to 80 characters).
cc                         May not be "command", as this specifies another
cc                         option.
cc                         The character "%" means the current input file, when
cc                         used in place of IN_FILE.  Used to move to a
cc                         different line in the current input file.

cc    IP1                See "IP1, IP2, ...".

cc    IP1, IP2, ...      In command "triangle", option "increment", the number
cc                         of times to increment (or decrement, if negative) the
cc                         names of the three vertex points of the base triangle
cc                         for each new triangle created.  May be positive, zero
cc                         or negative.  Each must be an integer or integer
cc                         variable.  See "increment names".

cc    IP1, IP2, ...      In command "tetrahedron", option "increment", the
cc                         number of times to increment (or decrement, if
cc                         negative) the names of the four vertex points of the
cc                         base tetrahedron for each new tetrahedron created.
cc                         May be positive, zero or negative.  Each must be an
cc                         integer or integer variable.  See "increment names".

cc    IP2                See "IP1, IP2, ...".

cc    IQ1                See "IQ1, IQ2, ...".

cc    IQ1, IQ2, ...      In command "zone", option "increment", the number of
cc                         times to increment (or decrement, if negative) the
cc                         names of the first, second, ..., bounding surface of
cc                         the base zone for each new zone created.  May be
cc                         positive, zero or negative.  See "increment names".
cc                         Each must be an integer or integer variable.

cc    IQ2                See "IQ1, IQ2, ...".

cc    IQU                See "IQU, IQV, IQW".

cc    IQU, IQV, IQW      In command "zone", the increments between the names of
cc                         successive quadric surfaces in a family.

cc    IQV                See "IQU, IQV, IQW".

cc    IQW                See "IQU, IQV, IQW".
cc                         See "increment names".

cc    IREM1              See "IREM1, IREM2, ...".

cc    IREM1, IREM2, ...
cc                       In command "crt", the remainders resulting from
cc                         dividing an unknown integer (to be found) by a
cc                         specified set of divisors, IDIV1, IDIV2, ..., IDIVN.

cc    IREM2              See "IREM1, IREM2, ...".

cc    iris               Command to find the series equal to a specified decimal
cc                         value, with the series consisting of an integer plus
cc                         a sum of reciprocals of integers.
cc
cc                         Command "iris" relates to objects:  symbol, variable.
cc
ccin                       help iris
ccin                       iris [help]
cc
cc                           Display the command options.
cc
ccin                       iris = VALUE
cc
cc                           Find the coefficients kc(1), kc(2), kc(3), kc(4),
cc                           ... of the series equivalent of VALUE:
cc                           VALUE = kc(1) + sum (n = 2, 3, 4, ...) 1 / kc(n),
cc                           and display them, with the partial sums of the
cc                           series.  The series will end with an accuracy
cc                           determined by TOL or by the biggest machine
cc                           integer.  The coefficients may be positive or
cc                           negative.
cc
cc                         Synonyms:  [help, h].

cg    isotropic          Isotropic means the same in every direction.

cg    iterate            Iterative methods of solution may be required by
cg                         commands "distance", "proximal" and "side", for
cg                         certain kinds of quadric surfaces, to find the
cg                         point on the quadric surface nearest a specified
cg                         point.  The convergence of such methods is based on
cg                         the value of TOL.  See "convergence".

cg    iteration          Iterative calculations may be made with command
cg                         "input" to repeat blocks of input lines
cg                         previously entered from your terminal or in
cg                         a specified input file.  Conditions for ending the
cg                         iteration may be specified with command "if",
cg                         followed by command "return" and command "endif".

cc    IVAL               In command "crt", the integer for which remainders are
cc                         to be found for division by either the first few
cc                         prime integers or by a set of specified integers.

cc    IVAR               In command "variable", the integer value of an
cc                         integer variable.  See "AVAR", "FVAR".
cc                         IVAR itself may be numerical, the name of an integer
cc                         variable, or a symbolic word which, after symbol
cc                         replacement, is an integer.

cc    IZU                See "IZU, IZV, IZW".

cc    IZU, IZV, IZW      In command "zone", the increments between zone names
cc                         in the direction of a family of quadric surfaces.
cc                         See "increment names".
cc
cc                         The extreme increments in ZNAME will be as follows.
cc                         IZU*(NQU - 1).
cc                         For NQV > 1, add
cc                         IZV*(NQV - 1),
cc                         IZU*(NQU - 1) + IZV*(NQV - 1).
cc                         For NQW > 1, add:
cc                         IZW*(NQW - 1),
cc                         IZV*(NQV - 1) + IZW*(NQW - 1),
cc                         IZU*(NQU - 1) + IZW*(NQW - 1),
cc                         IZU*(NQU - 1) + IZV*(NQV - 1) + IZW*(NQW - 1).
cc
cc                         For a continuous sequence of zone names:
cc                         IZU = 1.
cc                         For NQV > 1, make IZV = NQU.
cc                         For NQW > 1, make IZW = NQU * NQV.

cc    IZV                See "IZU, IZV, IZW".

cc    IZW                See "IZU, IZV, IZW".

J-------------------------------------------------------------------------------

cg    J(M)               The Euler totient function of modulo M, equal to the
cg                         number of integers in the set (1, 2, ..., (M - 1))
cg                         that are relatively prime to M (have no common factor
cg                         with M other than 1).

cg    jump               To jump to a command other than the next command in
cg                         sequence, when input is from a file, use command
cg                         "goto".

K-------------------------------------------------------------------------------

cg    K                  Kelvins, a temperature unit.
cg                         1 K = 8.617385E-5 eV (+/- 8.5 ppm).

cc    K                  In commands "mesh" and "point", the index k in a
cc                         logical array with indices (k,l,m).
cc                         K may have a value from 1 to kmax.
cc                         Alternates to indices (k,l,m) are (kl,m) or (klm),
cc                         where
cc                           kl  = k  + (l - 1) * kmax
cc                           klm = kl + (m - 1) * kmax * lmax.

cc    K                  In command "variable", an integer, to be used in the
cc                         equation (K [+, -, *, /, ^] L) mod M.

cg    k                  See "k and l lines".

cg    k and l lines      See k/l-line zoning.

cg    k-layer            All points with a fixed value of the index k, in a
cg                         3-D logical mesh with indices k, l and m.

cg    k-line             All points with a fixed value of the index k, in a
cg                         2-D logical mesh with only one l-layer or m-layer.

cg    k/l                See "k/l-line zoning".

cg    k/l-line           See "k/l-line zoning".

cg    k/l-line zoning    A family of zones symmetric around an axis, and bounded
cg                         by planar, cylindrical or conical surfaces specified
cg                         by k-lines and l-lines, may be constructed as
cg                         follows.
cg
cg                         Use command "plane", "cylinder" and/or "cone" to
cg                         create the surfaces sk(1), sk(2), sk(3), ...,
cg                         determined by the k-lines 1, 2, 3, ..., kmax, and
cg                         the surfaces sl(1), sl(2), sl(3), ..., determined by
cg                         the l-lines 1, 2, 3, ..., lmax.  This may be
cg                         simplified with the various options of creating
cg                         sets of surfaces with a single command.
cg
cg                         Name the zones by incrementing z(0) by j, where
cg                         j = k + kmax * (l - 1), where k and l are the larger
cg                         of the k and l values bounding the zone.  The zone
cg                         name will be indicated below by z(j).
cg                         The values of j at the vertices of the mesh will be
cg                         2 + kmax, 2*kmax, 2 + kmax*(lmax - 1) and kmax*lmax.
cg                         The total number of zones is (kmax - 1)*(lmax - 1).
cg
cg                         Other quantities in parentheses below indicate the
cg                         numerical value of the argument.
cg
cg                         Make the first zone, bounded by k = 1, k = 2, l = 1
cg                         and l = 2, with j = 2 + kmax:
cg
cg                         zone z(2+kmax) = - sk(1) sk(2) - sl(1) sl(2)
cg
cg                         Make the remaining kmax - 2 zones between l = 1 and
cg                         l = 2:
cg
cg                         zone z(3+kmax) = 1 (kmax-2) z(2+kmax) 1 1
cg
cg                         Make the remaining lmax - 2 zones between k = 1 and
cg                         k = 2:
cg
cg                         zone z(2+2*kmax) = (kmax) (lmax-2) z(2+kmax) 0 0 1 1
cg
cg                         Make all of the additional zones by repeating the
cg                         preceding command with arguments 2 and 5 incremented
cg                         by 1:
cg
cg                         alias r = 'indo zone 0 0 1 0 0 1'
cg                         r
cg                         r
cg                         r
cg                         ... (repeat until zone j = kmax * lmax is created)

cc    K1                 In command "mesh", the minimum value of the k index in
cc                         the mesh block.

cc    K2                 In command "mesh", the maximum value of the k index in
cc                         the mesh block.

cg    key                See "key word", "key words".

cg    key word           A key word is the first word of a command or comment
cg                         line or a word to be typed literally in a command.
cg                         See "key words".

cc    key words          The following command names and key words and their
cc                         synonyms are used.  See each entry for details.
cc
cc                         !                            Appended comment
cc                         ", '                         Delimit string (alias)
cc                         ", '                         Delimit text (marker)
cc                         ", '                         Delimit entry (define)
cc                         #, *, /                      Comment (column 1)
cc                         #                            Previous input file
cc                         $                            end-of-file (input)
cc                         $                            Last saved input line
cc                         %                            Current input file
cc                         &                            Add to previous data
cc                         &                            Respecify quadric data
cc                         *                            Multiply (big)
cc                         *                            Multiply (icalc, var, q)
cc                         *, rel                       Multiply length (vector)
cc                         ^, **                        Exponentiate (big)
cc                         ^, **                        Exponentiate (icalc,var)
cc                         +                            Add (big, icalc, var)
cc                         +                            Increment name last used
cc                         +                            Operator, not inverse
cc                         +                            Sign (no delimiter)
cc                         +                            Surface side (zone)
cc                         .                            No change in redo
cc                         -                            Delete (plot)
cc                         -                            Inverse operator
cc                         -                            Negative dir (brick)
cc                         -                            Remove (cl, mesh, pdf)
cc                         -                            Sign (no delimiter)
cc                         -                            Subtract (big)
cc                         -                            Subtract (icalc, var)
cc                         -                            Surface side (zone)
cc                         .                            Repeat preceding command
cc                         .and., .int.                 Boolean intersection
cc                         .nand.                       Boolean nand
cc                         .nxor., .eqv.                Boolean equivalence
cc                         .nor.                        Boolean nor
cc                         .not.                        Boolean not
cc                         .or., .un.                   Boolean or
cc                         .xor.                        Boolean exclusive or
cc                         /                            Divide (big)
cc                         /                            Divide (icalc, var, q)
cc                         /=, .ne.                     Not equal to (if)
cc                         0, zero                      Zero out big integers.
cc                         <,  .lt.                     Less than (if)
cc                         <=, .le.                     Less than or equal to
cc                         =                            Equals, is specified by
cc                         =/, .ne.                     Not equal to (if)
cc                         =,  .eq.                     Equals (if)
cc                         =<, .le.                     Less than or equal to
cc                         =>, .ge.                     Greater than or equal to
cc                         >,  .gt.                     Greater than (if)
cc                         >=, .ge.                     Greater than or equal to
cc                         ;                            Separates commands
cc                         ?                            Sort in random order.
cc                         ?                            List contents (input)
cc                         ?                            Random vector (walk)
cc                         [                            Prefix of optional input
cc                         ]                            Suffix of optional input
cc                         |                            Precedes default value
cc                         abs                          Absolute value function
cc                         abs                          Absolute length
cc                         accelerate, accel            Accelerate particle
cc                         acos                         Arc cosine (variable)
cc                         acosh                        Arc cosh (variable)
cc                         add, sum                     Vector sum
cc                         alias, al, a                 Alias
cc                         all                          All
cc                         alph, commands, comm         Alphabetic command list
cc                         angles, angle, ang           Angle
cc                         arc                          Curvatures of quadric
cc                         area                         Area of a polygon
cc                         arithmetic, arith            Arithmetic data
cc                         array                        Display plot array
cc                         array                        Rename array stem name
cc                         asin                         Arc sine function
cc                         asinh                        Arc sinh function
cc                         atan                         Arc tangent function
cc                         atan2                        Arc tangent (2 args)
cc                         atanh                        Arc tanh function
cc                         avg                          Average of two numbers
cc                         axial                        Relative to an axis
cc                         axial                        Axially dependent twist
cc                         axisym                       Axially symmetric
cc                         base                         Convert number base
cc                         big                          Big integers
cc                         bin                          Probability bin
cc                         binary, bin                  Turn on binary display
cc                         bisect                       Bisect an angle
cc                         block, bl                    Block of logical mesh
cc                         bound                        Bound point of vector
cc                         box                          Sudoku box digits
cc                         brick, br                    Six-faced solid body
cc                         c, C                         Comment
cc                         c...., C....                 Comment
cc                         c++++, C++++                 Comment
cc                         c----, C----                 Comment
cc                         c____, C____                 Comment
cc                         cartesian, cart, Cartesian   Cartesian coordinates
cc                         call, input                  Input medium
cc                         cat                          Concatenate big integers
cc                         cc, CC                       Comment
cc                         center                       Center point of plot
cc                         circle, circ                 Circle
cc                         cluster, cl                  Cluster of points
cc                         column                       Sudoku column digits
cc                         comb                         Combinations, C(M,N)
cc                         command                      Input lines in memory
cc                         commands, comm, alph         Display all commands
cc                         comp.                        Boolean complement
cc                         concentric, conc             Concentric spheres, cyls
cc                         cone                         Circular cone
cc                         contfr, cf                   Continued fractions
cc                         coordinate, coord, cs,
cc                           system, sys                Coordinate system
cc                         copy, cp, duplicate, dup     Copy an object
cc                         cos                          Cosine function
cc                         cosine, cos (project)        Cosine projection
cc                         cosine, cos (vector)         Cosine distribution
cc                         cosh                         Hyperbolic cosine funct
cc                         cross, outer                 Outer vector product
cc                         crt                          Chinese Remainder prob
cc                         curt                         Cube root function
cc                         cut, slice                   Major plane intersection
cc                         cute                         Double triangle size
cc                         cuts                         Cut triangle equally
cc                         cylinder, cyl                Circular cylinder
cc                         cylindrical, cyl             Cylindrical coordinates
cc                         data                         Beginning of data table
cc                         debug                        Display internal varbls
cc                         dec                          Decimal mode
cc                         define, def                  Display geom_base entry
cc                         deg/rad                      Degrees per radian
cc                         degrees, degree, deg         Degrees (0 to 360)
cc                         delete, del, remove, rm      Delete object(s)
cc                         delimiter, delimit, delim    Field delimiter
cc                         disk, disc, dk               Annular disk
cc                         distance, dist, int          Distance between objects
cc                         dms                          Degrees, minutes, sec
cc                         do                           Begin do loop.
cc                         dot, inner                   Dot product (variable)
cc                         dot, inner                   Dot product, angle
cc                         duplicate, dup, copy, cp     Copy an object
cc                         ebase                        Base of natural logs
cc                         ellipsoid, ellipse, ell, el  Ellipsoid
cc                         end, quit, split, exit       End of problem
cc                         enddo                        Repeat or end do loop.
cc                         endif                        End of "if-endif" block
cc                         environment, env, setup      Environment help
cc                         error, err                   Error help
cc                         euler, Euler                 Euler's constant
cc                         event                        Discrete event (bin)
cc                         exchange, exch               Exchange mesh indices
cc                         exp                          Exponential function
cc                         extrema, extr                Extreme points
cc                         fact                         Factorial (big)
cc                         fact                         Factorial (icalc, var)
cc                         fact, totient                Prime factors, totient
cc                         family, fam                  family of objects
cc                         file                         Input and output files
cc                         fit                          Fit to points, axis
cc                         focus, foc, focal            Focal point
cc                         gcm                          Greatest common divisor
cc                         geom_hsp                     Command summary file
cc                         goto                         Jump to new command
cc                         grads, grad                  Angle in grads.
cc                         h                            Horizontal axis (plot)
cc                         help, h                      Display help message
cc                         hex                          Hexadecimal input mode
cc                         hex                          Display hexadecimal
cc                         hyperb, saddle               Surface thru quadrangle
cc                         i/o                          Input and output files
cc                         icalc                        Do integer arithmetic
cc                         id, title                    Display/specify title
cc                         if                           Begin "if-endif" block
cc                         in                           Hex to decimal format
cc                         increment, incr              Increment name or string
cc                         indo                         Increment command args
cc                         inner, dot                   Inner vector product
cc                         input, in, i, read, rd, r    Input medium
cc                         input, call                  Input medium
cc                         insert, ins                  Insert mesh layer
cc                         int                          Integer function (var)
cc                         intcirc                      Intersect'n of 2 circles
cc                         intersect, int, inter (quadric), dist
cc                                                      Intersect'n of 2 objects
cc                         invert, inv                  Invert indices (mesh)
cc                         invert, inv                  Invert through point
cc                         iris                         Series of reciprocals
cc                         kiss                         Tangent circles, spheres
cc                         last                         Display last object
cc                         lcm                          Least common multiple
cc                         limits, lim                  Limits of plot axes
cc                         line, l, ln                  Line
cc                         linear                       Linear interpolation
cc                         linear                       Linear or axial scaling
cc                         linear                       Linear probability (bin)
cc                         list, print                  List objects (list)
cc                         lock, zzz                    Lock GEOM.
cc                         log                          Natural log function
cc                         log10                        Log to base 10 function
cc                         marker, m, mark              Marker and text
cc                         math                         Mathematics help
cc                         max                          Maximum function, value
cc                         maxwell, Maxwell             Maxwellian distribution
cc                         mcvol                        Estimate zone volume
cc                         mean                         Mean value
cc                         mesh                         Mesh
cc                         min                          Minimum function, value
cc                         misc                         Miscellaneous help
cc                         mod                          Modulus function
cc                         move, mv, trans, translate   Move a geometric object
cc                         move, mv, trans, translate   Move a mesh block
cc                         nest                         Nested cones
cc                         new                          New 9 x 9 Sudoku puzzle
cc                         nint                         Nearest integer function
cc                         nobin                        Turn off binary display
cc                         nomesh                       Not assigned to the mesh
cc                         none                         No preset synonyms
cc                         norepl                       Sample:  no replacement
cc                         normal, norm                 Normal vector direction
cc                         normal, norm                 Gaussian distribution
cc                         null                         Not created
cc                         object, obj                  Create/display help
cc                         oct                          Octal mode
cc                         off                          Stop storing plot points
cc                         on                           Start storing plot pnts
cc                         ortho                        Orthogonal quadrics
cc                         operator, op, oper           Tensor operator (3 by 3)
cc                         out                          Decimal to hex format
cc                         outer, cross                 Outer vector product
cc                         outline                      Outline of quadric
cc                         output, out, o, write        Output file
cc                         page                         Page restore in output
cc                         parallel, par                Parallel planes
cc                         parameter, param             Internal parameters
cc                         parse                        Interpret input lines
cc                         path                         Parabolic trajectory
cc                         pdf                          Probability dist funct
cc                         perm                         Sudoku permitted digits
cc                         perm                         Permutations, P(M,N)
cc                         perspective, pers            Perspective projection
cc                         phi                          polar angle direction
cc                         pi                           Math constant pi
cc                         planar                       Rotate plane onto plane
cc                         planck, Planck               Planck spectrum
cc                         plane, pl                    Plane (flat surface)
cc                         plot                         Plot points
cc                         point, p, pnt, pt            Point, fit to points
cc                         polar                        Polar projection
cc                         polygon, pg, polyg           Regular polygon
cc                         polyhedron, ph, polyh        Regular polyhedron
cc                         polynomial, poly             Polynomial function
cc                         power, pow                   Power-law distribution
cc                         prime                        Prime numbers
cc                         project, proj                Projection
cc                         prompt                       Input prompt
cc                         proximal, prox               Find proximal point
cc                         quadric, q, quad             Plane or quadric surface
cc                         quit                         End of problem
cc                         rad/deg                      Radians per degree
cc                         radial                       Radial scaling operator
cc                         radial                       Radially dependent twist
cc                         radians, rad, radian         Radians
cc                         ran                          Randomly sample
cc                         random, ?                    Random sampling
cc                         ratio, rat                   Ratio between terms
cc                         rcyl                         Cylindrical radius
cc                         read, rd, r, input, in, i    Input medium
cc                         rectangular, rect, xyz,      Cartesian coordinates
cc                         redo                         Replace command args
cc                         reflect, refl                Reflect in a plane
cc                         rel, *                       Multiply length (vector)
cc                         relax                        Relax coordinates
cc                         remove, rm, delete, del      Delete object(s)
cc                         rename, sw, switch           Rename object(s)
cc                         return, rtn                  Read previous input file
cc                         repack                       Repack object data
cc                         reset                        Reset Sudoku input
cc                         rgold                        Golden ratio
cc                         root                         Find a root (bi, var)
cc                         root                         Find real roots of polyn
cc                         root                         Find extrema, inflect pt
cc                         root                         Find polynomial coeffs
cc                         rootf                        Factor a polynomial eq
cc                         roots                        Find polynomial coeffs
cc                         roots                        Find roots of polynomial
cc                         rotate, rot                  Rotate an object
cc                         row                          Sudoku row digits
cc                         rsph                         Spherical radius
cc                         sample                       Randomly sample a pdf
cc                         save                         Save Sudoku puzzle input
cc                         scale, sc                    Scale an object
cc                         scale, uniform               Uniform scaling operator
cc                         search, find                 Search for objects
cc                         search, find                 Search for values
cc                         serial                       Sequential rotations
cc                         series                       Series of values
cc                         set, s, variable, var        Variable (scalar)
cc                         side                         Which side of quadric
cc                         side, edge                   Side of triangle
cc                         sign                         Sign function
cc                         sin                          Sine function
cc                         sinh                         Hyperbolic sine function
cc                         size                         Plot array size
cc                         sizes, tables                Object table sizes
cc                         slice, cut                   Major plane intersection
cc                         solve                        Solve 3 equations
cc                         solve                        Solve a Sudoku puzzle
cc                         sort                         Sort into ASCII order
cc                         sphere, sph                  Sphere
cc                         spherical, sph               Spherical coordinates
cc                         spin                         Spin random # sequence
cc                         split                        End of problem
cc                         sqrt                         Square root function
cc                         status, stat                 Display environment
cc                         stddev                       Standard deviation
cc                         steiner, stein               Triangle Steiner vertex
cc                         sudoku, su                   Sudoku puzzle solver
cc                         sum, add                     Vector sum
cc                         switch, sw, rename           Rename object(s)
cc                         symbol, sym, symb            Symbol with replacement
cc                         synonym, syn                 Synonyms
cc                         system, sys, coordinate      Coordinate system
cc                         tables, sizes                Object table sizes
cc                         tag                          Info about variable
cc                         tangent, tan                 Tangent function
cc                         tangent                      Plane tangent to surface
cc                         tangent                      Tangent circles
cc                         tanh                         Hyperbolic tangent
cc                         temp                         Temporary data tables
cc                         test                         Test mesh block geometry
cc                         tetrahedron, tet, tetra      Tetrahedron (4 points)
cc                         then                         End of command "if"
cc                         theta                        Azimuthal angle
cc                         thru                         Inclusive list
cc                         time, t                      Millisec machine time
cc                         time, t                      Time (accelerate)
cc                         title, id                    Display/specify title
cc                         title                        Title of plot
cc                         trace                        Trace return path
cc                         track, trk                   Track from point to obj
cc                         translate, trans, move, mv   Move an object
cc                         triangle, tri                Triangle (3 points)
cc                         trig                         Find triangle parts
cc                         triple                       3 orthogonal vectors
cc                         triple                       triple point
cc                         triple                       triple product
cc                         twist, tw                    Twist points
cc                         types                        Quadric surface types
cc                         undo, u                      Undo preceding commands
cc                         uniform                      Histogram (bin)
cc                         uniform, scale               Uniform scaling operator
cc                         unit                         Vector of unit length
cc                         v                            Vertical axis (plot)
cc                         vacuum                       Vacuum coordinates
cc                         value                        Discrete value (bin)
cc                         variable, var, set, s        Variable (scalar)
cc                         vector, v, vect              Vector (3 components)
cc                         vi                           Display/edit a file
cc                         viewfactor, view             Viewfactor projection
cc                         void                         Unassigned mesh indices
cc                         volume                       volume beteen mesh pts
cc                         volume                       volume of mesh elements
cc                         volume                       Volume of revolution
cc                         walk                         Random walk
cc                         when                         Code date and run date
cc                         where                        See if point in volume
cc                         wien, Wien                   Wien spectrum
cc                         write, wr, w, output         Output file
cc                         x                            X coordinate
cc                         xyz, rectangular, rect       Cartesian coordinates
cc                         y                            Y coordinate
cc                         yes                          OK to delete all objects
cc                         yes                          OK to write over file.
cc                         z                            Z coordinate
cc                         zone, zn, z                  Zone (volume element)
cc                         zoom                         Zoom of plot display

cc    kiss               Command to find the tangent points and center points
cc                         of three tangent circles or four tangent spheres,
cc                         given their radii, and in addition, to find two
cc                         additional circles or spheres tangent to each of the
cc                         initial circles or spheres.
cc
cc                         Command "kiss" relates to objects:  symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help kiss
ccin                       kiss [help]
cc
cc                           Display the command options.
cc
ccin                       kiss RAD1 RAD2 RAD3
cc
cc                           Find the center coordinates of three mutually
cc                           tangent circles with radii RAD1, RAD2 and RAD3.
cc                           A negative radius means that circle contains the
cc                           other two circles.  A very large radius, like
cc                           10^99, approximates a straight line.
cc                           Put the radii in increasing order of magnitude for
cc                           maximum accuracy.
cc                           The first center will be at the origin, the second
cc                           on the x axis, and the third at a positive y value
cc                           in the x-y plane.
cc
cc                           Also find the radii and center coordinates of two
cc                           additional circles, each tangent to the original
cc                           three circles.  See "tangent circles".
cc
cc                           This data is always displayed whenever a triangle
cc                           is displayed, for three tangent circles centered on
cc                           the vertices of the triangle.
cc                           See commands "triangle", "trig".
cc
cc                           This also applies to spheres all centered in the
cc                           x-y plane.
cc
ccin                       kiss RAD1 RAD2 RAD3 RAD4
cc
cc                           Find the center coordinates of four mutually
cc                           tangent spheres with radii RAD1, RAD2, RAD3 and
cc                           RAD4.  A negative radius means that sphere contains
cc                           the other three spheres.  A very large radius, like
cc                           10^99, approximates a plane.  Put the radii in
cc                           increasing order of magnitude for maximum accuracy.
cc                           The first center will be at the origin, the second
cc                           on the x axis, the third at a positive y value in
cc                           the x-y plane, and the fourth at a positive z
cc                           value.
cc
cc                           Also find the radii and center coordinates of two
cc                           additional spheres, each tangent to the original
cc                           four spheres.  See "tangent spheres".
cc
cc                         Synonyms:  [help, h].

cg    kissing            See "kissing circles", "kissing spheres".

cg    kissing circles    See "tangent circles", command "kiss".

cg    kissing spheres    See "tangent spheres", command "kiss".

cg    klm                A mesh point with mesh indices k, l and m, may also
cg                         be identified by a single index klm:
cg                           kl  = k  + (l - 1) * kmax
cg                           klm = kl + (m - 1) * kmax * lmax.

cc    KMAX               In command "mesh", the largest value of the index k in
cc                         a logical array with indices (k,l,m).

cc    KMOVE              In command "mesh move ...", the number of units to
cc                         move the mesh block in the k direction.

L-------------------------------------------------------------------------------

cc    L                  In commands "mesh" and "point", the index l in a
cc                         logical array with indices (k,l,m).
cc                         L may have a value from 1 to lmax.
cc                         Alternates to indices (k,l,m) are (kl,m) or (klm),
cc                         where
cc                           kl  = k  + (l - 1) * kmax
cc                           klm = kl + (m - 1) * kmax * lmax.

cc    L                  In command "variable", am integer to be used to find
cc                         L mod M, and if L and M have no common factor other
cc                         than 1, to find the order and multiplicative inverse
cc                         of L in modulo M.

cc    L                  In command "variable", an integer, to be used in the
cc                         equation (K [+, -, *, /, ^] L) mod M.  In the case
cc                         of option "^", L must not be negative.

cc    l                  A synonym for line.

cg    l-layer            All points with a fixed value of the index l, in a
cg                         3-D logical mesh with indices k, l and m.

cg    l-line             All points with a fixed value of the index l, in a
cg                         2-D logical mesh with only one m-layer or k-layer.

cc    L1                 In command "mesh", the minimum value of the l index in
cc                         the mesh block.

cc    L2                 In command "mesh", the maximum value of the l index in
cc                         the mesh block.

cg    label              A label for a table, plot or plot axis may be created
cg                         with command "marker".
cg
cg                         A descriptive tag may be assigned to a variable.
cg                         with command "variable", option "tag".
cg
cg                         A descriptive tag may be assigned to a big integer.
cg                         with command "big", option "tag".

cc    label              See "statement label".

cg    language           The commands and options in GEOM are based on English
cg                         words.  To create alternates in another language,
cg                         use commands "symbol" and/or "alias".  A group of
cg                         such commands could be put in a file, and command
cg                         "input" used at the beginning of a run to execute
cg                         them.

cg    last               See "last name", command "last".

cc    last               Command to display one or all base names or to create
cc                         the base name for any object type.  The base name is
cc                         either the last name used to create an object or the
cc                         base name assigned by use of command
cc                         "last OBJTYPE [=] OBJNAME"
cc                         described below, whichever occurs later.
cc                         Base names are the names that are incremented or
cc                         decremented by one character when creating, copying
cc                         or renaming an object, when "+" or "-" is used in
cc                         place of the new object name.  Deletion of the last
cc                         object created does not change the base name for that
cc                         object type.  See "increment names".
cc
cc                         Command "last" relates to objects:  alias, axisym,
cc                         big, bin, brick, cluster, cone, cylinder, disk,
cc                         ellipsoid, hyperb, line, marker, operator, pdf,
cc                         plane, point, polygon, polyhedron, quadric, sphere,
cc                         symbol, tetrahedron, triangle, variable, vector,
cc                         zone.
cc
cc                         Forms include:
cc
ccin                       help last
ccin                       last [help]
cc
cc                           Display the command options.
cc
ccin                       last all
cc
cc                           Display the base names for all object types.
cc
ccin                       last OBJTYPE
cc
cc                           Display the base name for objects of type OBJTYPE,
cc                           which may be alias, big, marker, variable, symbol,
cc                           vector, operator, point, cluster, line, triangle,
cc                           polygon, disk, plane, sphere, cylinder, cone,
cc                           axisym, hyperb, ellipsoid, quadric, tetrahedron,
cc                           polyhedron, brick, zone, bin, pdf.
cc
ccin                       last OBJTYPE [=] OBJNAME
cc
cc                           Change the base name for objects of type OBJTYPE to
cc                           OBJNAME.
cc                           See "OBJNAME", "increment names".
cc
ccin                       last OBJTYPE [=] random
cc
cc                           Change the base name for objects of type OBJTYPE
cc                           to a random string of lower case letters, with a
cc                           length equal to the maximum allowed length of the
cc                           object name.  See "OBJNAME".
cc
cc                         Note:  use command "icalc M ran N" to find a random
cc                         integer between M and N.
cc
cc                         Synonyms:  [brick, br], [cluster, cl],
cc                         [cylinder, cyl], [delete, del, remove, rm],
cc                         [disk, disc, dk], [ellipsoid, el, ell, ellipse],
cc                         [help, h], [hyperb, saddle], [line, l, ln],
cc                         [operator, op, oper], [plane, pl],
cc                         [point, p, pnt, pt], [polygon, pg, polyg],
cc                         [polyhedron, ph, polyh], [quadric, q, quad],
cc                         [random, ?], [sphere, sph], [symbol, sym, symb],
cc                         [tetrahedron, tet, tetra], [triangle, tri],
cc                         [variable, s, set, var], [vector, v, vect],
cc                         [zone, z, zn].

cg    last name          The base name for naming objects of a given type is the
cg                         last name used for creating an object of that type,
cg                         or the name specified with command "last", whichever
cg                         occurs later.  All base names may be displayed with
cg                         command "last".
cg                         The next name after or before the base name, in a
cg                         limited ASCII sequence (see "+", "-") may be used in
cg                         a command to create, copy, or rename an object of the
cg                         same type, by using the character "+" or "-" in place
cg                         of the new object name, but not in command "rename"
cc                         with option "array".

cg    LASTWORD           Last word of command "debug subscript LASTWORD".

cg    Lat                See "latus rectum".

cg    Lat rect           See "latus rectum".

cg    Latera             See "latus rectum".

cg    Latera recta       See "latus rectum".

cg    latus              See "latus rectum".

cg    latus rectum       On a parabola, hyperbola or ellipse, the latus rectum
cg                         is a straight line through a focus, perpendicular to
cg                         the straight line from the vertex to the focus, and
cg                         with end points on the curve.  Its length is the
cg                         width of the figure at the height of the focus.
cg
cg                         For the parabola or parabolic cylinder:
cg                         QY * y + QXX * x^2 = 0,
cg                         the length of the latus rectum is abs (QY / QXX).
cg
cg                         For the hyperbola or hyperbolic cylinder:
cg                         QC + QXX * x^2 + QYY * y^2 = 0
cg                         (QXX > 0, QYY < 0),
cg                         the length of the latus rectum is:
cg                          2 * sqrt (-QC * QYY) / QXX, (QC > 0) or
cg                         -2 * sqrt (-QC * QXX) / QYY, (QC < 0).
cg
cg                         For the ellipse or elliptic cylinder:
cg                         QC + QXX * x^2 + QYY * y^2 = 0
cg                         (QC < 0, QXX => QYY > 0),
cg                         the length of the latus rectum is:
cg                         2 * sqrt (-QC * QYY) / QXX.
cg
cg                         For the circular or elliptic paraboloid:
cg                         QZ * z + QXX * x^2 + QYY * y^2 = 0,
cg                         (QXX => QYY > 0),
cg                         the length of the latus rectum is:
cg                         abs (QZ / QYY), in the plane x = 0, and
cg                         abs (QZ / QXX), in the plane y = 0.
cg
cg                         For the hyperboloid of one sheet:
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QC < 0, QXX => QYY > 0, QZZ < 0),
cg                         the length of the latus rectum is:
cg                         -2 * sqrt (-QC * QYY) / QZZ, in the plane x = 0, and
cg
cg                         For the hyperboloid of two sheets:
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0,
cg                         (QC > 0, QXX => QYY > 0, QZZ < 0),
cg                         the length of the latus rectum is:
cg                         2 * sqrt (-QC * QZZ) / QYY, in the plane x = 0, and
cg                         2 * sqrt (-QC * QZZ) / QXX, in the plane y = 0.
cg
cg                         For the ellipsoid:
cg                         QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0
cg                         (QC < 0, QXX => QYY > QZZ > 0),
cg                         the length of the latus rectum is:
cg                         2 * sqrt (QC * QZZ) / QYY, in the plane x = 0
cg                         2 * sqrt (QC * QZZ) / QXX, in the plane y = 0
cg                         2 * sqrt (QC * QYY) / QXX, in the plane z = 0

cg    Law                See "Law of Cosines", "Law of Sines".

cg    Law of Cosines     The Law of Cosines for a plane triangle with edges
cg                         a, b and c, and opposite angles A, B and C is:
cg                         a^2 = b^2 + c^2 - 2*b*c*cos(A)
cg                         b^2 = c^2 + a^2 - 2*c*a*cos(B)
cg                         c^2 = a^2 + b^2 - 2*a*b*cos(C)
cg                         cos(A) = (a^2 - b^2 - c^2) / (2*b*c)
cg                         cos(B) = (b^2 - c^2 - a^2) / (2*c*a)
cg                         cos(C) = (c^2 - a^2 - b^2) / (2*a*b)
cg                         Given three edges or two edges and the included
cg                         angle, the remaining edges and angles may be found.
cg                         See "Law of Sines", "triangle solution", "trig".

cg    Law of Sines       The Law of Sines for a plane triangle with edges
cg                         a, b and c, and opposite angles A, B and C is:
cg                         a / sin(A) = b / sin(B) = c / sin(C)
cg                         Given one edge and the two adjacent angles, the
cg                         remaining angle and the other two edges may be found.
cg                         See "Law of Cosines", triangle solution", "trig".

cc    lcm                An option in commands "icalc" and "variable", to find
cc                         the least common multiplier of a pair of integers.
cc                         For more than two numbers, repeat the command with
cc                         the result and the next number.
cc                         See "common multiplier".

cg    least              See "least common", "minimum".

cg    least common       See "common multiple", "lcm", commands "icalc" and
cg                         "variable".

cg    left               See "left side".

cg    left arrow         The "less than" or left arrow character, "<".

cg    left bracket       The left square bracket, "[" or the left curly
cg                         bracket, "{".

cg    left side          Aliases may be created, so that when a delimited
cg                         character string up to 24 characters long, including
cg                         any subscripts (the left side) is the first word of
cg                         an input line, it is replaced by the specified string
cg                         (the right side).  See command "alias".
cg
cg                         Markers may be created, with a single character
cg                         (the left side), assigned to a text of up to 72
cg                         characters (the right side).

cg    length             The length of a vector specified by two points is
cg                         either (no option, or option "rel" or "*") a multiple
cg                         RELLEN (defaults to 1) of the distance between the
cg                         two points, or (option "abs") an absolute distance
cg                         ABSLEN (defaults to 1), or 1 (option "unit").
cg                         A warning will be displayed if the vector has no
cg                         components larger than TOL.  The length will not
cg                         include any components less than TOL, or less than
cg                         TOL times the length, so may truncate to zero.
cg
cg                         To multiply the length of a vector VNAME by factor
cg                         RATIO, use one of the two equivalent commands:
cg
cg                         vector VNAME rel RATIO
cg                         vector VNAME * RATIO
cg
cg                         The length of a line is the distance between the two
cg                         points that specify the line.  See "TOL".
cg
cg                         To display the length of the edge of a regular
cg                         polygon, use command "polygon".
cg                         To display the length of the edge of a regular
cg                         polyhedron, use command "polyhedron".
cg                         To display the lengths of the edges of a solid
cg                         bounded by surfaces of the coordinate system, use
cg                         command "brick".
cg
cg                         To create a floating point variable VARNAME, with a
cg                         value equal to the length of a vector VNAME, use
cg                         command  "variable VARNAME vector VNAME".

cg    length             See "name length", "length units", "multiply length".

cg    length units       See "conv.mac", "conversion factors", commands
cg                         "operator", "scale".

cg    less than          The "less than" or left arrow character, "<".

cn    libapt.a           The AR library of APT subroutine binary files.
cn                         in Open YANA Cluster:  ~edwards/work/apt/src
cn                         in /users/u47/edwards/apt/src on archives.
cn                         See "aptflibe".

cg    light              The velocity of light is exactly c = 299,792,458 m / s.

cg    light speed        The velocity of light is exactly c = 299,792,458 m / s.

cc    lim                A synonym for limits.

cg    limit              Tolerance limit for numerical truncation.  See "TOL".
cg                         Also see "limit cases", "error estimate".

cg    limit cases        Certain types of quadric surfaces are limit cases of
cg                         other more general quadric surfaces, and may be
cg                         created accidentally when one or more coefficients
cg                         of the implicit quadric equation are very small or
cg                         large relative to other coefficients:
cg
cg                         General Quadric Surface     Limit Cases
cg
cg                         Ellipsoid                   Sphere, elliptic
cg                                                     cylinder,
cg                                                     parallel planes
cg                         Elliptic cylinder           Parallel planes,
cg                                                     circular cylinder
cg                         Elliptic paraboloid         Plane or parabolic
cg                                                     cylinder
cg                         Hyperbolic cylinder         Parallel planes,
cg                                                     intersecting planes
cg                         Hyperboloid of one sheet    Cone, elliptic cylinder,
cg                                                     hyperbolic cylinder
cg                         Hyperboloid of two sheets   Cone, parallel planes,
cg                                                     hyperbolic cylinder
cg                         Hyperbolic paraboloid       Plane, parabolic
cg                                                     cylinder
cg                         Intersecting planes         Coincident planes
cg                         Parabolic cylinder          Simple plane,
cg                                                     coincident planes
cg                         Parallel planes             Coincident planes

cg    limits             For the maximum numbers of various objects, see
cg                         "maximum number" or use command "tables".

cc    limits             An option in command "plot", to display the plot axis
cc                         limits or specify the horizontal or vertical plot
cc                         axis limits.
cc                         Synonyms:  [limits, lim].

cc    LINE               In commands "redo" and "indo", the number of the
cc                         input line saved in memory to be redone, after
cc                         optionally replacing or incrementing arguments.
cc                         Must be an integer, not an integer variable.
cc                         See "incrementing names".

cg    line               To create a family of points equally spaced in a
cg                         straight line, use command "point", option "move".

cg    line               A geometric object bounded by the two points used to
cg                         specify it, and having the shortest path between the
cg                         two points.  If a point is moved, the directions and
cg                         lengths of all lines bounded by that point change.
cg                         A point may not be deleted while a line exists that
cg                         is specified by that point.  However, changing the
cg                         name of such a point will not change the line.
cg                         A line may be used to create a vector parallel to
cg                         the line.
cg                         The maximum number of lines is now 1000.
cg
cg                         Lines may be sorted by name with command "sort".
cg
cg                         The distance between a straight line and a point may
cg                         be found with command "distance".
cg
cg                         The distance between a straight line and a the
cg                         parabolic trajectory of a uniformly accelerated
cg                         particle may be found with command "accelerate".
cg
cg                         The intersection between a straight line and a plane
cg                         or other nonplanar quadric surface may be found with
cg                         command "track".
cg
cg                         At any point on some quadric surfaces (all planes,
cg                         cylinders, hyperbolic paraboloids, and hyperboloids
cg                         of one sheet) one or more straight lines can be
cg                         drawn, lying entirely on the surface.  See commands
cg                         "hyperb", "arc".
cg
cg                         The following commands relate to lines:
cg                         accelerate, cone, copy, debug, delete, distance,
cg                         help, last, line, list, rename, repack, search, sort,
cg                         symbol, synonym, tables, track, vector.

cc    line               An option in command "debug", to display GEOM internal
cc                         parameters and variables for lines: nlinem, nlines,
cc                         aline, alipt1, lalipt1, alipt2, lalipt2,
cc                         alines, lalines.
cc                         Synonyms:  [line, l, ln].

cc    line               Command to display one or more lines or to create a
cc                         line.  Creating a line replaces any existing line
cc                         having the same name.  The display for a line
cc                         includes the names and coordinates of the end points,
cc                         the length, and the components of the length.
cc                         See "do loop use".
cc
cc                         Command "line" relates to objects:  line, point,
cc                         symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help line
ccin                       line [help]
cc
cc                           Display the command options.
cc
ccin                       line [all,list]
cc
cc                           Display all lines.
cc
ccin                       line list ALNAME1 ALNAME2 ALNAME3 ...
cc
cc                           Display lines ALNAME1, ALNAME2, ALNAME3, ..., with
cc                           or without subscripts, with points, directions and
cc                           lengths.
cc
ccin                       line ALNAME
cc
cc                            Display line ALNAME, with points, direction and
cc                            length.
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         ALNAME to form the name by incrementing or
cc                         decrementing the base name for lines.
cc                         See command "last", "incrementing names".
cc
ccin                       line ALNAME = PNAME1 PNAME2
cc
cc                           Create line ALNAME, from point PNAME1 to point
cc                           PNAME2.  Any later change in the names or
cc                           coordinates of points PNAME1 or PNAME2 will also
cc                           change the definition of line ALNAME.
cc                           This command could be inside a do loop, using
cc                           subscripted names, to create a family of lines.
cc
ccin                       line LINE(2) incr NUMLINE INC LINE(1) [IP1 [IP2]]
cc
cc                           Create a family of NUMLINE lines LINE(2), ..., with
cc                           names incremented by INC characters, by
cc                           incrementing the names of the two end points of the
cc                           preceding line by the number of characters
cc                           indicated by the integers IP1, IP2, starting from
cc                           line LINE(1).  Final arguments of "0" may be
cc                           omitted.  See "increment names", "incrementing
cc                           names".
cc
cc                           This command can be replaced, using a do loop.
cc                           See "do loop use", commands "do", "enddo".
cc
cc                         Synonyms:  [increment, incr], [help, h],
cc                         [line, l, ln].

cc    line               An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more lines.
cc                         Synonyms:  [line, l, ln].

cg    line               See "line element", "line length".

cg    line arrays        Lines may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    line element       A line element in the mesh is a pair of point elements
cg                         with the same value of two of the three mesh indices,
cg                         and two adjacent values of the other mesh index.
cg                         See "point element", "surface element",
cg                         "volume element".

cg    line length        All input lines are restricted to 80 characters or
cg                         less.  This includes input lines that have been
cg                         expanded by use of an alias or commands "indo" or
cg                         "redo".

cc    LINE1              In command "input", the number of the initial line to
cc                         be read in a specified input file or in the input
cc                         lines saved in memory.  LINE1 must be an integer or
cc                         an integer variable.
cc                         When used with option "command", a negative value
cc                         of LINE1 means the line preceding the last
cc                         input line saved in memory by that many lines.
cc                         When used with option "command", may not be less than
cc                         the lowest command number not written over by a new
cc                         command when the total number of input lines saved in
cc                         memory exceeds the storage space, and wraps around.

cc    LINE2              In command "input", the number of the final line to be
cc                         read in a specified input file or in the input lines
cc                         saved in memory.  LINE2 must be an integer or an
cc                         integer variable.
cc                         When used with option "command", a negative value
cc                         of LINE2 means the line preceding the last
cc                         input line saved in memory by that many lines.
cc                         If not specified, LINE2 defaults to LINE1.  If LINE2
cc                         is the same as LINE1, a single line will be read.

cc    linear             An option in command "bin", to create a probability bin
cc                         with a linear probability distribution.

cc    linear             An option in command "mesh", to linearly interpolate
cc                         the x, y, z coordinates of all mesh points in a
cc                         mesh block, based on the vertex values.

cc    linear             An option in command "operator", to create a tensor
cc                         operator that performs a linear scaling in the
cc                         direction of a specified vector VNAME, by a specified
cc                         scale factor RATIO.  When RATIO is -1, this is
cc                         equivalent to a reflection in a plane through the
cc                         origin with the normal vector VNAME.
cc                         This type of scaling operator can change an axially
cc                         symmetric quadric surface to a axially unsymmetric
cc                         quadric surface, if the scaling axis is not through
cc                         the center of the sphere being scaled or parallel to
cc                         the axis of any other axially symmetric quadric
cc                         surface being scaled.
cc                         See "scale", "radial", "uniform".

cg    linear             A linear probability distribution is a type of
cg                         probability bin or probability distribution function
cg                         (pdf) for which the relative differential probability
cg                         is a linear function of the value of the random
cg                         variable, for a specified range of such values.
cg                         See "discrete event", "discrete object",
cg                         "discrete value", "uniform", "power-law",
cg                         "exponential".
cg                         See commands "bin", "pdf", "sample".

cg    linear             See "linear acceleration", "linear equations",
cg                         "linear interpolation", "linear recursion".

cg    linear acceleration
cg                         See command "accelerate".

cg    linear equations   To solve three simultaneous linear equations in three
cg                         unknowns or two simultaneous linear equations in two
cg                         unknowns, use command "solve".

cg    linear interpolation
cg                         See "interpolation".

cg    linear motion      Linear motion is defined by an initial point p1,
cg
cg                           p1 = (x1 , y1 , z1)        (Cartesian),
cg                              = (rc1, th1, z1)        (cylindrical),
cg                              = (rs1, th1, ph1)       (spherical),
cg
cg                         (where rc and th are short for rcyl and theta,
cg                         and rs and ph are short for rsph and phi),
cg                         and by an initial unit direction vector u1,
cg
cg                           u1 = (ux1 , uy1 , uz1)     (Cartesian).
cg                              = (urc1, uth1, uz1)     (cylindrical),
cg                              = (urs1, uth1, uph1)    (spherical).
cg
cg                         To obtain the coordinates in cylindrical and
cg                         spherical systems from x1, y1 and z1, see
cg                         "coordinate conversion".
cg
cg                         To obtain the vector components in cylindrical and
cg                         spherical systems from ux1, uy1 and uz1, see
cg                         "vector conversion".
cg
cg                         Direction cosines ux. uy, and uz remain constant
cg                         along the track, but direction cosines urc, uth, urs
cg                         and uph may change continuously along the track.
cg
cg                         For a movement of distance s along the track,
cg
cg                           p2   = p1 + the integral from 0 to s of u * ds
cg
cg                         In Cartesian coordinates, the direction cosines
cg                         ux, uy and uz are constant along the track, so
cg
cg                           p2  = p1 + u * s
cg                           x2  = x1 + ux * s
cg                           y2  = y1 + uy * s
cg                           z2  = z1 + uz * s
cg                           u2  = u1
cg                           ux2 = ux1
cg                           uy2 = uy1
cg                           uz2 = uz1
cg
cg                         The distance s satisfies the equations
cg
cg                           s = (x2 - x1) / ux
cg                           s = (y2 - y1) / uy
cg                           s = (z2 - z1) / uz
cs                           s = sqrt ((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
cg
cg                         The minimum cylindrical radius rcmin along the path
cg                         occurs at distance dcmin, at point pcmin, where
cg
cg                           rc1   = sqrt (x1^2 + y1^2)
cg                           urc1  = (x1 * ux + y1 * uy) / rc1
cg                           uth1  = (x1 * uy - y1 * ux) / rc1
cg                           rcmin =  rc1 * uth1 / sqrt (1 - uz^2)
cg                           dcmin = -rc1 * urc1 / sqrt (1 - uz^2)
cg                           pcmin = (xcmin, ycmin, zcminnn)
cg                           xcmin = uy * (x1 * uy - y1 * ux) / sqrt (1 - uz^2)
cg                           ycmin = ux * (y1 * ux - x1 * uy) / sqrt (1 - uz^2)
cg                           zcmin = z1 - uz*(x1*ux + y1*uy)  / sqrt (1 - uz^2)
cg
cg                         The minimum spherical radius rsmin along the path
cg                         occurs at distance dsmin, at point psmin, where
cg
cg                           rs1   = sqrt (x1^2 + y1^2 + z1^2)
cg                           urs1  = (x1 * ux + y1 * uy + z1 * uz) / rs1
cg                           rsmin = rs1 * sqrt (1 - urs1^2)
cg                           dsmin = -rs1 * urs1
cg                           psmin = (xsmin, ysmin, zsmin)
cg                           xsmin = x1 - sqrt (rs1^2 - rsmin^2) * ux
cg                           ysmin = y1 - sqrt (rs1^2 - rsmin^2) * uy
cg                           zsmin = z1 - sqrt (rs1^2 - rsmin^2) * uz
cg
cg                         In cylindrical coordinates, the direction cosines
cg                         urc and uth vary continuously along the track, and
cg
cg                           rc2 = sqrt ((rc1 + s * urc1)^2 + s^2 * uth1^2)
cg                               = sqrt (rc1^2 + 2 * rc1 * s * urc1 +
cg                                       s^2 * (1 - uz^2))
cg                           sin (th2) = ((rc1 + s * urc1) * sin (th1) +
cg                                          s * uth1 * cos (th1)) / rc2
cg                           cos (th2) = ((rc1 + s * urc1) * cos (th2) -
cg                                          s * uth1 * sin (th1)) / rc2
cg                           z2 = z1 + uz * s
cg
cg                           urc2 = (rc1 * ur1 + s * (1 - uz^2)) / rc2
cg                           uth2 = rc1 * uth1 / rc2
cg                           uz2  = uz1
cg
cg                         The minimum cylindrical radius rcmin along the path
cg                         occurs at distance dcmin, where
cg
cg                           rcmin =  rc1 * uth1 / sqrt (1 - uz^2)
cg                           dcmin = -rc1 * ur1  / sqrt (1 - uz^2)
cg
cg                         The minimum spherical radius rsmin along the path
cg                         occurs at distance dsmin, where
cg
cg                           rsmin =  sqrt (rc1^2 + z1^2) -
cg                                          (rc1 * ur1 + z1 * uz))
cg                           dsmin = -(rc1 * ur1 + z1 * uz)
cg
cg                         MORE !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1

cg    linear recursion   See "series".

cg    link               See "linked input", "trace".

cg    linked             See "linked input", "trace".

cg    linked input       Any input file may specify input from one or more
cg                         additional input files with command "input".
cg                         The initial and final lines in each input file may be
cg                         defaulted to the first and last line or specified by
cg                         either line number or initial delimited character
cg                         string.  An input file may return to the previous
cg                         input file with command "return" or "input #".
cg                         Input is also returned to the previous input file
cg                         when the specified last line is reached or when an
cg                         end-of-file is reached.
cg
cg                         Input files may be linked by this means in any
cg                         order, including recursively and in infinite loops.
cg                         For example, input file AAA may contain command
cg                         "input AAA", or input file AAA may contain the
cg                         command "input BBB", and input file BBB may contain
cg                         command "input AAA".  Commands "if" and "endif"
cg                         may be used to break these loops, based on testing
cg                         the value of some variable.
cg
cg                         If an error is found in command "input", input is
cg                         returned to the user's terminal.  The user may then
cg                         use command "trace", to trace the return path from
cg                         the current input file, and then correct the error,
cg                         if possible, and use command "return" or "input" to
cg                         return input to the file where the error occurred.

cg    list               See "list commands", "command list".

cc    list               Command to display the names of big integers,
cc                         variables, points, clusters, mesh points, lines,
cc                         triangles, regular polygons, annular disks, vectors,
cc                         tensor operators, planes, spheres,
cc                         circular cylinders, circular cones,
cc                         hyperbolic paraboloids, ellipsoids,
cc                         axially symmetric quadric surfaces, general quadric
cc                         surfaces, regular polyhedrons, tetrahedrons, bricks,
cc                         zones, probability bins, probability distribution
cc                         functions (pdfs), aliases and symbols.
cc                         See command "search".
cc
cc                         Command "list" relates to objects:  alias, axisym,
cc                         big, bin, brick, cluster, cone, cylinder, disk,
cc                         ellipsoid, hyperb, line, marker, mesh, operator, pdf,
cc                         plane, point, polygon, polyhedron, quadric, sphere,
cc                         symbol, tetrahedron, triangle, variable, vector,
cc                         zone.
cc
cc                         Forms include:
cc
ccin                       help list
ccin                       list [help]
cc
cc                           Display the command options.
cc
ccin                       list all
cc
cc                           List all objects and mesh points.
cc
ccin                       list OBJTYPE
cc
cc                           List all objects of type OBJTYPE, which may be
cc                           alias, axisym, bin, brick, cluster, cone, cylinder,
cc                           disk, ellipsoid, hyperb, line, marker,
cc                           operator, pdf, plane, point, polygon, polyhedron,
cc                           quadric, sphere, symbol, tetrahedron, triangle,
cc                           variable, vector or zone, or a synonym of any of
cc                           these.
cc
ccin                       list quadric
cc
cc                           List all quadric surfaces, including
cc                           planes, spheres, circular cylinders, circular
cc                           cones, hyperbolic paraboloids, ellipsoids,
cc                           axially symmetric and general quadric surfaces.
cc
ccin                       list mesh
cc
cc                           List all mesh points.
cc
ccin                       list OBJ1 OBJ2 OBJ3 ...
cc
cc                           List all objects of types OBJ1, OBJ2, OBJ3, ...,
cc                           where OBJ1, OBJ2, OBJ3, ...  may be any of the
cc                           object types OBJTYPE given above, or may be mesh.
cc
cc                         If a list of specific objects is requested, and none
cc                         exist, the word "^NONE^" will be displayed.
cc
cc                         Synonyms:  [brick, br], [cluster, cl],
cc                         [cylinder, cyl], [disk, disc, dk],
cc                         [ellipsoid, el, ell, ellipse], [help, h],
cc                         [hyperb, saddle], [line, l, ln],
cc                         [list, display, print], [marker, m, mark],
cc                         [operator, op, oper], [plane, pl],
cc                         [point, p, pnt, pt], [polygon, pg, polyg],
cc                         [polyhedron, ph, polyh], [quadric, q, quad],
cc                         [symbol, sym, symb], [sphere, sph],
cc                         [tetrahedron, tet, tetra], [triangle, tri],
cc                         [variable, s, set, var], [vector, v, vect],
cc                         [zone, z, zn].

cc    list               An option in commands other than command "list", to
cc                         display the data for the list of objects named, of
cc                         the type specified by the first word of command
cc                         (alias, marker, variable, symbol, point, cluster,
cc                         line, triangle, polygon, disk, vector, operator,
cc                         plane, sphere, cylinder, cone, hyperb, ellipsoid,
cc                         axisym, quadric, polyhedron, tetrahedron, brick,
cc                         zone).  If a listed name has no subscript, all
cc                         objects with that name and one or more subscripts
cc                         will be displayed.
cc                         If no object names follow "list", all objects
cc                         of the specified type will be displayed.
cc
cc                         Do not name any object "list", but if you do, use
cc                         command "rename" to rename it or command "delete" to
cc                         delete it.

cg    list commands      To list all commands and their synonyms in alphabetic
cg                         order, use commands "commands" and "synonym".


cc    LMAX               In command "mesh", the largest value of the index l in

cc    LMOVE              In command "mesh move ...", the number of units to
cc                         move the mesh block in the l direction.
cc                         a logical array with indices (k,l,m).

cc    ln                 A synonym for line.

cg    local              See "local coordinates".

cg    local coordinates
cg                       The local coordinates of a point on a line, or in a
cg                         triangle, polygon or polyhedron are a measure of the
cg                         coordinates of the point relative to the coordinates
cg                         of the vertices, the edges of the object or the
cg                         faces of the object.
cg
cg                         In a triangle or a tetrahedron, good local
cg                         coordinates are the vertex weights.
cg
cg                         In a brick, good local coordinates are the fractional
cg                         distances between opposite faces, on a volume basis,
cg                         measuring from the minimum to the maximum
cg                         coordinates.
cg
cg                         See "fractional dist", "vertex weights".

cg    location           To find where a point is relative to one or all quadric
cg                         surfaces, use command "distance", "proximal" or
cg                         "side".
cg
cg                         To find if one point, a cluster of points or all
cg                         points are inside one or all bricks, tetrahedrons or
cg                         zones, use command "distance" or "where".
cg
cg                         To find if a point PNAME is in the mesh, and if so,
cg                         its mesh indices, use command "point PNAME".
cg
cg                         To find if the points in a cluster CLNAME are
cg                         assigned mesh indices, use command "cluster CLNAME".
cg
cg                         To put all points inside a volume element (brick,
cg                         tetrahedron or zone) into a cluster, use command
cg                         "cluster".
cg
cg                         To put all mesh points into cluster CLNAME, use
cg                         command "cluster CLNAME mesh all".
cg
cg                         To put all non-mesh points into cluster CLNAME, use
cg                         command "cluster CLNAME nomesh".
cg
cg                         To put all mesh points in the mesh block into cluster
cg                         CLNAME, use command "cluster CLNAME mesh block".

cc    lock               Command to prompt the user for a password, and not
cc                         continue until the user retypes it.
cc                         WARNING:  NOT VERY SECURE!
cc
cc                         Forms include:
cc
ccin                       help lock
ccin                       lock [help]
cc
cc                           Display the command options.
cc
ccin                       lock [all,list]
cc
cc                           Lock GEOM, after prompting the user to specify
cc                           a password and verify it.
cc                           The user must retype the password to restart
cc                           GEOM.  Can only be used when input is from the
cc                           user's terminal.
cc
cc                         Synonyms:  [help, h], [lock, zzz].

cg    log                A log of changes to GEOM is in file geom_log, in
cg                         YANA directory ~edwards/work/geom/doc .

cc    log                An option in command "variable", argument FUNCTION,
cc                         to indicate the natural logarithm function.
cc                         A log B means A * log (B).

cf    log                The log function.  Example:  y = log (x) means y is
cf                         the natural logarithm of x, and x = e^y, where e is
cf                         the base of natural logarithms.

cc    log10              An option in command "variable", argument FUNCTION,
cc                         to indicate the base 10 logarithm function.
cc                         A log10 B means A * log10 (B).

cf    log10              The log function, to the base 10.
cf                         Example:  y = log10 (x) means y is the logarithm of
cf                         x to the base 10, and x = 10^y.

cg    log10(e)           The value of log_10 (e) is:
cg                         0.43429448190325182765112891891660508229439700580366
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cg    log(10)            The value of log_e (10) is
cg                         2.30258509299404568401799145468436420760110148862877
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cg    logical            See "mesh", "Boolean".

cg    long               See "long commands".

cg    long commands      If you want to use a long command repeatedly, create a
cg                         short alias for the command.  See "alias".
cg                         If a command exceeds 80 characters, replace any long
cg                         numerical arguments with variables with short names,
cg                         or replace long variable names with short ones.

cg    loop               See "do loop", commands "if", "endif" and "goto".

cg    lowest             See "minimum".

M-------------------------------------------------------------------------------

cc    M                  In command "icalc", the integer (not a variable) to be
cc                         displayed or the first argument of a two-argument
cc                         function, in the current icalc integer mode.

cc    M                  In command "variable", am integer to be used to find
cc                         K mod M, and if K and M have no common factor other
cc                         than 1, to find the order and multiplicative inverse
cc                         of K in modulo M.

cc    M                  In command "variable", an integer, to be used in the
cc                         equation (K [+, -, *, /, ^] L) mod M.  M must not
cc                         be less than 2.

cc    M                  In commands "mesh" and "point", the index m in a
cc                         logical array with indices (k,l,m).
cc                         M may have a value from 1 to mmax.
cc                         Alternates to indices (k,l,m) are (kl,m) or (klm),
cc                         where
cc                           kl  = k  + (l - 1) * kmax
cc                           klm = kl + (m - 1) * kmax * lmax.

cc    m                  A synonym for marker.

cg    m-layer            All points with a fixed value of the index m, in a
cg                         3-D logical mesh with indices k, l and m.

cg    m-line             All points with a fixed value of the index m, in a
cg                         2-D logical mesh with only one k-layer or l-layer.

cc    M1                 In command "mesh", the minimum value of the m index in
cc                         the logical mesh.

cc    M1                 See "M1, M2, M3, ...".

cc    M1, M2, M3, ...    In command "factor", integers for which all prime
cc                         factors are to be found.  Negative values are changed
cc                         to positive, which must be from 1 to 10^18.

cc    M2                 See "M1, M2, M3, ...".

cc    M2                 In command "mesh", the maximum value of the m index in
cc                         the logical mesh.

cc    M3                 See "M1, M2, M3, ...".

cg    machine            See "machine time", "machine words".

cg    machine time       To find the machine time charged to the problem, use
cg                         command "time".

cg    machine words      The default storage of data in machine words in GEOM,
cg                         on the open YANA Cluster is as follows:
cg
cg                         Type character:  8 characters, stored in 8-bit bytes,
cg                         from right to left in 64-bit machine words.
cg                         E.g.:  'ABCDEFGH' is stored as 4847464544434241 hex.
cg
cg                         Type integer:  64-bit format, with 1's-complement
cg                         for negative integers.  The largest possible absolute
cg                         value of an integer is 2^63 - 1 or the 19 digits
cg                         9223372036854775807.  GEOM only allows integers with
cg                         up to 18 decimal digits.
cg                         E.g.:   1   is stored as 0000000000000001 hex.
cg                         E.g.:  -1   is stored as FFFFFFFFFFFFFFFF hex.
cg
cg                         Type real (floating point):  64-bit format, with
cg                         exponents from -308 to +308. Counting from the right
cg                         bit (bit 0), the mantissa is in bits 0-52, with the
cg                         leading 1 missing, the exponent is in bits 53-62,
cg                         with the "zero" value 3FF hex, and the sign bit is
cg                         in bit 63.
cg                         E.g.:   1.0 is stored as 3FF0000000000000 hex.
cg                         E.g.:  -1.0 is stored as BFF0000000000000 hex.
cg
cg                         To display the machine word for a character string,
cg                         or a decimal or hexadecimal integer or floating point
cg                         value, use command "hex".

cg    macro              A macro is an input file containing GEOM commands and
cg                         comments.  To execute macro file MACROFILE:
cg
cg                         input MACROFILE
cg
cg                           Execute the commands in file MACROFILE.  If the
cg                           last line is "return" or if an end-of-file is
cg                           reached, the input medium will become the preceding
cg                           input medium.
cg
cg                         To use as a macro with arguments, replace any
cg                         desired arguments in the macro with variables
cg                         or symbols, and recreate the variables or
cg                         symbols as needed, before each execution of the
cg                         macro.
cg
cg                         Use "+" or "-" in place of the name of an object
cg                         created by the macro, to form the name by
cg                         incrementing or decrementing the base name for the
cg                         object type.  See "increment names", command "last".
cg
cg                         Many macros may be kept in a single file, by heading
cg                         each macro with a comment line that begins with a
cg                         unique word (with "#", "*" or "/" in column 1),
cg                         which may be specified in command "input", and
cg                         ending each macro with "return".  For example:
cg
cg                         input MACROFILE #macro.123 #macro.123.end
cg
cg                           Open file MACROFILE, and position it at the first
cg                           line that begins with the string "#macro.123".
cg                           Execute commands until command "return" or an
cg                           end-of-file, or a line that begins with
cg                           "#macro.123.end" is reached, then return to the
cg                           previous input medium.

cg    magnitude          The scalar value of the length of a vector.
cg                         See "length".

cg    major              Major refers to one of the following:
cg                         A major coordinate direction of the coordinate system
cg                         (see x, y, z, rcyl, rsph, theta, phi).
cg                         A major surface, with a constant value of one of the
cg                         major coordinates of the coordinate system.
cg                         See "slice", "cut", "brick".
cg                         A major axis of a geometric object having one or more
cg                         axes of symmetry, which need not be parallel to a
cg                         major coordinate direction of the coordinate system.
cg                         See "quadric".

cd    make               Directory for making GEOM.
cd                         In Open YANA Cluster:  ~edwards/work/geom/make
cd                         Archives:  /users/u47/edwards/geom/make
cd                         See "doc", "test".

cg    map                For subroutine calls, see files geom_link, geom_to,
cg                         geom_from, geom_aptcomms, geom_commapts.

cc    mark               A synonym for marker.

cg    marker             A marker is a single character created by command
cg                         "marker", with up to 72 characters of text assigned
cg                         to it.  The marker and its assigned text may be used
cg                         for general notes or for plot titles, axis labels
cg                         and plot point characters.
cg
cg                         The following commands relate to markers.
cg                         copy, debug, delete, help, last, list, marker, plot,
cg                         rename, repack, search, sort, symbol, synonym,
cg                         tables.

cc    marker             An option in command "debug", to display GEOM internal
cc                         parameters and variables for markers:
cc                         nmarkm, nmarks, amark, amarkt, amarks.
cc                         Synonyms:  [marker, m, mark].

cc    marker             Command to display or create a 1-character marker and
cc                         up to 72 characters of text assigned to the marker.
cc                         Creating a marker replaces any existing marker with
cc                         the same character.  The display for a marker
cc                         includes the assigned text.  See "definition".
cc                         See "do loop use".
cc
cc                         Command "marker" relates to objects:  marker, symbol.
cc
cc                         Forms include:
cc
ccin                       help marker
ccin                       marker [help]
cc
cc                           Display the command options.
cc
ccin                       marker [all,list]
cc
cc                           Display all markers and their texts.
cc
ccin                       marker list AMNAME1 AMNAME2 AMNAME3 ...
cc
cc                           Display the marker and text for all listed markers
cc                           (the default is all markers).
cc
ccin                       marker AMNAME
cc
cc                           Display the marker and text for marker AMNAME.
cc
ccin                       marker AMNAME = " "
cc                         delete marker AMNAME
cc
cc                           Remove marker AMNAME.
cc
ccin                       marker AMNAME = "TEXT"
cc
cc                           Create the 1-character marker AMNAME with text TEXT
cc                           (bracketed by ' or ").  TEXT may contain blanks.
cc
cc                           If TEXT is bracketed by single quotes, repeat any
cc                           internal single quotes.  If TEXT is bracketed by
cc                           double quotes, repeat any internal double quotes.
cc
cc                           Use "+" or "-" instead of AMNAME to form the name
cc                           by incrementing or decrementing the base name for
cc                           markers.  See command "last", "increment names".
cc
cc                         Synonyms:  [help, h], [marker, m, mark].

cc    marker             An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more markers.
cc                         Synonyms:  [marker, m, mark].

cc    marker             An option in command "plot", to display or specify the
cc                         character to be used in the plot array to represent
cc                         any plot points subsequently stored.

cg    math               Simple math may be done with commands "variable" and
cg                         "icalc".  Also see command "math".

cc    math               Command to display a help message for mathematics
cc                         commands.
cc
cc                         Forms include:
cc
ccin                       help math
ccin                       math [help]
cc
cc                           Display a help message for mathematics commands.
cc
cc                         Synonyms:  [help, h].

cc    math               An option in command "help", to display help messages
cc                         for mathematics commands.

cg    math               See "math constants", "math topics".

cg    math constants     See "ebase", "pi", "rgold", "deg/rad", "rad/deg",
cg                         "euler".

cg    math topics        Some of the math topics which may be investigated with
cg                         GEOM include:
cg
cg                         Topic                            Commands
cg
cg                         Chinese remainder problem        crt
cg                         continued fractions              contfr
cg                         Euler totient functions          factor
cg                         geometric series                 ratio
cg                         modular arithmetic               variable
cg                         prime factors                    factor
cg                         prime numbers                    variable
cg                         reciprocal series                iris
cg                         roots of polynomials             root, rootf, roots
cg                         simultaneous equations           solve, triple

cg    Max                In the output display, indicates a maximum value on a
cg                         quadric surface, in the direction of a specified
cg                         axis.

cc    max                An option in commands "icalc" and "variable", argument
cc                         FUNCTION, to indicate the maximum function,
cc                         A min B means min (A, B), or the maximum of a group
cc                         of variables, found by command "mean".

cf    max                The maximum function.  Example:  z = max (x, y) means
cf                         that z is the more positive or lesser negative of
cf                         x and y.

cg    max                In the display of the edge lengths of a brick,
cg                         indicates the maximum value of one of the other two
cg                         coordinates perpendicular to the direction of the
cg                         edge.

cg    Maximum            In the output display, indicates:
cg                         a maximum value of a quadric surface, in the
cg                         direction of a specified axis; or
cg                         a maximum value of a set of randomly sampled values,
cg                         or a maximum coordinate in a point cluster.

cg    maximum            To find the maximum value of a coordinate on a quadric
cg                         curve in a major plane, use command "quadric" or
cg                         "slice".
cg
cg                         To find the maximum distance between the points in a
cg                         cluster and another point or cluster, a line, a
cg                         triangle or a plane, use command "distance".
cg
cg                         To find the maximum and minimum points in any
cg                         specified direction on a quadric surface, use
cg                         command "quadric" or "extrema".
cg
cg                         To find the maximum coordinates of a set of
cg                         points, create a cluster of the set of points.
cg
cg                         To find the maxima of a polynomial equation, use
cg                         command "root" or "roots".  See "extrema".

cg    maximum            See "maximum number".

cg    maximum number     The current maximum numbers of various objects, and the
cg                         commands to create them (see command "tables"):
cg
cg                         Object                              Limit  Command
cg                         ----------------------------------  -----  -------
cg
cg                         Aliases                               256  alias
cg                         Annular disks                        1000  disk
cg                         Bounding surfaces per zone             16  zone
cg                         Bricks (coordinate solids)           1000  brick
cg                         Clusters of points                    100  cluster
cg                         Lines                                1000  line
cg                         Operators                             100  operator
cg                         Point-cluster pairs                  1000  cluster
cg                         Points                               1000  point
cg                         Polygons                             1000  polygon
cg                         Polyhedrons                          1000  polyh
cg                         Probability bin-pdf pairs            1000  pdf
cg                         Probability bins                     1000  bin
cg                         Probability distribution functions    100  pdf
cg                         Quadric surfaces                     1000  (all)
cg                           Planes                                   plane
cg                           Spheres                                  sphere
cg                           Circular cylinders                       cylinder
cg                           Circular cones                           cone
cg                           Hyperbolic paraboloids                   hyperb
cg                           Ellipsoids                               ellipsoid
cg                           Axially symmetric quadric surfaces       axisym
cg                           General quadric surfaces                 quadric
cg                         Saved input lines (internal)    last 2000  input
cg                         Saved input lines (file geom_cmd)    none  input
cg                         Markers                               256  marker
cg                         Tetrahedrons                         1000  tetra
cg                         Triangles                            1000  triangle
cg                         Variables                            1000  variable
cg                         Symbols                              1000  symbol
cg                         Vectors                              1000  vector
cg                         Vertices per polygon                 1000  polygon
cg                         Zones                                1000  zone

cc    Maxwell            A synonym for maxwell.  Also see "Maxwellian".

cc    maxwell            An option in command "bin", to create a relativistic
cc                         Maxwellian probability distribution.
cc                         Synonyms:  [maxwell, Maxwell].
cc                         See "Maxwellian", "velocity".

cg    Maxwellian         A relativistic Maxwellian probability distribution is a
cg                         pdf for which the differential probability
cg                         p(BETA, RATIO) of a value BETA of the random
cg                         variable, at a specified dimensionless ratio RATIO of
cg                         the gas temperature to the particle rest mass,
cg                         is the relativistic Maxwellian distribution:
cg                         p(BETA, RATIO) = A * GAMMA^5 * exp (-GAMMA / RATIO)
cg                         where BETA is the ratio of the particle velocity to
cg                         the speed of light, v / c (c = 299,792,458 m / s),
cg                         GAMMA = 1 / sqrt( 1 - BETA^2),
cg                         A = BETA^2 / (RATIO * KB2(1 / RATIO),
cg                         and KB2 is the modified Bessel function of the second
cg                         kind.  The expected values are approximately:
cg                         For RATIO << 1, <BETA>     = 1.60 * sqrt (RATIO)
cg                                         Std Dev    = 0.67 * sqrt (RATIO)
cg                                         <BETA^2>  = 3.00 * RATIO
cg                         For RATIO = .1, <BETA>     = 1.47 * sqrt (RATIO)
cg                                         Std Dev    = 0.49 * sqrt (RATIO)
cg                                         <BETA^2>  = 2.40 * RATIO
cg                         For RATIO =  1, <BETA>     = 0.9056
cg                                         Std Dev    = 0.1067
cg                                         <BETA^2>  = 0.8315
cg                                         <GAMMA>    = 3.369
cg                                         Std Dev    = 1.661
cg                                         <GAMMA^2> = 14.11
cg                         For RATIO >> 1, <BETA>     = 1 - 0.23 / RATIO^2
cg                                         Std Dev    = 1 - 1.26 / RATIO^2
cg                                         <BETA^2>  = 1 - 0.46 / RATIO^2
cg                                         <GAMMA>    = 3.00 * RATIO
cg                                         Std Dev    = 1.73 * RATIO
cg                                         <GAMMA^2> = 12.0 * RATIO^2
cg                         1 K = 8.617385E-5 eV (+/- 8.5 ppm),
cg                         1 amu = 9.3149432E8 eV (+/- 0.3 ppm).
cg                         See commands "bin", "pdf", "sample".
cg                         See "velocity".

cc    mcvol              Command to estimate the volume of a zone.
cc                         If the zone is a body of revolution of a polygonal
cc                         area around an axis, the area may be found exactly
cc                         with command "volume".
cc
cc                         Command "mcvol" relates to objects:  brick, symbol,
cc                         variable, zone.
cc
cc                         Forms include:
cc
ccin                       help mcvol
ccin                       mcvol [help]
cc
cc                           Display the command options.
cc
ccin                       mcvol ZNAME BRNAME NSAMP
cc
cc                           Estimate the volume of any part of zone ZNAME in
cc                           brick BRNAME, using NSAMP randomly sampled points
cc                           in BRNAME.  The estimated percent error is
cc                           100 * sqrt ((1 - f)/(f * (NSAMP - 1))), where
cc                           f = npin / NSAMP is the fraction of sampled points
cc                           that are in zone ZNAME.  For example, for f = 0.5
cc                           and NSAMP = 10000, the estimated error is 1%.
cc                           You may make your own error estimate by repeating
cc                           command "mcvol" many times.  Use command "alias" to
cc                           save typing.  The ranges of the coordinates of
cc                           points in zone ZNAME are displayed, to make it
cc                           possible to choose a smaller brick to make a more
cc                           accurate estimate.
cc                           Also consider moving and rotating the zone, to make
cc                           it easier to surround it with a smaller brick.
cc                           Required machine time on a SUN4 is about 1 second
cc                           per 3000 points.
cc
cc                         Synonyms:  [help, h].

cc    mean               Command to find the mean value and standard deviation
cc                         of specified sets of variables.  The results may
cc                         be saved with command "variable", options "mean",
cc                         "stddev", "min" and "max".
cc
cc                         Forms include:
cc
ccin                       help mean
ccin                       mean [help]
cc
cc                           Display the command options.
cc
ccin                       mean all
cc
cc                           Find the mean value, standard deviation, minimum
cc                           and maximum of all variables.
cc
ccin                       mean VARNAME all
cc
cc                           Find the mean value, standard deviation, minimum
cc                           and maximum of all variables with the stem name
cc                           VARNAME, and with one or more subscripts with any
c                            value.
cc
ccin                       mean VARNAME1 thru VARNAME2
cc
cc                           Find the mean value, standard deviation, minimum
cc                           and maximum of all variables with names (including
cc                           stem names and any subscripts) in the range from
cc                           VARNAME1 to VARNAME2.
cc                           Names VRANAME1 and VARNAME2 may be the same.
cc
cc                           NOTE!  No object may have the name "thru".
cc
cc                         Synonyms:  [help, h].

cc    mean               An option in command "variable", to create a variable
cc                         with the mean value of a group of variables, as
cc                         previously found with command "mean".

cg    mean               The mean <x> of N values of x is equal to the sum of
cg                         the x values divided by N.
cg                         The mean <x^2> of the squares of x is equal to the
cg                         sum of the x^2 values divided by N.
cg                         The standard deviation of the x values from the mean
cg                         is sigma = sqrt (<x^2> - <x>^2).
cg                         See command "mean".

cg    mean               See "mean free path", "mesh aliases", "mesh block".

cg    mean free path     To randomly sample distances to an event with a
cg                         specified mean free path (mfp) DMEAN:
cg
cg                         bin BINAME 1 0 VRANR exp SIGMA
cg
cg                           Create a probability bin BINAME with an
cg                           exponentially decaying probability with decay
cg                           constant SIGMA = 1 / DMEAN, and VRANR = 100 * DMEAN
cg                           (negligible probability).  If the distance is
cg                           actually limited, use that limit for VRANR.
cg
cg                         pdf PDFNAME = BINAME
cg
cg                           Create a probability distribution function (pdf)
cg                           PDFNAME, consisting only of the probability bin
cg                           BINAME.
cg
cg                         sample PDFNAME [NSAMP|1]
cg
cg                           Sample from pdf PDFNAME NSAMP times.

cg    measure            See "side measure".

cg    median             The median of an array of values is the value for which
cg                         there are as many smaller values as there are larger
cg                         values.

cg    median             A median of a triangle is a line from a vertex to the
cg                         center of the opposite edge.  The three medians
cg                         intersect at the centroid of the triangle.
cg                         See "altitude", "bisector", "trig".

cg    medium             The input medium.  See "input".

cg    mesh               A mesh is an logical array with indices (k,l,m), with
cg                         k from 1 to kmax, l from 1 to lmax, and m from 1 to
cg                         mmax.  Each set of indices k, l and m may be assigned
cg                         to physical points (mesh points) with specified
cg                         spatial coordinates in a 3-D coordinate system.
cg                         The spatial coordinates of the mesh points are the
cg                         physical representation of the geometry of the system
cg                         represented by the mesh.
cg
cg                         Unassigned mesh indices may be displayed with
cg                         command "mesh list void" or "mesh list block void".
cg
cg                         Mesh indices assigned to preliminary mesh points with
cg                         x, y, z coordinates (-10^99, -10^99, -10^99) may be
cg                         displayed with command "point list" or "mesh list",
cg                         and looking for the label "VACUUM".
cg
cg                         A part or all of the mesh may be included in a mesh
cg                         block, to be used as the object of various commands.
cg
cg                         To reflect the logical mesh of mesh points in the
cg                         mesh block in the direction of one of the indices,
cg                         use command "mesh", option "invert".
cg
cg                         To rotate the logical mesh of mesh points in the mesh
cg                         block by plus or minus 90 degrees around an axis in
cg                         the direction of one of the indices, use command
cg                         "mesh", option "exchange" to exchange the other two
cg                         indices, then use command "mesh", option "invert", to
cg                         invert one of the two exchanged indices.
cg
cg                         To rotate the logical mesh of mesh points in the mesh
cg                         block by 180 degrees around an axis in the direction
cg                         of one of the indices, use command "mesh", option
cg                         "invert" twice, once for each of the other indices.
cg
cg                         The following commands relate to the mesh:
cg                         cluster, delete, help, list, mesh, point, repack,
cg                         sort, symbol, synonym and any commands that affect
cg                         points.
cg
cg                         See "transform the mesh".

cc    mesh               An option in command "debug", to display GEOM internal
cc                         parameters and variables for the mesh:
cc                         kmax, kbl1, kbl2, lmax, lbl1, lbl2, mmax, mbl1,
cc                         mbl2, npointm, npoints, apoint, lapoint, kpoint,
cc                         lpoint, mpoint, x, y, z.

cc    mesh               Command to display or specify a 3-D mesh.
cc                         See command "point" to create points with geometric
cc                         coordinates.  Such points may be assigned mesh
cc                         indices, thus becoming mesh points.
cc                         See "aliases", for suggestions for short commands.
cc                         See "do loop use".
cc
cc                         Command "mesh" relates to objects:  cluster, mesh,
cc                         point, symbol, variable.
cc
cc                         The mesh and its assigned points may be involved in
cc                         the following commands:
cc                         cluster, help, list, mesh, point, repack, sort,
cc                         tables, and by any commands that affect points.
cc
cc                         Forms include:
cc
ccin                       help mesh
ccin                       mesh [help]
cc
cc                           Display the command options.
cc
ccin                       mesh size
cc
cc                           Display the mesh size (initially zero).
cc
ccin                       mesh size [=] KMAX LMAX MMAX
cc
cc                           Change the mesh size to k = 1 to KMAX, l = 1 to
cc                           LMAX and m = 1, MMAX, and release any mesh points
cc                           with mesh indices k > KMAX, l > LMAX or m > MMAX.
cc
ccin                       mesh block
cc
cc                           Display the ranges of the mesh block indices
cc                           (initially 0) and the indices of the vertices of
cc                           the mesh block.
cc
ccin                       mesh block all
cc
cc                           Change the mesh block to the entire mesh.
cc
ccin                       mesh block K1 K2  L1 L2  M1 M2
cc
cc                           Change the mesh block indices to the ranges k = K1
cc                           to K2, l = L1 to L2 and m = M1 to M2.  From the
cc                           right, missing final arguments default to 1.
cc
ccin                       mesh [all,list [all]]
cc
cc                           Display the mesh size, block indices, and all mesh
cc                           points.  "Vacuum" mesh points with x, y, z
cc                           coordinates (-10^99, -10^99, -10^99) will be
cc                           labeled "VACUUM".
cc
cc
ccin                       mesh list void
cc
cc                           Display the mesh size, block indices, and all mesh
cc                           indices not assigned to mesh points.
cc
ccin                       mesh list block
cc
cc                           Display all mesh points in the mesh block.
cc
ccin                       mesh list block void
cc
cc                           Display all mesh indices in the mesh block not
cc                           assigned to mesh points.
cc
ccin                       mesh list K L M
cc
cc                           Display all mesh points with mesh indices K, L and
cc                           M.  Any missing final arguments after K default to
cc                           1.
cc
ccin                       mesh point PMESH
cc
cc                           For each set of mesh coordinates (k, l, m) for
cc                           which no point is currently assigned, assign any
cc                           existing point PMESH(k,l,m) with its current
cc                           coordinates, or create a new "vacuum" point
cc                           PMESH(k,l,m) with x, y, z coordinates
cc                           (-10^99, -10^99, -10^99).  Name PMESH may not be
cc                           subscripted.  See command
cc                           "point PMESH brick BRNAME mesh block [volume]".
cc
ccin                       mesh point PMESH block
cc
cc                           For each set of mesh block coordinates (k, l, m)
cc                           for which no point is currently assigned, assign
cc                           any existing point PMESH(k,l,m) with its current
cc                           coordinates, or create a new "vacuum" point
cc                           PMESH(k,l,m) with x, y, z coordinates
cc                           (-10^99, -10^99, -10^99).  Name PMESH may not be
cc                           subscripted.  See command
cc                           "point PMESH brick BRNAME mesh block [volume]".
cc
ccin                       mesh point PMESH(K,L,M)
cc
cc                           Assign mesh indices k = K, l = L and m = M to
cc                           existing point PMESH(K,L,M), after releasing any
cc                           other mesh point with those indices.
cc                           Any missing final arguments after K default to 1.
cc                           Any point previously assigned to the the same mesh
cc                           indices will be renamed by replacing the first
cc                           character of its name with "-".  That must not
cc                           create a duplicate name.
cc
ccin                       mesh point PNAME K L M
cc
cc                           Assign mesh indices k = K, l = L and m = M to
cc                           existing point PNAME after releasing and renaming
cc                           any other mesh point with those indices.
cc                           Any missing final arguments after K default to 1.
cc                           If PNAME has three subscripts, they must be K, L
cc                           and M.
cc                           Any point previously assigned to the the same mesh
cc                           indices will be renamed by replacing the first
cc                           character of its name with "-".  That must not
cc                           create a duplicate name.
cc
ccin                       mesh - all
cc
cc                           Release all mesh points (set their assigned mesh
cc                           indices to zero), and change their names by
cc                           replacing the first character with "-".  That must
cc                           not create any duplicate names.
cc
ccin                       mesh - block
cc
cc                           Release all mesh points in the mesh block (set
cc                           their assigned mesh indices to zero), and change
cc                           their names by replacing the first character with
cc                           "-".  That must not create any duplicate names.
cc
ccin                       mesh - K L M
cc
cc                           Release any mesh point with mesh indices K, L and M
cc                           (set its assigned mesh indices to zero), and change
cc                           its name by replacing the first character with "-".
cc                           That must not create a duplicate name.
cc                           Any missing final arguments after K default to 1.
cc
ccin                       mesh - point PMESH(K,L,M)
cc
cc                           Release mesh point PMESH(K,L,M) (set its assigned
cc                           mesh indices to zero), and change its name by
cc                           replacing the first character with "-".
cc                           That must not create a duplicate name.
cc
ccin                       mesh - cluster CLNAME
cc
cc                           Release all mesh points in cluster CLNAME
cc                           (set their assigned mesh indices to zero), and
cc                           change their names by replacing the first character
cc                           with "-".  That must not create a duplicate name.
cc
ccin                       mesh exchange block [k,l,m] [l,m,k]
cc
cc                           Exchange the [k, l, m]-layers in the mesh block
cc                           with the [l, m, k]-layers in the mesh block:
cc                           replace mesh point index i with jbl1 + i - ibl1
cc                           and index j with ibl1 + j - jbl1), where
cc                           i = [k, l, m] and j = [l, m, k], and ibl1 and jbl1
cc                           are the first i and j indices in the mesh block).
cc                           The number of layers in the mesh block in the
cc                           directions of the two indices must be the same.
cc                           See "transform the mesh".
cc
ccin                       mesh invert block [k,l,m]
cc
cc                           Invert the [k, l, m]-layers in the mesh block:
cc                           replace mesh point index n with (nbl1 + nbl2 - n),
cc                           where n = [k, l, m].  See "transform the mesh".
cc
ccin                       mesh move block KMOVE LMOVE MMOVE
cc
cc                           Move the mesh block, and all mesh points in the
cc                           mesh block, by KMOVE, LMOVE and MMOVE units in the
cc                           k, l and m directions, after first releasing and
cc                           renaming any mesh points not in the mesh block, but
cc                           in the new location of the mesh block.
cc                           Such points will be renamed by replacing the first
cc                           character of their names with "-".  That must not
cc                           create any duplicate names.
cc                           See "transform the mesh".
cc
ccin                       mesh insert [k,l,m] N1 [N2|N1]
cc
cc                           Insert new [k, l, m]-layers with indices from N1 to
cc                           N2 into the mesh, preceding the current layer N1.
cc                           Shift mesh point [k, l, m] indices of N1 or more
cc                           up to make room for new mesh layers with [k, l, m]
cc                           indices from N1 to N2.  This will change the mesh
cc                           size, and may change any mesh block.
cc
ccin                       mesh delete [k,l,m] N1 [N2|N1]
cc
cc                           Delete the current [k, l, m]-layers with indices
cc                           from N1 to N2 from the mesh.  Release all mesh
cc                           points with assigned [k, l, m] mesh indices from
cc                           N1 to N2 and rename them by replacing the first
cc                           characters of their names with "-", That must not
cc                           create any duplicate names.
cc                           Shift the subscripts and assigned mesh indices of
cc                           all points with higher assigned mesh point
cc                           [k, l, m] indices down to fill the gap.  This will
cc                           change the mesh size, and may change the size of
cc                           any mesh block.
cc
ccin                       mesh linear block
cc
cc                           Linearly interpolate the x, y and z coordinates of
cc                           all mesh points in the mesh block, between the
cc                           x, y and z coordinates of the mesh points at the
cc                           vertices of the mesh block.
cc                           WARNING:  poor results if mesh has primarily
cc                           cylindrical or spherical surfaces.
cc
ccin                       mesh relax [-] block
cc
cc                           Find the x, y and z coordinates of all mesh points
cc                           [not] in the mesh block, except for exterior points
cc                           of [the mesh and] the mesh block, by a relaxation
cc                           method, which moves each mesh point to the average
cc                           coordinates of its nearest neighbors.  Exterior
cc                           points have exactly zero or one neighbor in one or
cc                           more of the k, l and m directions.
cc                           WARNING:  poor results if mesh has primarily
cc                           cylindrical or spherical surfaces.
cc
ccin                       mesh test block
cc
cc                           See if any mesh indices in the mesh block have not
cc                           been assigned to mesh points, and if any linear,
cc                           surface or volume elements in the mesh block
cc                           violate certain geometric criteria.  See "mesh
cc                           tests".
cc
cc                         Synonyms:  [block, bl], [cluster, cl],
cc                         [delete, del, remove, rm], [exchange, exch],
cc                         [insert, ins], [move, mv, trans, translate],
cc                         [point, p, pnt, pt].

cc    mesh               An option in command "sort", to sort all points by
cc                         mesh indices.
cc                         An option in command "list", to list the names of
cc                         all mesh points.
cc                         An option in command "cluster", to create a cluster
cc                         of all mesh points or mesh points in the mesh block.
cc                         An option in command "point", to create mesh points.
cc                         An option in command "repack", to repack mesh points.

cg    mesh               See "mesh aliases", "mesh block", "mesh coordinate",
cg                         "mesh indices", "mesh klm", "mesh point",
cg                         "mesh reflection", "mesh rotation", "mesh size",
cg                         "mesh tests", "mesh translation".

cg    mesh aliases       See "aliases".

cg    mesh block         A mesh block is a part of the logical mesh, with
cg                         specified ranges of each of the mesh indices k, l
cg                         and m.  There may be only one mesh block at a time.
cg                         The mesh block is initially null, and may be
cg                         specified or changed with command "mesh block ...".
cg                         The mesh block may also be changed if command
cg                         "mesh" with option "insert" or "delete" is used.
cg
cg                         If you want to reverse the order of one of the mesh
cg                         indices of all mesh points in the mesh block, use
cg                         command "mesh invert ...".
cg
cg                         If you want to exchange any pair of the mesh indices
cg                         in all mesh points in the mesh block, use command
cg                         "mesh exchange ...".
cg
cg                         If you want to move the mesh block, and all of its
cg                         assigned mesh points, by specified increments in the
cg                         k, l and m directions, use command "mesh move ...".
cg
cg                         If you want to create and assign mesh points at
cg                         equal spatial or volume intervals in  each of the
cg                         three coordinate directions, see command
cg                         "point PMESH brick BRNAME mesh block [volume]".

cg    mesh coordinate    See "mesh indices", "mesh points".

cg    mesh edge          An edge element in the mesh is defined by a pair of
cg                         mesh points with the same value of two of the three
cg                         mesh indices, and two adjacent values of the other
cg                         mesh index, e.g., p(k,l,m) and p(k,l,m+1).
cg                         The edge is a straight line, with a length equal to
cg                         the distance between the two point, or the magnitude
cg                         of the vector defined by the two points.  See command
cg                         "vector VNAME point PNAME1 PNAME2".

cg    mesh indices       The indices in a mesh are k, l and m, specifying a
cg                         k-layer, an l-layer and an m-layer.  If the set of
cg                         mesh indices (k, l, m) is assigned to a point, the
cg                         point is called a mesh point, and has a position in
cg                         3-D space specified by three coordinates in the
cg                         current physical coordinate system, which may be
cg                         rectangular (x, y, z), cylindrical (rcyl, theta, z),
cg                         or spherical (rsph, theta, phi).
cg
cg                         See "mesh point", "mesh edge", "mesh surface",
cg                         "mesh volume".
cg
cg                         For a single value of the index m, a single composite
cg                         index kl may be used to refer to the mesh indices
cg                         k and l:
cg                           kl  = k + (l - 1) * kmax
cg
cg                         Given kl, the indices k and l are as follows:
cg                           k = 1 + mod (kl - 1, kmax)
cg                           l = 1 + (kl - 1) / kmax
cg
cg                         For the entire mesh of kmax * lmax * mmax points,
cg                         a single composite index klm may be used to refer
cg                         to the mesh indices k, l and m:
cg                           klm = k  + (l - 1) * kmax + (m - 1) * kmax * lmax
cg                           klm = kl + (m - 1) * kmax * lmax
cg
cg                         Given klm, the indices kl, k, l and m are as
cg                         follows:
cg                           kl = 1 + mod (I  - 1, kmax * lmax)
cg                           k  = 1 + mod (kl - 1, kmax)
cg                           l  = 1 + (kl - 1) /  kmax
cg                           m  = 1 + (I  - 1) / (kmax * lmax)

cg    mesh klm           A mesh point with mesh indices k, l and m, may also
cg                         be identified by a single index klm:
cg                           kl  = k  + (l - 1) * kmax
cg                           klm = kl + (m - 1) * kmax * lmax,
cg                         where kmax is the maximum value of k in the mesh,
cg                         and lmax is the maximum value of l in the mesh.
cg                         If the maximum value of m in the mesh is 1, then
cg                         kl = klm.

cg    mesh point         A mesh point is a point with specified spatial
cg                         coordinates, created with command "point", and
cg                         assigned a specified set of mesh indices k, l and m,
cg                         specifying the k-layer, l-layer and m-layer
cg                         containing the point, with command "mesh".
cg                         A mesh point named p(k,l,m) can only be assigned
cg                         to mesh indices k, l and m.
cg
cg                         To create the point or points to be assigned mesh
cg                         indices, use command "point" (many options), and
cg                         especially see command
cg                         "point PMESH brick BRNAME mesh block [volume]".
cg
cg                         To assign or reassign mesh indices to mesh points,
cg                         use commands "mesh point PMESH",
cg                         "mesh point PMESH(K,L,M)", "mesh point PNAME K L M"
cg                         or "mesh point PMESH block".
cg
cg                         To change the mesh indices of mesh points, use
cg                         command "mesh" with options "exchange", "invert",
cg                         "move", "insert" or "delete".
cg
cg                         To release mesh points from their mesh index
cg                         assignments, use command "mesh - ...".
cg
cg                         Any command that operates on clusters of points can
cg                         be used to operate on mesh points or non-mesh points,
cg                         if command "cluster" is first used to collect the
cg                         points into a cluster.
cg
cg                         This includes commands "area", "copy", "delete",
cg                         "distance", "invert", "last", "list", "mesh", "move",
cg                         "project", "reflect", "rename", "repack", "rotate",
cg                         "scale", "sort", "twist" and "where".
cg
cg                         See commands "cluster", "mesh", "point".

cg    mesh points        An array of mesh points may be created in 1, 2 or 3
cg                         dimensions, in rectangular, cylindrical or spherical
cg                         coordinates, uniformly spaced over each coordinate.
cg
cg                         Specify the coordinate system:
cg
cg                           coordinate cartesian
cg                           coordinate cylindrical
cg                           coordinate spherical
cg
cg                         Specify the limits in each coordinate direction
cg                         ("s" is a synonym for "variable"):
cg
cg                           s UMIN = [XMIN | RCYLMIN  | RSPHMIN ]
cg                           s UMAX = [XMAX | RCYLMAX  | RSPHMAX ]
cg                           s VMIN = [YMIN | THETAMIN | THETAMIN]
cg                           s VMAX = [YMAX | THETAMAX | THETAMAX]
cg                           s WMIN = [ZMIN | ZMIN     | PHIMIN  ]
cg                           s WMAX = [ZMAX | ZMAX     | PHIMAX  ]
cg
cg                         Specify the number of points in each coordinate
cg                         direction:
cg
cg                           s NU = [NX | NRCYL  | NRSPH ]
cg                           s NV = [NY | NTHETA | NTHETA]
cg                           s NW = [NZ | NZ     | NPHI  ]
cg
cg                         Specify the mesh size:
cg
cg                           mesh size = NU NV NW
cg
cg                         The following commands could be put in an input file,cg                         to be used with different values of the data above.
cg
cg                         Create the coordinate values in each direction.
cg
cg                           s U(1) = UMIN
cg                           s DU = UMAX - UMIN
cg                           s NINTU = NU - 1
cg                           s DU = DU / NINTU
cg                           do NLU = 2 NU
cg                           s NLUM = NLU - 1
cg                           s U(NLU) = U(NLUM) + DU
cg                           enddo
cg
cg                           s V(1) = VMIN
cg                           s DV = VMAX - VMIN
cg                           s NINTV = NV - 1
cg                           s DV = DV / NINTV
cg                           do NLV = 2 NV
cg                           s NLVM = NLV - 1
cg                           s V(NLV) = V(NLVM) + DV
cg                           enddo
cg
cg                           s W(1) = WMIN
cg                           s DW = WMAX - WMIN
cg                           s NINTW = NW - 1
cg                           s DW = DW / NINTW
cg                           do NLW = 2 NW
cg                           s NLWM = NLW - 1
cg                           s W(NLW) = W(NLWM) + DW
cg                           enddo
cg
cg                         Create the points, with the name PMESH appended by
cg                           the mesh indices as subscripts, and assign them to
cg                           the mesh:
cg
cg                           do NLU = 1 NU
cg                           do NLV = 1 NV
cg                           do NLW = 1 NW
cg                           point PMESH(NLU,NLV,NLW) = U(NLU) V(NLV) W(NLW)
cg                           enddo
cg                           enddo
cg                           enddo
                             mesh point PMESH
cg
cg                         See macro file "meshpoint.mac".

cg    mesh reflection    See command "mesh", options "invert" and "exchange".
cg                         See "transform the mesh".

cg    mesh rotation      See command "mesh", options "invert" and "exchange".
cg                         See "transform the mesh".

cg    mesh size          The size of the mesh is initially zero, and may be
cg                         specified or changed with command
cg                         "mesh size KMAX LMAX MMAX".
cg                         The size of the mesh will also be changed if command
cg                         "mesh delete ..." or "mesh insert ..." is used.

cg    mesh surface       A surface element in the mesh is defined by a set of
cg                         four point elements with the same value of one of the
cg                         mesh indices, and two adjacent values of each of the
cg                         other two indices, for example p(k,l,m), p(k,l+1,m),
cg                         p(k,l+1,m+1), and p(k,l,m+1).  Note that only one
cg                         index may be incremented or decremented by one unit
cg                         in traversing the four edges of the surface element.
cg                         The surface element is part of a a hyperbolic
cg                         paraboloid passing though the four corner points,
cg                         the four midpoints of each edge, and the point at the
cg                         intersection of the lines joining the midponts of
cg                         opposite edges, which is also the mean value of the
cg                         four corner points.  The equation of the surface may
cg                         be found by using the command
cg                         "hyperb HPNAME fit PNAME1 PNAME2 PNAME3 PNAME4".
cg                         The area of a surface element is equal to the mean
cg                         value of the magnitudes of the vector cross products
cg                         at any two opposite vertices of the surface element.
cg                         See command "vector VNAME cross VNAME1 VNAME2".

cg    mesh tests         Command "mesh test block" tests the mesh block for
cg                         several geometric conditions.
cg
cg                         Each combination of k, l and m indices in the mesh
cg                         block is tested to see if is assigned to a (physical)
cg                         point.
cg
cg                         A line element is bounded by two mesh points for
cg                         which two of the mesh indices are the same, and the
cg                         other mesh index has two consecutive values.
cg                         Each pair of two sequential line elements in the
cg                         same index direction is tested to see if the angle
cg                         between them exceeds 90 degrees.
cg
cg                         A surface element is bounded by the line elements
cg                         connecting four mesh points for which one of the mesh
cg                         indices is the same, and the other two mesh indices
cg                         each have two consecutive values.  At each vertex of
cg                         such a surface element, a normal vector is defined by
cg                         the vector product of the two adjacent line element
cg                         vector.  Each surface element is tested to see if
cg                         either of the two angles between opposite normal
cg                         vectors exceeds 90 degrees.  This indicates a
cg                         boomerang or bow-tie, if the vertices are coplanar.
cg
cg                         A volume element is bounded by the six surface
cg                         elements defined by eight mesh points for which each
cg                         of the three mesh indices has two consecutive values.
cg                         Each volume element is tested to see if any of the
cg                         vertex volumes, defined by the scalar triple product
cg                         of the line element vectors at the vertex, is
cg                         negative, indicating an inverted or concave vertex.

cg    mesh translation   See command "mesh", option "move".
cg                         See "transform the mesh".

cg    mesh volume        A volume element in the mesh is defined by a set of
cg                         eight mesh points, with two adjacent values of each
cg                         of the three mesh indices, for example p(k,l,m),
cg                         p(k,l+1,m), p(k,l+1,m+1), p(k,l,m+1), p(k+1,l,m)
cg                         p(k+1,l+1,m), p(k+1,l+1,m+1), p(k+1,l,m+1).
cg                         The volume of a volume element is equal to the mean
cg                         value of the scalar triple products at the eight
cg                         vertices of the volume element.  See command
cg                         "variable VARNAME triple VNAME1 VNAME2 VNAME3".

cn    meshpoint.mac      An input file to create mesh points evenly spaced in
cn                         one, two or three dimensions in a rectangular,
cn                         cylindrical or spherical coordinate system.

cg    message            See "help", "define", "error", "id".

cg    metric             See "metric conversion".

cg    metric conversion
cg                       Selected metric conversion factors are in macro file
cg                         conv.mac in directory ~edwards/work/geom/test .

cg    mfp                See "mean free path".

cg    midpoint           To create a point at the midpoint of an edge of a
cg                         triangle or tetrahedron, use one of the commands
cg
cg                         point PNAME triangle TRNAME W1 W2 W3
cg                         point PNAME tetrahedron TRNAME W1 W2 W3 W4
cg
cg                         with weight factors of 0.5 at the vertices bounding
cg                         the edge, and zero elsewhere.

cg    milliseconds       To find the machine time charged to the problem, use
cg                         command "time".

cg    Min                In the output display, indicates a minimum value on a
cg                         quadric surface, in the direction of a specified
cg                         axis.

cc    min                An option in commands "icalc" and "variable", argument
cc                         FUNCTION, to indicate the minimum function,
cc                         A min B means min (A, B), or the minimum of a group
cc                         of variables, found by command "mean".

cf    min                The minimum function.  Example:  z = min (x, y) means
cf                         that z is the more negative or lesser positive of
cf                         x and y.

cg    min                In the display of the edge lengths of a brick,
cg                         indicates the minimum value of one of the other two
cg                         coordinates perpendicular to the direction of the
cg                         edge.

cg    min/max            In output from commands "root" and "roots", indicates
cg                         an inflection point at which the slope is zero,
cg                         for a polynomial equation.  See "extrema",
cg                         "inflection point".

cg    Minimum            In the output display, indicates:
cg                         a minimum value on a quadric surface, in the
cg                         direction of a specified axis; or
cg                         a minimum value of a set of randomly sampled values,
cg                         or a minimum coordinate in a point cluster.

cg    minimum            To find the minimum value of a coordinate on a quadric
cg                         curve in a major plane, use command "quadric" or
cg                         "slice".
cg
cg                         To find the minimum distance between the points in a
cg                         cluster and another point or cluster, a line, a
cg                         triangle or a plane, use command "distance".
cg
cg                         To find the minimum and maximum points in any
cg                         specified direction on a quadric surface, use
cg                         command "quadric" or "extrema".
cg
cg                         To find the minimum distance from a point to a
cg                         quadric surface, use command "distance", "proximal"
cg                         or "side".
cg
cg                         To find the minimum coordinates of a set of points,
cg                         create a cluster of the set of points.
cg
cg                         To find the minima of a polynomial equation, use
cg                         command "root" or "roots".  See "extrema".

cg    minimum            See "minimum field".

cg    minimum field      To find the minimum field width for output of floating
cg                         point numbers, see "format".

cg    minus              The minus sign, "-".

cc    MINUTES            In command "angles", the integer number of minutes of
cc                         an angle specified in degree, minutes and second
cc                         units.
cc                         May be integer, floating point or a variable.
cc                         See "DEGREES", "SECONDS".

cc    misc               Command to display a help message for miscellaneous
cc                         commands.
cc
cc                         Forms include:
cc
ccin                       help misc
ccin                       misc [help]
cc
cc                           Display a help message for miscellaneous commands.
cc
cc                         Synonyms:  [help, h].

cc    misc               An option in command "help", to display help messages
cc                         for miscellaneous commands.

cc    miscellaneous      Miscellaneous commands include all commands not
cc                         listed by the help messages displayed with commands
cc                         "environment", "object", "error", "math", "random".
cc                         See command "misc".

cc    MMAX               In command "mesh", the largest value of the index m in
cc                         a logical array with indices (k,l,m).

cc    MMOVE              In command "mesh move ...", the number of units to
cc                         move the mesh block in the m direction.

cc    mod                An option in commands "icalc" and "variable", argument
cc                         FUNCTION, to indicate the modulus function.
cc                         In command "icalc", M mod N means amod (M, N).
cc                         which may be written as M modulo N, or M (mod N),
cc                         In command "variable", A mod B means amod (A, B),
cc                         which may be written as A modulo B, or A (mod B).
cc                         The result will always be positive, in the range from
cc                         zero to one less than the modulus.

cf    mod                The modulus function.  Example:  z = mod (x, y) means
cf                         that z is the remainder of x / y, with y added if x
cf                         is negative.
cf                         That is, mod (x, y) = (x/y - int (x/y)) * y + k * y,
cf                         where int (x/y) is the integer part of x / y, and
cf                         k = 1 if x is negative, otherwise k = 0.
cf                         Also may be written as z = x modulo y,
cf                         z = x (mod y) or z = x mod y.

cg    mode               The mode of an array of values is the value which
cg                         occurs most often in the array.

cg    mode               The mode of a character string may be ASCII, integer
cg                         or floating point.

cg    mode               The mode for input of integers in command "icalc"
cg                         may be decimal, hexadecimal or octal.  Output from
cg                         command "icalc" is all of those.  Binary output is
cg                         optional.  See commands "icalc", "hex".

cg    modification       To repeat a GEOM run with modifications, make a new
cg                         input file from one of the output files from the GEOM
cg                         run, either geom_cmd or geom_hsp.
cg                         See "input file".

cg    modular            See "modular arithmetic".

cg    modular arithmetic
cg                       Modular (Gaussian) arithmetic modulo N operates on a
cg                         finite set of integers (0, 1, ..., N - 1) (and their
cg                         negatives), compared with regular (Euclidean)
cg                         arithmetic, which operates on an infinite set of
cg                         integers.  Some other differences are listed here.
cg                         See command "icalc" and "variable", option "mod".
cg
cg                         Modular equivalents: given any positive integer x,
cg                         and a modulus N, the modular equivalent of x is y, or
cg                         x = y mod N = x - m * N, for some non-negative
cg                         integer m, just sufficient to put x - m * N in the
cg                         range from 0 to N - 1.
cg
cg                         Zeros:  a * b = 0 mod N, when in regular arithmetic
cg                         a * b = m * N, for any m.  Neither a nor b need be
cg                         zero.
cg
cg                         Negative integers:  any negative integer (-c) has a
cg                         positive equivalent b, (-c) + m * N = b mod N, for
cg                         some positive integer m, just sufficiently large that
cg                         m * N > c, and b is between 0 and N - 1.
cg                         Commands "icalc" and "variable" always return a
cg                         non-negative result for modular arithmetic.
cg
cg                         Addition and subtraction:  to add a and b, with
cg                         subtraction the same as addition of a negative
cg                         integer, which can be made positive as above,
cg                         (a + b) mod N = ((a mod N) + (b mod N)) mod N.
cg
cg                         Multiplication:  to multiply a and b, with negative
cg                         numbers replaced as above,
cg                         (a * b) mod N = ((a mod N) * (b mod N)) mod N.
cg
cg                         Division:  division is best treated as a provisional
cg                         multiplication problem.  E.g. to find x when
cg                         x = (a / b) mod N, rewrite the equation as
cg                         x = (a * B) mod N, where B is the multiplicative
cg                         inverse of b, such that b * B = 1 mod N.  B only
cg                         exists if b is not zero, and b and N are relatively
cg                         prime, that is, have no common factors other than 1.
cg                         B may be found by trial and error, with the command
cg                         "icalc b mod N" or the command
cg                         "variable x = b mod N".  In both cases, the
cg                         multiplicate inverse of b is displayed, if it exists.
cg                         If N is prime, all division problems have a solution.
cg                         See "inverse, modular".
cg
cg                         Another way is to find an m, if any exists, such that
cg                         x = (a + m * N) / b is an integer.
cg
cg                         Exponentiation:  in modular arithmetic, modulo N,
cg                         (K^M) mod N = (K mod N) ^ (M mod J(N)), where J(N) is
cg                         the Euler's totient function for modulo N arithmetic.
cg                         J(N) is the number of integers in the set
cg                         (1, ..., N - 1) that are relatively prime to N (have
cg                         no common factor other than 1).
cg                         See "totient function", command "factor".
cg                         See "group, modular".
cg
cg                         Polynomial roots:  even a linear equation may have
cg                         no roots, one root or multiple roots, depending
cg                         on the equations:
cg                           3 * x + 1 = 0 mod 6 has no solution.
cg                           3 * x + 1 = 0 mod 5 has one solution, x = 3.
cg                           3 * x - 6 = 0 mod 9 has 3 solutions, x = 2, 5, 8.
cg
cg                         Simultaneous equations:  simultaneous linear
cg                         equations may have no roots, one root or multiple
cg                         roots, depending on the equations.
cg                         See "Chinese Remainder".

cg    modulo             See "modular arithmetic".

cg    modulus            See "mod".

cg    Monte Carlo        To display, create and use Monte Carlo objects, and
cg                         randomly sample from probability distributions, see
cg                         the entries and commands:  icalc, variable, point,
cg                         vector, bin, pdf, random, sample, spin.

cg    motion             See commands "accelerate", "track".

cg    mouse              See "cut and paste".

cg    move               Points, clusters, planes, quadric surfaces and vectors
cg                         may be moved (translated) with command "move".
cg                         Lines, triangles, regular polygons, annular disks,
cg                         regular polyhedrons, and tetrahedrons may be moved by
cg                         moving the points used to create them.  This may be
cg                         done by first putting the points into a cluster.
cg                         Zones may be moved by moving the planes and quadric
cg                         surfaces used to create them.

cc    move               Command to move one or all vectors, points, clusters,
cc                         planes, spheres, circular cylinders, circular cones,
cc                         hyperbolic paraboloids, ellipsoids, axially symmetric
cc                         quadric surfaces or general quadric surfaces in the
cc                         direction of a vector VMOVE, by a distance equal to a
cc                         multiple VMULT of the length of the vector.
cc                         If not specified, VMULT defaults to 1.
cc                         See commands "undo", "redo", "indo".
cc
cc                         Command "move" relates to objects:  axisym, cluster,
cc                         cone, cylinder, ellipsoid, hyperb, plane, point,
cc                         sphere, symbol, variable, vector.
cc
cc                         Forms include:
cc
ccin                       help move
ccin                       move [help]
cc
cc                           Display the command options.
cc
ccin                       move point PNAME VMOVE [VMULT|1]
cc
cc                           Move point PNAME by vector VMOVE multiplied by
cc                           VMULT (defaults to 1).
cc
ccin                       move cluster CLNAME VMOVE [VMULT|1]
cc
cc                           Move cluster CLNAME by vector VMOVE multiplied by
cc                           VMULT (defaults to 1).
cc
ccin                       move point all VMOVE [VMULT|1]
cc
cc                           Move all points by vector VMOVE multiplied by
cc                           VMULT (defaults to 1).
cc
ccin                       move vector VNAME VMOVE [VMULT|1]
cc
cc                           Move vector VNAME by vector VMOVE multiplied by
cc                           VMULT (defaults to 1).
cc
ccin                       move vector all VMOVE [VMULT|1]
cc
cc                           Move all vectors by vector VMOVE multiplied by
cc                           VMULT (defaults to 1).
cc
ccin                       move QTYPE QNAME VMOVE [VMULT|1]
cc
cc                           Move the quadric surface of type QTYPE (plane,
cc                           sphere, cylinder, cone, hyperb, ellipsoid, axisym
cc                           or quadric) with name QNAME, by vector VMOVE
cc                           multiplied by VMULT (defaults to 1).
cc                           See command "quadric", option "move".
cc
ccin                       move QTYPE all VMOVE [VMULT|1]
cc
cc                           Move all quadric surfaces of type QTYPE (plane,
cc                           sphere, cylinder, cone, hyperb, ellipsoid, axisym
cc                           or quadric) by vector VMOVE multiplied by VMULT
cc                           (defaults to 1).
cc                           See command "quadric", option "move".
cc
cc                         Synonyms:  [cluster, cl], [cylinder, cyl],
cc                         [ellipsoid, el, ell, ellipse], [help, h],
cc                         [hyperb, saddle], [move, mv, trans, translate],
cc                         [plane, pl], [point, p, pnt, pt],
cc                         [quadric, q, quad], [sphere, sph], [vector, v, vect].

cc    move               An option in command "mesh", to move a mesh block and
cc                         any mesh points in the mesh block.
cc                         Synonyms:  [move, mv, trans, translate].

cc    move               An option in command "point", to create a set of points
cc                         by repeated displacement of a base point by a
cc                         specified translation vector.
cc                         See command "vector".
cc                         Synonyms:  [move, mv, trans, translate].
cc
cc                         This command option can be replaced, using a do
cc                         loop.  See "do loop use", commands "do", "enddo".

cc    move               An option in commands "plane", "sphere", "cylinder",
cc                         "cone", "ellipsoid", "axisym", "quadric", to create
cc                         a family of quadric surfaces by repeated displacement
cc                         of a base quadric surface by a specified translation
cc                         vector.
cc                         See "concentric", "nest", "parallel", "rotate",
cc                         "scale".
cc                         Synonyms:  [move, mv, trans, translate].
cc
cc                         This command option can be replaced, using a do
cc                         loop.  See "do loop use", commands "do", "enddo".

cg    MOVED              In the display for a point, indicates a mesh point
cg                         in a mesh block for which a mesh index has been
cg                         incremented.

cg    multiple           See "multiple commands", "multiple objects",
cg                         "multiply length", "common multiplier".

cg    multiple commands
cg                       More than one command and/or comment may appear on
cg                         one input line, if separated by the character ";",
cg                         which need not be delimited.
cg                         The character ";" will not be recognized as a
cg                         command or comment separator if between single or
cg                         double quotes, as in commands "alias", "define" and
cg                         "marker" or within a comment.
cg                         Can not be done after commands "indo", "input",
cg                         "read", "redo", "return" or"undo", because chaos
cg                         might ensue.
cg
cg                         NOTE!  An alias can only replace the first word on
cg                         an input line containing multiple commands.

cg    multiple objects   See "axisymmetrics", "bricks", "cones", "cylinders",
cg                         "planes", "points", "quadric families",
cg                         "quadric surfaces", "spheres", "tetrahedrons",
cg                         "vectors", "zones".
cg                         See "family of objects".

cg    multiplicative     See "group, modular", "inverse, modular",

cg    multiplier         See "common multiplier", "lcm".

cg    multiply           To multiply real or integer numbers, see commands
cg                         "variable", "icalc".

cg    multiply length    See "scale", "scale a vector", "vector".

cg    musical            See "musical scale".

cg    musical scale      See "equally-tempered", "frequency ratios",
cg                         "scale.mac", "cents".

cc    mv                 A synonym for move.

N-------------------------------------------------------------------------------

cc    N                  In command "factor", an integer for which all prime
cc                         factors are to be found.

cc    N                  In command "icalc", the second integer (not a variable)
cc                         argument of a two-argument function, in the current
cc                         icalc integer mode.

cc    N                  In command "spin", the number of random numbers to go
cc                         past in the random number sequence.
cc                         Must be an integer or integer variable.

cc    N1                 In command "do", the initial value of the loop variable
cc                         VARNAME.  Must be an integer or integer variable.

cc    N1                 See "N1, N2", "N1, N2, N3".

cc    N1, N2             In command "mesh", indices of K, l or m layers.

cc    N1, N2, N3         In command "plot", indices of plot points.
cc                         In command "mesh", indices of K, l or m layers.

cc    N2                 In command "do", the final value of the loop variable
cc                         VARNAME, if N2 - N1 is an integer multiple of INC.

cc    N2                 See "N1, N2", "N1, N2, N3".

cc    N3                 See "N1, N2, N3".

cg    name               See "name length", "name sequence".

cg    name conflicts     Tests are made for conflicts between the names of
cg                         preset synonyms and symbols and aliases created by
cg                         the user.  If any are found, a warning message is
cg                         displayed, and the objects are displayed.
cg                         To find any existing object with the name OBJNAME,
cg                         use command "search OBJNAME".

cg    name length        Operational, geometric and Monte Carlo object names may
cg                         have up to 24 characters, including any subscripts,
cg                         except markers, which may only have names of one
cg                         character, and big integers, which may only have
cg                         names of one upper case alphabetic character.

cg    name sequence      Object names may be given an ASCII sequence by
cg                         creating a base name for the object type with command
cg                         "last", or by choosing a name with enough
cg                         characters for the full sequence when creating the
cg                         first object of a given type, and then using the
cg                         character "+" or "-" in place of the new object name
cg                         in creating subsequent objects of the same type.
cg                         The sequence may be changed by creating a new base
cg                         name, using command "last".
cg                         This does not apply to the names of big integers.
cg                         See "increment names".
cg                         Also see command "point", options "polygon",
cg                         command "vector", options "rotate" and "scale";
cg                         "polyhedron", "move", "rotate" and "scale";
cg                         command "plane", options "parallel", "rotate" and
cg                         "scale";
cg                         commands "cylinder" and "sphere", option
cg                         "concentric";
cg                         command "cone", option "nest";
cg                         commands "sphere", "cylinder", "cone", "ellipsoid",
cg                         "axisym" and "quadric", options "move", "rotate" and
cg                         "scale".

cg    named              See "named objects".

cg    named objects      The following commands are used to create and display
cg                         named objects:
cg
cg                         alias (replaces the initial word of a command)
cg                         axisym (an axisymmetric quadric surface)
cg                         big (a big integer)
cg                         bin (a probability bin)
cg                         brick (a volume bounded by coordinate surfaces)
cg                         cluster (a set of points)
cg                         cone (a circular cone)
cg                         cylinder (a circular cylinder)
cg                         disk (an annular disk)
cg                         ellipsoid (a triaxial ellipsoid)
cg                         hyperb (a hyperbolic paraboloid)
cg                         line (a straight line)
cg                         marker (a character string)
cg                         operator (a tensor operator)
cg                         pdf (a probability distribution function)
cg                         plane (a plane)
cg                         point (a point)
cg                         polygon (a regular polygon)
cg                         polyhedron (a regular polyhedron)
cg                         quadric (a quadric surface)
cg                         sphere (a sphere)
cg                         symbol (replaces any word, stem or subscript
cg                           in a command)
cg                         tetrahedron (a tetrahedron)
cg                         triangle (a triangle)
cg                         variable (an integer or floating point variable)
cg                         vector (a three-dimensional vector)
cg                         zone (a 3-D zone bounded by quadric surfaces)
cg
cg                       See "object types".

cg    names              The objects aliases, markers, variables, points,
cg                         clusters, lines, triangles, regular polygons, annular
cg                         disks, vectors, tensor operators, quadric surfaces,
cg                         regular polyhedrons, tetrahedrons, bricks, zones,
cg                         bins probability distribution functions (pdfs) and
cg                         symbols may have arbitrary names, with or without
cg                         suscripts, with the following restrictions:
cg
cg                         Each name must be unique for its object type.
cg                         The object type "quadric surface" includes planes,
cg                         spheres, circular cylinders, circular cones,
cg                         hyperbolic paraboloids, ellipsoids, axially symmetric
cg                         quadric surfaces and general quadric surfaces.
cg                         Axially symmetric quadric surfaces include planes,
cg                         spheres, circular cylinders, circular cones, circular
cg                         paraboloids, and circular hyperboloids.
cg                         To find any existing object with the name OBJNAME,
cg                         use command "search OBJNAME".
cg
cg                         No name may have more than 24 characters, except for
cg                         markers, which may have only one character, and
cg                         big integers, which may have only one upper case
cg                         alphabetic character.
cg                         No name may have any leading or trailing blanks or
cg                         any included blanks unless used only with a non-blank
cg                         field delimiter.  No name may have any leading,
cg                         trailing or included character when used with that
cg                         character as a field delimiter.
cg
cg                         Variable names must begin with a lower or upper case
cg                         letter or one of the characters "$", "%" or "@", may
cg                         not be "random" or "tol", and must not contain ";".
cg                         Big integer names must consist of a single upper case
cg                         alphabetic character ("A" to "Z").
cg
cg                         No name may be "+", "-", "all", "comp.", "h", "help",
cg                         "list" or "thru", begin with "!" or contain ";".
cg                         See the entries for particular names to find other
cg                         prohibited names.
cg                         Control characters should be avoided, as they may
cg                         cause problems with displaying data in the output
cg                         files or in viewing files with a text editor.
cg
cg                         Subscripted names may be created with any of the
cg                         commands for creating objects or with command
cg                         "symbol".  See "subscripted names", "array".
cg                         Use of subscripts minimizes the need for different
cg                         character strings for names.
cg
cg                         Objects may be sorted into increasing, decreasing or
cg                         random order of their ASCII or integer names, with
cg                         any completly integer names or any integer subscripts
cg                         treated as integers rather than ASCII strings, using
cg                         command "sort".
cg                         Any other numerical sorting requires
cg                         names of equal length, with the numerical part
cg                         right-adjusted and left-filled with the same
cg                         character (e.g., "0" or "_").
cg
cg                         Words used in the description of commands, where you
cg                         must provide the name of an object, include:
cg
cg                         aliases:      ANAME, ANAME1, ANAME2, ANAME3, ...
cg                         annular disks:
cg                                       DKNAME, DKNAME1, DKNAME2
cg                         axisyms:      AXNAME, AXNAME1, AXNAME2,
cg                                       AXQ(1), AXQ(2), ...
cg                         big integers: BIGNAME, BIGNAME1, BIGNAME2, BIGNAME3
cg                         bins:         BINAME, BINAME1, BINAME2, BINAME3
cg                         bricks:       BRNAME, BRNAME1, BRNAME2,
cg                                       BR(1), BR(2), ...
cg                         circular cylinders:
cg                                       CYLNAME, CYLNAME1, CYLNAME2
cg                                       CYL(1), CYL(2), ...
cg                         circular cones:
cg                                       CNNAME, CNNAME1, CNNAME2,
cg                                       CONE(1), CONE(2), ...
cg                         clusters:     CLNAME, CLNAME1, CLNAME2, CLNAME3
cg                         ellipsoids:   ELNAME, ELNAME1, ELNAME2,
cg                                       ELL(1), ELL(2), ...
cg                         hyperbolic paraboloids:
cg                                       HPNAME, HPNAME1, HPNAME2, ...
cg                         lines:        ALNAME, ALNAME1, ALNAME2
cg                         markers:      AMNAME, AMNAME1, AMNAME2, AMNAME3, ...
cg                         variables:
cg                                       VARNAME, VARNAME1, VARNAME2
cg                         pdfs:         PDFNAME, PDFNAME1, PDFNAME2
cg                         planes:       PLNAME, PLNAME1, PLNAME2
cg                                       PL(1), PL(2), PL(3), ...
cg                         points:       PNAME, PNAME(1), PNAME(2), PNAME(3),
cg                                       PNAME1, PNAME2, PNAME3,
cg                                       PAXIS, PCEN, PCOP, PINIT, PINV, PVER
cg                         quadrics:     QNAME, QNAME1, QNAME2,
cg                                       QNAME(1), QNAME(2), ...
cg                                       AQU, AQV, AQW
cg                         regular polygons:
cg                                       PGNAME, PGNAME1, PGNAME2
cg                         regular polyhedrons:
cg                                       PHNAME, PHNAME1, PHNAME2
cg                         spheres:      SPHNAME, SPHNAME1, SPHNAME2
cg                                       SPH(1), SPH(2), ...
cg                         symbols:      SYMBNAME, SYMBNAME1, SYMBNAME2,
cg                                       REPLNAME
cg                         tensor operators:
cg                                       OPNAME, OPNAME1, OPNAME2
cg                         tetrahedrons:
cg                                       TETNAME, TETNAME1, TETNAME2
cg                                       TET(1), TET(2), ...
cg                         triangles:    TRNAME, TRNAME1, TRNAME2
cg                         vectors:      VINIT, VNAME, VNAME1, VNAME2, VNAME3,
cg                                       VNAME4, VACC, VAXIS, VMOVE, VNORM
cg                         zones:        ZNAME, ZNAME1, ZNAME2
cg                                       ZN(1), ZN(2), ...
cg
cg                         See "object types", "search".

cc    NBASE              In commands "base" and "big", an integer number base.

cpar  nbrickm            Maximum number of bricks.  Now 1000.

cpar  nclustm            Maximum number of point clusters.  Now 100.

cpar  ncmdm              Maximum number of saved input lines.  Now 2000.

cc    NCOLUMNS           In command "plot", option "size", the number of plot
cc                         array columns [NCOLUMNS (<= 164)].  Used with NLINES.
cc                         The combination of (NCOLUMNS, NLINES) of 65 by 41 is
cc                         almost square, and fits nicely in a standard 80 by 47
cc                         window.  Other almost square sizes are 81 by 51,
cc                         101 by 61, 129 by 81 and 161 by 101.

cpar  ndiskm             Maximum number of annular disks.  Now 1000.

cg    nearest            To find the point on a line, a triangle, a plane or a
cg                         quadric surface nearest to a specified point, or the
cg                         points on two lines where they are nearest each
cg                         other, use command "distance".
cg
cg                         To find the point on a plane or quadric surface
cg                         nearest to a specified point, use command "distance",
cg                         "proximal" or "side".
cg
cg                         To find the nearest points on a plane and a sphere,
cg                         or on two spheres, or on a sphere and a circular
cg                         cylinder, or on two circular cylinders, use command
cg                         "distance".
cg
cg                         To find the point on a quadric surface nearest a
cg                         specified plane, use command "distance".

cg    negligible         A value is negligible if it is less than TOL times
cg                         some standard or average value.
cg                         See "error estimate".

cc    nest               An option in command "cone", to create a family of
cc                         nested cones, with a common vertex and axis, with
cc                         vertex half-angles equally spaced between two
cc                         specified values.
cc                         See "concentric", "move", "parallel", "rotate",
cc                         "scale".
cc
cc                         This command option can be replaced, using a do loop.
cc                         See "do loop use", commands "do", "enddo".

cg    nest               Blocks of commands bracketed by commands "if" and
cg                         "endif" may be nested.  See command "if".

cg    nested             Refers to a family of concentric spheres or cylinders,
cg                         or a set of nested cones.
cg                         See "parallel", "move", "rotate", "scale".

cg    nested             See "nested cones".

cg    nested cones       A family of nested cones have the same vertex and axis,
cg                         but with equally spaced vertex half-angles.  Such a
cg                         family may be created with command "cone", option
cg                         "nest".
cg                         See "concentric", "move", "parallel", "rotate",
cg                         "scale".

cg    net                The net effect of a process or operation on a system,
cg                         is based on a comparison between the final and
cg                         initial conditions of the system, ignoring any
cg                         intermediate conditions.

cg    new                An option in command "sudoku", to start a new Sudoku
cg                         puzzle.

cc    nint               An option in command "variable", argument FUNCTION,
cc                         to indicate the nearest integer function.
cc                         A nint B means A * nint (B).

cf    nint               The nearest integer function.  Example:  y = nint (x)
cf                         means y is the nearest integer to x.  If x has a
cf                         non-integer part of 1/2, y will be the integer next
cf                         larger in magnitude, regardless of sign.

cpar  nlinem             Maximum number of lines.  Now 1000.

cc    NLINES             In command "plot", option "size", the number of plot
cc                         array lines [NLINES (<= 102)].  Used with NCOLUMNS.
cc                         The combination of (NCOLUMNS, NLINES) of 65 by 41 is
cc                         almost square, and fits nicely in a standard 80 by 47
cc                         window.  Other almost square sizes are 81 by 51,
cc                         101 by 61, 129 by 81 and 161 by 101.

cc    nobin              Option in command "icalc", to turn off the display of
cc                         results of command "icalc" in binary mode.
cc                         This is the default option.

cc    nomesh             An option in command "cluster", to create a cluster of
cc                         all points not assigned to the mesh.

cg    non-mesh           A non-mesh point is a point that is not assigned to
cg                         the mesh.  See "nomesh".  See command "cluster".

cg    non-simple         See "non-simple planes".

cg    non-simple planes
cg                       Non-simple planes are two coincident planes, two
cg                         parallel planes, or two intersecting planes,
cg                         represented by a single quadric surface.
cg                         The long display for such quadric surfaces will
cg                         include the simple planes.
cg
cg                         Two simple planes may have the implicit equations
cg                           F1 = ac + ax * x + ay * y + az * z = 0,
cg                           F2 = bc + bx * x + by * y + bz * z = 0.
cg
cg                         The two simple planes have the normal vectors
cg                           N1 = (ax, ay, az), N2 = (bx, by, bz), respectively.
cg
cg                         Any line of intersection of the two simple planes
cg                           has the direction vector
cg                           N = (NX, NY, NZ) = N1 x N2 (the cross product),
cg                         which is zero if the planes are coincident or
cg                         parallel.
cg                           NX = ay * bz - az * by
cg                           NY = az * bx - ax * bz
cg                           NZ = ax * by - ay * bx
cg
cg                         The equation of the line of intersection may be
cg                         represented in several ways:
cg                           x = ((ay * bc - ac * by) + NX * z) / NZ (NZ not 0)
cg                           y = ((ac * bx - ax * bc) + NY * z) / NZ (NZ not 0)
cg                           y = ((az * bc - ac * bz) + NY * x) / NX (NX not 0)
cg                           z = ((ac * by - ay * bc) + NZ * x) / NX (NX not 0)
cg                           z = ((ax * bc - ac * bx) + NZ * y) / NY (NY not 0)
cg                           x = ((ac * bz - az * bc) + NX * y) / NY (NY not 0)
cg
cg                         A single quadric surface representing both planes
cg                         may have the equation
cg                           F = F1 * F2 = sc + sx * x + sy * y + sz * z
cg                               + sxy * x * y + syz * y * z + szx * z * x
cg                               + sxx * x^2   + syy * y^2   + szz * z^2 = 0,
cg                         where
cg                           sc  = ac * bc
cg                           sx  = ac * bx + ax * bc
cg                           sy  = ac * by + ay * bc
cg                           sz  = ac * bz + az * bc
cg                           sxy = ax * by + ay * bx
cg                           syz = ay * bz + az * by
cg                           szx = az * bx + ax * bz
cg                           sxx = ax * bx
cg                           syy = ay * by
cg                           szz = az * bz
cg
cg                         The quadric surface has the normal vector
cg                           N12  = (N12x, N12y, N12z) = F2 * N1 + F1 * N2
cg                           N12x = sx + 2*sxx*x + sxy * y + szx * z
cg                           N12y = sy + sxy * x + 2*syy*y + syz * z
cg                           N12z = sz + szx * x + syz * y + 2*szz*z

cc    none               An option in command "synonym", to display all command
cc                         words and key words for which there is no preset
cc                         synonym.

cpar  noperm             Maximum number of tensor operators.  Now 100.

cc    norepl             An option in command "sample", to sample without
cc                         replacement (do not sample the same probability bin
cc                         more than once).  See "random reordering".

cg    Norm               In an output display, indicates a normal vector.

cc    norm               A synonym for normal.

cc    normal             An option in command "bin", to create a normal
cc                         (Gaussian) probability distribution.
cc                         Synonyms:  [normal, norm].

cc    normal             An option in command "project", to project in the
cc                         direction normal to a plane.
cc                         Synonyms:  [normal, norm].
cc                         Also see "normal vector".

cc    normal             An option in command "reflect", to reflect a vector
cc                         in a surface with a specified normal vector.  The
cc                         reflection direction VREFL of a track with initial
cc                         direction vector VINIT, reflected in a surface with
cc                         normal vector VNORM, is found as follows:
cc                         UNORM = VNORM / |VNORM|,
cc                         VREFL = VINIT - 2 * (VINIT dot UNORM) * UNORM

cc    normal             An option in command "vector", to create a vector
cc                         normal at a specified point to a specified family
cc                         of quadric surfaces.
cc                         Synonyms:  [normal, norm].

cg    normal             A normal probability distribution is a type of
cg                         probability bin or probability distribution function
cg                         (pdf) for which the differential probability p(V) of
cg                         a value V of the random variable is a normal
cg                         (Gaussian) distribution with a mean value VMEAN and a
cg                         standard deviation DEV:
cg                         p(V) = exp (-f(V)) / (DEV * sqrt (2 * pi)), where
cg                         f(V) = ((V - VMEAN) / DEV)^2 / 2
cg                         See commands "bin", "pdf", "sample".

cg    normal             See "normal vector".

cg    normal vector      The normal vector for a surface F(x,y,z) = 0 is the
cg                         vector N = (dF/dx, dF/dy, dF/dz), (partial
cg                         derivatives) which may be evaluated anywhere in
cg                         space, not just on the surface.  The normal vector
cg                         field is everywhere perpendicular to the family of
cg                         surfaces F(x,y,z) = C, for arbitrary C.
cg
cg                         The positive side of a surface is in the direction
cg                         of the normal vector at the surface.
cg
cg                         A surface is a positive boundary of a zone if the
cg                         positive side of the surface is outside the zone.
cg
cg                         For a small change in coordinates, the distance in
cg                         the normal direction is dF / |N|, where
cg                         dF = F(x+dx,y+dy,z+dz) - F(x,y,z).
cg
cg                         To find the normal vector, at a point PNAME, of the
cg                         family of quadric surfaces associated with a quadric
cg                         surface QNAME, execute command
cg                         "vector VNAME normal PNAME QNAME".
cg
cg                         The normal vector of spheres, circular cylinders,
cg                         circular cones and ellipsoids is usually in the
cg                         "outward" direction, but may be made "inward" by
cg                         creating them with command "quadric QNAME = ...",
cg                         with the signs of all coefficients reversed from the
cg                         standard form.
cg
cg                         Indicated in commands by VNORM.
cg
cg                         See "radius of curvature".

cg    notes              See "definition", commands "alias", "marker".

cpar  npairm             Maximum number of entries in cluster pair table.
cpar                       Now 1000.

cg    npin               The number of randomly sampled points actually found to
cg                         be in a zone, when command "mcvol" is used.

cpar  npointm            Maximum number of points.  Now 1000.

cpar  npolygm            Maximum number of regular polygons.  Now 1000.

cpar  npolyhm            Maximum number of regular polyhedrons.  Now 1000.

cc    NPOW               In command "big", the power to which a big integer
cc                         is to be raised.

cc    NQU                See "NQU, NQV, NQW".

cc    NQU, NQV, NQW      In command "zone", the number of quadric surfaces in
cc                         a family of quadric surfaces.

cpar  nquadm             Maximum number of quadric surfaces.  Now 1000.
cpar                       Includes planes, spheres, circular cylinders,
cpar                       circular cones, hyperbolic paraboloids, ellipsoids,
cpar                       axially symmetric quadric surfaces, and other quadric
cpar                       surfaces.

cc    NQV                See "NQU, NQV, NQW".

cc    NQW                See "NQU, NQV, NQW".

cc    NROOT              In command "big", the root to be found, of a big
cc                         integer.

cc    NSAMP              Number of points to randomly sample in a brick, to
cc                         estimate the volume of a zone.
cc                         Used in command "mcvol".
cc                         The number of events/objects/values to randomly
cc                         sample from a probability distribution function
cc                         (pdf).  Used in command "sample".
cc                         NSAMP must be a positive integer or integer variable.

cc    NSET               Number of sets of NSAMP events/objects/values to
cc                         sample, without replacement, from probability
cc                         distribution function PDFNAME, in command "sample".
cc                         If not specified, the default is 1.
cc                         NSET must be a positive integer or integer variable.

cpar  ntetm              Maximum number of tetrahedrons.  Now 1000.

cpar  ntrigm             Maximum number of triangles.  Now 1000.

cg    null               A null input line has no non-null arguments.
cg                         A null argument has zero length or contains only
cg                         blank characters, but is delimited by the beginning
cg                         of an input line and a non-blank field delimiter,
cg                         by two non-blank field delimiters, by a field
cg                         delimiter and the end of an input line or by the
cg                         beginning and end of an input line.
cg                         A null character is represented by binary "0000".
cg
cg                         A null vector has no components greater than TOL.

cc    null               An option in command "if", to see if a variable has
cc                         been created.  Useful in a macro, to see if it has
cc                         been specified as the input medium without necessary
cc                         arguments.

cc    NUMAX              In command "axisym", the number of axially symmetric
cc                         quadric surfaces to create.  See "INC".
cc                         NUMAX must be a positive integer or integer variable.

cg    number             The number or pound character, "#".

cg    number             See "number base", "prime number", command "base".

cc    number base        See command "base".

cc    NUMBR              In command "brick", the number of bricks to create.
cc                         See "INC".
cc                         NUMBR must be a positive integer or integer variable.

cc    NUMCONE            In command "cone", the number of circular cones to
cc                         create.  See "INC".
cc                         NUMCONE must be a positive integer or integer
cc                         variable.

cc    NUMCYL             In command "cylinder", the number of circular
cc                         cylinders to create.  See "INC".
cc                         NUMCYL must be a positive integer or integer
cc                         variable.

cc    NUMDIR             In command "proximal", the number of directions to
cc                         randomly sample from an isotropic distribution to
cc                         try to find the proximal point in a quadric surface,
cc                         relative to a specified point, if the standard method
cc                         fails.  Defaults to 100.
cc                         The next stage is to randomly sample 10 * NUMDIR
cc                         directions from a cosine-power distribution directed
cc                         toward the last best direction, with the power
cc                         geometrically increasing from 1 to 10^10.

cc    NUMELL             In command "ellipsoid", the number of ellipsoids
cc                         to create.  See "INC".
cc                         NUMELL must be a positive integer or integer
cc                         variable.

cg    numerical          A numerical command argument is one which must specify
cg                         an integer or floating point value.
cg
cg                         If the argument must specify an integer value, it may
cg                         only be input as an integer, the name of an integer
cg                         variable or a symbolic word which, after symbol
cg                         replacement, is an integer.
cg
cg                         Otherwise, it may be input as an integer, a floating
cg                         point number, the name of an integer or floating
cg                         point variable or a symbolic name which, after
cg                         symbol replacement, is either integer or floating
cc                         point.
cg
cg                         See "ARGUMENT", "arguments", "symbol", "value",
cg                         "variable".
cg
cg                         Numerical tolerance is determined by TOL.
cg                         See command "tol".

cc    NUMLINE            In command "line", the number of lines to create.
cc                         See "INC".
cc                         NUMLINE must be a positive integer or integer
cc                         variable.

cc    NUMPL              In command "plane", the number of planes to create.
cc                         See "INC".
cc                         NUMPL must be a positive integer or integer variable.

cc    NUMPRIME           In command "variable", the number of prime numbers to
cc                         save as variables.

cc    NUMPT              In command "point", the number of points to create.
cc                         See "INC".
cc                         NUMPT must be a positive integer or integer variable.

cc    NUMQ               In command "quadric", the number of quadric surfaces
cc                         to create displaced from a base quadric surface in
cc                         increments of a specified vector, or scaled from a
cc                         base quadric surface by repeated scaling with a
cc                         specified scaling operator and invariant point.
cc                         See "INC".
cc                         NUMQ must be a positive integer or integer variable.

cc    NUMSPH             In command "sphere", the number of spheres to
cc                         create.  See "INC".
cc                         NUMSPH must be a positive integer or integer
cc                         variable.

cc    NUMSTR             In command "increment", the number of delimited strings
cc                         to display.  See "INC".
cc                         NUMSTR must be a positive integer or integer
cc                         variable.

cc    NUMT               In command "ratio", the number of terms in a geometric
cc                         series, for which the first term is TERM1 and the
cc                         sum is SUM.
cc                         NUMT must be a positive integer or integer variable.

cc    NUMTET             In command "tetrahedron", the number of tetrahedrons to
cc                         create.  See "INC".
cc                         NUMTET must be a positive integer or integer
cc                         variable.

cc    NUMTR              In command "triangle", the number of triangles to
cc                         create.

cc    NUMV               In command "vector", the number of vectors to create.
cc                         See "INC".
cc                         NUMV must be a positive integer or integer variable.

cc    NUMVAR             In command "variable", option "series", the number of
cc                         variables to create.  See "INC".
cc                         NUMVAR must be a positive integer or integer
cc                         variable, or a symbolic word which, after symbol
cc                         replacement, is an integer number.

cc    NUMZN              In command "zone", the number of zones to create.
cc                         See "INC".
cc                         NUMZN must be a positive integer or integer variable.

cpar  nvarm              Maximum number of variables.  Now 1000.

cpar  nvectm             Maximum number of vectors.  Now 1000.

cc    NVER               Number of vertices in a regular polygon:  must be
cc                         an integer or integer variable between 3 and nverpgm
cc                         (now 1000).  Used in command "polygon".
cc                         Number of vertices in a regular polyhedron:  must be
cc                         4, 6, 8, 12 or 20.  Used in command "polyhedron".
cc                         NVER must be a positive integer or integer variable.

cpar  nverpgm            Maximum number of vertices on a regular polygon.
cpar                       Now 1000.

cg    nx                 Number of points with different x values resulting from
cg                         command "slice".

cg    ny                 Number of points with different y values resulting from
cg                         command "slice".

cg    nz                 Number of points with different z values resulting from
cg                         command "slice".

cpar  nzonem             Maximum number of zones.  Now 1000.

cpar  nzquadm            Maximum number of bounding surfaces of a zone.
cpar                       Now 16.

O-------------------------------------------------------------------------------

cc    o                  A synonym for output.

cc    obj                A synonym for object.

cc    OBJ1               See "OBJ1, OBJ2, OBJ3".

cc    OBJ1, OBJ2, OBJ3   In command "list", any geometrical or operational
cc                         object types.

cc    OBJ2               See "OBJ1, OBJ2, OBJ3".

cc    OBJ3               See "OBJ1, OBJ2, OBJ3".

cg    object             See "object types", "named objects".

cc    object             Command to display a help message for commands that
cc                         create and display objects, which include:
cc                         aliases, annular disks, axially symmetric quadric
cc                         surfaces, big integers, bricks, circular cones,
cc                         circular cylinders, clusters of points, ellipsoids,
cc                         general quadric surfaces, hyperbolic paraboloids,
cc                         lines, logical meshes, markers, mesh blocks,
cc                         variables, operators (to reflect, rotate, invert an
cc                         scale), planes, points, probability bins, probability
cc                         distribution functions, regular polygons, regular
cc                         polyhedrons, spheres, symbols, tetrahedrons,
cc                         triangles, vectors and zones.  See "named objects".
cc
cc                         Forms include:
cc
ccin                       help object
ccin                       object [help]
cc
cc                           Display a help message for commands that create and
cc                           display objects.
cc
cc                         Synonyms:  [help, h], [object, obj].

cc    object             An option in command "help", to display help messages
cc                         for commands that create and display objects.

cg    object             See "object names", "object types", "multiple objects".

cg    object names       See "named objects", "names", "object types".

cg    object types       An object may be an environmental, operational,
cg                         geometric or Monte Carlo object.
cg                         See "named objects".
cg
cg                         Environmental objects and the relevant commands
cg                         include:
cg
cg                         help messages ("alph", "help", "env", "err", "obj",
cg                           "math", "misc"),
cg                         definitions in geom_base ("define"),
cg                         the prompt message ("prompt"),
cg                         the code date and the run date ("when"),
cg                         the problem title ("title"),
cg                         the machine time ("time"),
cg                         the input files, including the user's terminal
cg                           and external files ("input"),
cg                         the input lines saved in memory ("input"),
cg                         the input lines saved in file geom_cmd ("input"),
cg                         the output files, including the user's
cg                           terminal, geom_cmd, geom_hsp, and any other output
cg                           files ("output"),
cg                         the numerical tolerance limit ("tol"),
cg                         the integer display mode ("icalc").
cg                         the angle units ("angles"),
cg                         the coordinate system ("coordinate"),
cg                         the field delimiter ("delimiter"),
cg                         synonyms ("symbol", "alias"),
cg                         symbols ("symbol"),
cg                         aliases ("alias"),
cg                         markers ("marker"),
cg                         plot specifications ("plot").
cg
cg                         Operational objects and the commands to display and
cg                         create them include:
cg
cg                         variables ("variable"),
cg                         big integers ("big"), with up to 1001 digits,
cg                         vectors ("vector"), which may also be used as
cg                           translation operators,
cg                         tensor operators or symmetry operators, including
cg                           reflection, rotation, inversion and scaling
cg                           operators ("operator").
cg
cg                         Geometric objects and the commands to display and
cg                         create them include:
cg
cg                         points ("point"),
cg                         clusters of points ("cluster"),
cg                         a logical mesh ("mesh"),
cg                         lines ("line"),
cg                         triangles ("triangle"),
cg                         regular polygons ("polygon)",
cg                         annular disks ("disk"),
cg                         quadric surfaces:
cg                           planes ("plane") ("quadric"),
cg                           spheres ("sphere", "quadric"),
cg                           circular cylinders ("cylinder", "quadric"),
cg                           circular cones ("cone", "quadric"),
cg                           hyperbolic paraboloids ("hyperb", "quadric"),
cg                           ellipsoids ("ellipsoid", "quadric"),
cg                           axially symmetric quadric surfaces ("axisym",
cg                           "plane", "sphere", "cylinder", "cone", "ellipsoid",
cg                           "quadric"),
cg                           general quadric surfaces ("quadric"),
cg                         regular polyhedrons ("polyhedron"),
cg                         tetrahedrons ("tetrahedron"),
cg                         six-faced solid bricks ("brick"),
cg                         zones ("zone").
cg
cg                         Monte Carlo objects and the commands to display and
cg                         create them include:
cg
cg                         probability bins ("bin"),
cg                         probability distribution functions ("pdf").
cg
cg                         Object types OBJTYPE for which the objects are
cg                         assigned names OBJNAME are listed below:
cg
cg                         Object type                OBJTYPE       OBJNAME
cg                         -------------------------  ---------     --------
cg                         alias                      alias         ALNAME
cg                         axisymmetric quadric       axisym        AXNAME
cg                         big integer                big           BIGNAME
cg                         probability bin            bin           BINAME
cg                         six-faced solid brick      brick         BRNAME
cg                         cluster of points          cluster       CLNAME
cg                         circular cone              cone          CNNAME
cg                         annular disk               disk          DKNAME
cg                         circular cylinder          cylinder      CYLNAME
cg                         ellipsoid                  ellipsoid     ELNAME
cg                         hyperbolic paraboloid      hyperb        HPNAME
cg                         line between two points    line          ALNAME
cg                         plot marker                marker        AMNAME
cg                         tensor operator            operator      OPNAME
cg                         probability dist function  pdf           PDFNAME
cg                         planar surface             plane         PLNAME
cg                         point                      point         PNAME
cg                         regular polygon            polygon       PGNAME
cg                         regular polyhedron         polyhedron    PHNAME
cg                         quadric surface            quadric       QNAME
cg                         sphere                     sphere        SPHNAME
cg                         symbol                     symbol        SYMBNAME
cg                         tetrahedron                tetrahedron   TETNAME
cg                         triangle                   triangle      TRNAME
cg                         variable                   variable      VARNAME
cg                         vector                     vector        VNAME
cg                         zone                       zone          ZNAME
cg
cg                         Objects may be listed, sorted, copied, deleted,
cg                         renamed, repacked and searched for.
cg
cg                         See commands "copy", "debug", "delete", "last",
cg                         "list", "rename", "repack", "search", "sort",
cg                         "symbol", "synonym" and "table".
cg
cg                         For subscripted object names, see
cg                         "subscripted names".

cg    object use         See the entry in this file for each command.

cc    OBJNAME            In commands "copy", "delete", "rename", the name of an
cc                         object of type OBJTYPE.

cc    OBJNAME            In command "last", the base name for objects of type
cc                         OBJTYPE.
cc
cc                         No object name may be "+", "-", "all", "h", "help",
cc                         "list" or "thru", begin with "!" or contain ";".
cc
cc                         No object name may exceed 24 characters, except for
cc                         names of markers, which may only have 1 character.

cc    OBJNAME1           In command "search", the name of an object to be
cc                         searched for among the objects with assigned names,
cc                         marker text, symbol replacements, alias replacements,
cc                         or numerical values to be searched for among the
cc                         values of variables, big integers, the coordinates of
cc                         points or vector bound points, and the components of
cc                         vectors.  Names and values may both be specified in
cc                         the same command.
cc                         If any OBJNAME is unsubscripted, it also includes all
cc                         objects with the subscripted name
cc                         OBJNAME(NSUB1,NSUB2,...).
cc                         Likewise for OBJNAME2, OBJNAME3, ...

cc    OBJNAME2           In commands "copy" and "rename", the name of a new or
cc                         renamed object of type OBJTYPE.
cc
cc                         No object name may be "+", "-", "all", "h", "help",
cc                         "list" or "thru", begin with "!" or contain ";".
cc
cc                         No object name may exceed 24 characters, except for
cc                         names of markers, which may only have 1 character.

cc    OBJTYPE            In commands to create and display objects, and in
cc                         commands, "copy", "delete", 'last", "list",
cc                         "rename", "repack" and "sort", an object type with
cc                         name OBJNAME.  See "object types".

cg    oblate             See "oblate spheroid".

cg    oblate spheroid    An oblate spheroid is a circular ellipsoid for which
cg                         the symmetry axis is along the smaller semiaxis.
cg                         See "circular ellipsoid".
cg
cg                         The surface area of a oblate spheroid with equal
cg                         semiaxes SAY = SAZ, shorter semiaxis SAX, and
cg                         eccentricity e = sqrt (1 - (SAX / SAZ)^2), is
cg                         area = 2 * pi * SAZ^2 * (1 + (SAX / SAZ)^2 * f(e)),
cg                         where f(e) = log ((1 + e)/(1 - e)) / (2 * e).

cc    oct                A synonym for octal, in commands "big" and "icalc".

cg    octahedron         A regular octahedron has 6 vertices, 12 edges, and
cg                         8 equilateral triangular faces.
cg                         The edge length, face area, volume, radius of
cg                         inscribed sphere, dihedral angle and central edge
cg                         angle, for a circumscribed sphere radius of 1.0 are:
cg
cg                         edge    = sqrt (2.0)              = 1.414213562373
cg                         area    = 0.5 * sqrt (3.0)        = 0.8660254037844
cg                         volume  = 4.0 / 3.0               = 1.333333333333
cg                         rinsc   = 1.0 / sqrt (3.0)        = 0.5773502691896
cg                         angdih  = acos (-1.0 / 3.0)       = 109.4712206345
cg                         angcent = acos (0.0)              = 90.00000000000
cg
cg                         See "polyhedron".

cc    octal              An option in command "big", to display a big integer
cc                         in octal.

cc    octal              An option in command "icalc", to set the input mode for
cc                         integers in command "icalc" to octal.

cg    octal              The octal form of an integer M, specified in the
cg                         current integer mode (see command "icalc"),
cg                         may be displayed with command:
cg
cg                         icalc M
cg
cg                           Display M in decimal, hexadecimal, octal and
cg                           (optionally) binary mode.

cc    off                An option in command "plot", to stop storing plot
cc                         points.  They will still be saved for later plotting.
cc                         See commands "plot off" and "plot delete".

cg    off                Some environmental parameters may be turned on and off.
cg                         See commands "icalc", "plot", "prompt", "tol".

cg    offset             See commands "move", "plane ... parallel ...".

cc    on                 An option in command "plot", to start or resume storing
cc                         plot points.  Previously stored points will still be
cc                         saved.  See commands "plot on" and "plot delete".

cg    on                 Some environmental parameters may be turned on and
cg                         off.

cg    onto               See "project".

cc    op                 A synonym for operator.

cc    oper               A synonym for operator.

cg    operational        To create, display and use operational objects, see
cg                         the entries and commands:  variable, vector,
cg                         operator.
cg                         Also see commands:  last, list, repack, search, sort.

cg    operations         See "operator", "vector operations".

cg    operator           A tensor operator is a 3 by 3 tensor, with the 9
cg                         components (where the first digit indicates the row,
cg                         and the second digit the column):
cg                         OP11, OP12, OP13, OP21, OP22, OP23, OP31, OP32, OP33.
cg                         The inverse of the tensor operator has the 9
cg                         components:
cg                         IN11, IN12, IN13, IN21, IN22, IN23, IN31, IN32, IN33.
cg
cg                         The tensor operators used in GEOM are Hermitian,
cg                         i.e., each row and each column represents a unit
cg                         vector, and the inverse is obtained by switching row
cg                         and column indices.
cg
cg                         A tensor operator is used to invert, reflect, rotate
cg                         or scale points, clusters, vectors, planes and
cg                         quadric surfaces in three dimensions.
cg
cg                         Tensor operators are specified with command
cg                         "operator".
cg                         The maximum number of operators is now 100.
cg
cg                         NOTE!  A tensor operator is NOT dependent on the
cg                         subsequent movement of any point(s) used to create
cg                         it.
cg
cg                         The following commands relate to tensor
cg                         operators:
cg                         axisym, cone, copy, cylinder, debug, delete,
cg                         ellipsoid, help, invert, last, list, operator, plane,
cg                         point, quadric, reflect, rename, repack, rotate,
cg                         scale, search, sort, sphere, symbol, synonym, tables,
cg                         vector.

cc    operator           An option in command "debug", to display GEOM internal
cc                         parameters and variables for tensor operators:
cc                         noperm, nopers, aoper, aoptype, aoptyps, toper,
cc                         topin, aopers, laopers.
cc                         Synonym:  [operator, oper, op].

cc    operator           Command to display one or more tensor operators, or
cc                         create a tensor operator to rotate, reflect, scale or
cc                         invert points, vectors and quadric surfaces.
cc                         Creating an operator replaces any existing operator
cc                         having the same name.
cc                         The display for a tensor operator includes the type,
cc                         the 9 components, the 9 components of the inverse
cc                         operator, if different, and other data.
cc
cc                         Tensor operators are generated based on an invariant
cc                         point at the origin, but the actual invariant point
cc                         may be specified when the tensor operator is used in
cc                         a command.
cc
cc                         See "RATIO" for a warning about truncation error.
cc
cc                         Command "operator" relates to objects:  axisym, cone,
cc                         cylinder, ellipsoid, hyperb, operator, plane, sphere,
cc                         symbol, variable, vector.
cc
cc                         Forms include:
cc
ccin                       help operator
ccin                       operator [help]
cc
cc                           Display the command options.
cc
ccin                       operator [all,list]
cc
cc                           Display all tensor operators.
cc
ccin                       operator list OPNAME1 OPNAME2 OPNAME3 ...
cc
cc                           Display operators OPNAME1, OPNAME2, OPNAME3, ...,
cc                           with or without subscripts.
cc
ccin                       operator OPNAME
cc
cc                           Display tensor operator OPNAME.
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         OPNAME to form the name by incrementing or
cc                         decrementing the base name for tensor operators.
cc                         See command "last", "increment names".
cc
cc                         Create a scaling operator:
cc
ccin                       operator OPNAME [scale,uniform] RATIO
cc
cc                           Create scaling operator OPNAME to scale uniformly
cc                           by factor RATIO (scale and uniform are synonyms).
cc                           RATIO = -1 is the same as inverting through the
cc                           fixed point.
cc                           This option may be used to convert from one length
cc                           unit to another, such as centimeters, inches, feet
cc                           and meters.
cc
cc                         WARNING:  the following two types of scaling
cc                         operators may change an axially symmetric quadric
cc                         surface to axially unsymmetric, if the scaling
cc                         operator has an axis that is not parallel to that of
cc                         the quadric.
cc
ccin                       operator OPNAME radial VAXIS RATIO
cc
cc                           Create scaling operator OPNAME to scale radially,
cc                           perpendicular to an axis though the invariant point
cc                           in the direction of vector VAXIS, by factor RATIO.
cc                           RATIO = -1 is the same as inverting through the
cc                           axis.
cc
ccin                       operator OPNAME linear VAXIS RATIO
cc
cc                           Create scaling operator OPNAME to scale linearly,
cc                           parallel to the vector VAXIS, by factor RATIO.
cc                           RATIO = -1 is the same as reflection in a plane
cc                           perpendicular to the axis.
cc
cc                         Create a rotation operator (the three row vectors are
cc                         the orthogonal unit vector triple to be rotated to
cc                         the x, y and z axes, and the three column vectors are
cc                         the orthogonal unit vector triple the x, y and z axes
cc                         are to be rotated to):
cc
ccin                       operator OPNAME QTYPE QNAME
cc
cc                           Create rotation operator OPNAME to rotate the
cc                           quadric surface of type QTYPE (plane, sphere,
cc                           cylinder, cone, hyperb, ellipsoid, axisym, quadric)
cc                           with name QNAME to make its axes, if any, parallel
cc                           to the major axes.
cc
ccin                       operator OPNAME axial VAXIS ANGLE
cc
cc                           Create rotation operator OPNAME to rotate around
cc                           axis vector VAXIS by angle ANGLE (counterclockwise,
cc                           with the axis pointed at the observer).
cc
ccin                       operator OPNAME serial U UANGLE V VANGLE W WANGLE
cc
cc                           Create rotation operator OPNAME to rotate around
cc                           major axis U by angle UANGLE, around major axis V
cc                           by angle VANGLE, around major axis W by angle
cc                           WANGLE, where U, V and W are x, y or z (no two the
cc                           same), and 5, 7 or 9 words may be used.
cc
ccin                       operator OPNAME vector VNAME1 VNAME2
cc
cc                           Create rotation operator OPNAME to rotate vector
cc                           VNAME1 to be parallel to vector VNAME2, around an
cc                           axis perpendicular to both vectors.
cc
ccin                       operator OPNAME planar PLNAME1 PLNAME2
cc
cc                           Create rotation operator OPNAME to rotate plane
cc                           PLNAME1 to be parallel to plane PLNAME2, around an
cc                           axis parallel to both planes.
cc
ccin                       operator OPNAME triple VNAME1 VNAME2 VNAME3 VNAME4
cc
cc                           Create rotation operator OPNAME to rotate vector
cc                           VNAME1 to be parallel to vector VNAME3, and the
cc                           plane containing vectors VNAME1 and VNAME2 to be
cc                           parallel to the plane containing vectors VNAME3 and
cc                           VNAME4.
cc
cc                         Create a reflection or inversion operator:
cc
ccin                       operator OPNAME reflect PLNAME
cc
cc                           Create reflection operator OPNAME to reflect in a
cc                           plane through the origin parallel to plane PLNAME.
cc
ccin                       operator OPNAME invert
cc
cc                           Create inversion operator OPNAME to invert through
cc                           the origin.  Replace all points (x, y, z) with
cc                           (-x, -y, -z).
cc
cc                         Synonyms:  [cylinder, cyl],
cc                         [ellipsoid, el, ell, ellipse], [help, h],
cc                         [hyperb, saddle], [operator, op, oper], [plane, pl],
cc                         [quadric, q, quad], [reflect, refl],
cc                         [scale, uniform], [sphere, sph], [vector, v, vect].

cc    operator           An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more tensor operators.

cg    operator arrays    Operators may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    OPij               A component of a tensor operator, with i = 1, 3,
cg                         and j = 1, 3.

cc    OPNAME             The name of a 3 by 3 tensor operator to invert,
cc                         reflect, rotate or scale.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.  No operator name may be "+", "-", "all",
cc                         "h", "help", "list" or "thru", begin with "!" or
cc                         contain ";".
cc                         Specified with command "operator".
cc
cc                         Also referred to as OPNAME1, OPNAME2, OPNAME3, ...
cc
cc                         Tensor operators may appear in commands:
cc                         axisym, cone, copy, cylinder, delete, ellipsoid,
cc                         help, invert, last, list, operator, plane, point,
cc                         quadric, reflect, rename, repack, rotate, scale,
cc                         search, sort, sphere, vector.

cg    options            Options in a command are either shown as required
cg                         arguments in optional forms of the command, or
cg                         enclosed in square brackets, in lower case letters.
cg
cg                         [a]           Indicates an optional command argument
cg                                       a, which has no default, unless
cg                                       specified in the description of the
cg                                       command.
cg                         [a|b]         Indicates an optional command argument
cg                                       a, which defaults to b.
cg                         [a,b,c|d]     Indicates an optional command argument
cg                                       a, b or c, which defaults to d.

cc    OPTYPE             The type of 3 by 3 tensor operator to invert, reflect,
cc                         rotate or scale, with the invariant point at the
cc                         origin.  Must be one of the following:
cc
cc                         axial         Rotate by an angle around an axis
cc                                       through the origin.
cc                         planar        Rotate a plane to be parallel to
cc                                       another plane, through an axis parallel
cc                                       to both, and through the origin.
cc                         serial        Rotate by 3 angles around 3 major axes.
cc                         triple        Rotate a vector triple to be aligned
cc                                       with another one.
cc                         vector        Rotate a vector to be parallel to
cc                                       another vector, around an axis
cc                                       perpendicular to both, and through the
cc                                       origin.
cc                         linear        Scale parallel to an axis through the
cc                                       origin.
cc                         radial        Scale perpendicularly to an axis
cc                                       through the origin.
cc                         uniform       Scale uniformly in 3-D space, around
cc                                       the origin.
cc                         invert        Invert through the origin.

cg    orbit              See "planetary orbits".

cg    order              See "order modulo N", "random order".

cg    order              To put objects in ASCII order, use command "sort".
cg                         The ASCII order of keyboard characters is:
cg                         blank !"#$%&'()*+,-./ 0-9 :;<=>?@ A-Z [\]^_` a-z {|}~
cg                         Entries in this file are ordered without regard to
cg                         case.  See "character set".

cg    order              In referring to a polynomial equation in z,
cg                         P(z) = sum (n = 0, N) {a(n) * z^n}, the order is N,
cg                         the largest exponent of z.  In referring to a root r
cg                         of such an equation, the order is the power M on the
cg                         term (z - r)^M that may be factored out of the
cg                         equation (see command "rootf").  At the root r, P(z)
cg                         and its first M - 1 derivatives are all zero, and the
cg                         Mth derivative is not.

cg    order              See "order modulo N".

cg    order modulo N     In modular arithmetic, with modulus N, the order m of
cg                         an integer K less than N is the least value of m for
cg                         which K^m = 1 modulo N.
cg                         Displayed by commands "icalc" and "variable", when
cg                         option "mod" is used.

cc    ortho              An option in command "quadric", to find the quadric
cc                         surface containing all points at which the two
cc                         vector fields based on two quadric surfaces are
cc                         orthogonal.
cc                         If one of the quadric surfaces is a plane, the
cc                         resulting quadric surface passes through the
cc                         outline of the other quadric surface, as projected
cc                         perpendicularly to the plane.
cc                         If one of the quadric surfaces is a cylinder, the
cc                         resulting quadric surface passes through the
cc                         outline of the other quadric surface, as projected
cc                         perpendicularly to the axis of the cylinder.
cc                         If one of the quadric surfaces is a sphere, the
cc                         resulting quadric surface passes through the
cc                         outline of the other quadric surface, as projected
cc                         toward the center of the sphere.

cg    orthocenter        When a triangle is created or displayed, the
cg                         orthocenter is displayed.
cg                         The orthocenter is at the intersection of the
cg                         altitudes, the lines from each vertex perpendicular
cg                         to the opposite edge.
cg                         See "circumscribed circle", "inscribed circle",
cg                         "centroid", "trig".

cg    orthogonal         To find out if two vectors are orthogonal:
cg
cg                         variable VARNAME dot VNAME1 VNAME2
cg
cg                           Create variable VARNAME with the value of the dot
cg                           (inner) product of the two vectors VNAME1 and
cg                           VNAME2.
cg
cg                         If VARNAME is zero, then vectors VNAME1 and VNAME2
cg                         are orthogonal.
cg
cg                         To find out if three vectors are mutually orthogonal:
cg
cg                         vector VNAME123 triple VNAME1 VNAME2 VNAME3
cg
cg                           Create vector VNAME, the vector triple product of
cg                           vectors VNAME1, VNAME2 and VNAME3:
cg                           VNAME1 cross (VNAME2 cross VNAME3) =
cg                           (VNAME3 cross VNAME2) cross VNAME1.
cg
cg                         vector VNAME321 triple VNAME3 VNAME2 VNAME1
cg
cg                           Create vector VNAME, the vector triple product of
cg                           vectors VNAME3, VNAME2 and VNAME1:
cg                           VNAME3 cross (VNAME2 cross VNAME1) =
cg                           (VNAME1 cross VNAME2) cross VNAME3.
cg
cg                         If both vectors VNAME123 and VNAME321 have zero
cg                         length, then vectors VNAME1, VNAME2 and VNAME3 are
cg                         mutually orthogonal.

cg    orthogonal         To find the locus of points where the two vector
cg                         fields, or any members of the two families of quadric
cg                         surfaces represented by two specified quadric
cg                         surfaces, are orthogonal to each other, use command
cg                         "quadric", option "ortho".

cc    out                A synonym for output.

cc    out                An option in command "hex", to display the hexadecimal
cc                         machine words and hexadecimal floating point forms,
cc                         if any, of one or more character strings, constants
cc                         or variables.

cc    outer              A synonym for cross.

cg    outer product      See "cross product".

cc    outline            An option in command "plane", to find the plane
cc                         containing the outline of a quadric surface, as seen
cc                         in perspective from a specified point.
cc                         See "projection".

cg    outline            The outline of a quadric surface, as seen from a
cg                         specified external point, is the locus of any points
cg                         on the quadric surface for which the line from the
cg                         external point to the surface point is tangent to the
cg                         quadric surface.  This outline is a planar quadric
cg                         curve, in a plane whose normal vector is the normal
cg                         vector associated with the quadric surface, evaluated
cg                         at the specified external point.  The plane may be
cg                         be found with command:
cg
cg                         plane PLNAME outline PNAME QNAME
cg
cg                           Create plane PLNAME, containing any outline of
cg                           quadric surface QNAME, as seen from point PNAME.
cg                           Display the "central point" of the outline plane,
cg                           nearest the center of QNAME.
cg
cg                         Points on a real outline satisfy the equation of the
cg                         quadric surface and the equation of the plane, but
cg                         may be hidden by part of the surface.  If no points
cg                         satisfy both equations, the outline is imaginary.
cg
cg                         The cylindrical quadric surface through the outline
cg                         may be found with command "quadric", option
cg                         "intersect".
cg
cg                         Points on a real outline may be found by using
cg                         command "move" to translate the outline plane and the
cg                         quadric surface to put the "central point" of the
cg                         outline plane at the origin, then using commands
cg                         "operator" and "rotate" to rotate the outline plane
cg                         and the quadric surface to make the normal vector of
cg                         the outline plane point in the x direction, then
cg                         using command "slice" to find y and z points in the
cg                         quadric surface at x = 0, then doing the inverse
cg                         rotation and the inverse translation on those points.

cg    output             See "output data", "output file", "output of data",
cg                         command "output".

cc    output             Command to display the name of the current output file,
cc                         or specify the name of a new output file.
cc
cc                         Forms include:
cc
ccin                       help output
ccin                       output help
cc
cc                           Display the name of the current output file,
cc                           and command options.
cc
ccin                       output
cc
cc                           Display the name of the current output file.
cc                           Initially geom_hsp.
cc
ccin                       output OUT_FILE
cc
cc                           Close and save the current output file, and open
cc                           new output file OUT_FILE, which must not be the
cc                           the current input file or the command summary file,
cc                           "geom_cmd".
cc
cc                           NOTE!  If OUT_FILE already exists, and the input is
cc                           from the user's terminal, the user will be asked to
cc                           type either "yes" to allow OUT_FILE to be written
cc                           over, which will destroy any existing data in it,
cc                           or "add" to append any new output to the end of the
cc                           file, or "no" to name a new output file, or to keep
cc                           using the current output file.  The latter is the
cc                           default option if the current input is not from the
cc                           user's terminal.
cc
cc                         Synonyms:  [add, a], [help, h],
cc                         [output, o, out, w, wr, write], [yes, y].

cg    output             Each non-null input line read from the user's
cg                         terminal or from the input file is echoed in the
cg                         output file, preceded by the character ">", except
cg                         for commands beginning with "input", "read", "undo",
cg                         "indo", "redo" or any synonyms of these, or an
cg                         alias.
cg                         All output generated as a result of an input line is
cg                         sent to the user's terminal and to the current
cg                         output file.
cg                         See "output file", "record of session".
cg
cg                         Floating point output from GEOM is usually in decimal
cg                         floating point format 1pE20.12 (a sign, a nonzero
cg                         integer, a decimal point, 12 decimal digits, the
cg                         character "E", a sign, and two exponent digits).
cg                         If the exponent requires three digits, the character
cg                         "E" is omitted.
cg
cg                         For hexadecimal output, see command "hex".
cg
cg                         The following commands may affect or involve the
cg                         output file:  geom, output, tables.

cg    output data        For a definition of any item in the output data, see
cg                         the entry for the command that produced the data, or
cg                         for the label accompanying the data, or use the
cg                         command:
cg
cg                         define 'DATANAME'
cg
cg                           Display any entry DATANAME in file geom_base (this
cg                           file).

cg    output file        The output file is initially geom_hsp.  the user
cg                         may change the output file with command "output".
cg                         All non-null input lines are echoed in the output
cg                         file, preceded by the character ">".  All output sent
cg                         to the user's terminal is also sent to the
cg                         output file.  To convert the output file into an
cg                         input file that will produce the same output, see
cg                         "input file".
cg
cg                         The following commands may affect or involve the
cg                         output file:  debug, geom, output.
cg
cg                         See "redirect output", "standard output".

cg    output of data     To find the minimum field width for output of floating
cg                         point numbers, see "format".

cc    OUT_FILE           An output file specified on the GEOM execution line,
cc                         to be used in place of geom_hsp, until and unless
cc                         changed later by command "output".

cc    OUT_FILE           In command "output", the name of an output file,
cc                         which must not be the name of the current input file
cc                         or geom_cmd.  If you name an existing file, you will
cc                         be asked to type "yes" to allow the existing file to
cc                         be written over.

cg    overflow           A floating point overflow occurs when a floating point
cg                         number has an exponent exceeding the available
cg                         storage space in computer memory.  On DEC or YANA
cg                         machines, with 64-bit floating point words, the
cg                         maximum exponent is about plus or minus 308.
cg                         See "floating point".
cg                         Functions "*", "/", "^", "!" and "exp" in command
cg                         "variable" are tested for overflow to prevent GEOM
cg                         from crashing.
cg
cg                         An integer overflow occurs when an integer number
cg                         exceeds the available storage space in computer
cg                         memory.  On DEC or YANA machines with 64-bit integers,
cg                         the maximum number of digits is about 18.
cg                         See "integer".

P-------------------------------------------------------------------------------

cc    p                  A synonym for point.

cc    P1                 See "P1, P2, P3".

cc    P1, P2, P3         In command "accelerate", option "fit", the names of
cc                         points on the parabolic trajectory of a uniformly
cc                         accelerated particle, at times T1, T2, T3.

cc    P2                 See "P1, P2, P3".

cc    P3                 See "P1, P2, P3".

cc    page               Command to write a line consisting of the page restore
cc                         character "^L" (ctrl-L) into the output file.
cc
ccin                       page
cc
cc                           Write a line consisting of the page restore
cc                           character "^L" (ctrl-L) into the output file.

cg    palindrome         A palindrome is a character string that reads the same
cg                         forwards and backwards.
cg                         All 688 prime numbers between 1 and 10000000 that are
cg                         palindromes are listed in file
cg                         ~/work/geom/test/prime_palindromes .
cg                         All prime palindromes have an odd number of digits,
cg                         since all palindromes with an even number of digits
cg                         are divisible by 11.  Proof:  (10^n + 1) is divisible
cg                         by 11 for all odd n.

cc    par                A synonym for parallel.

cg    parabolic          See "parabolic cylinder", "parabolic path".

cg    parabolic cylinder
cg                       A parabolic cylinder is a quadric surface for which the
cg                         standard equation is:  -QZ * z + x^2 = 0.
cg                         This is a limit case of a hyperbolic paraboloid or an
cg                         elliptic paraboloid.
cg
cg                         To find the distance from a point to a parabolic
cg                         cylinder, use command "distance", "proximal" or
cg                         "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and a parabolic cylinder, use command "distance".

cg    parabolic path     A uniformly accelerated particle follows a parabolic
cg                         path.  To find points on such a path for various
cg                         times, or to find the intersection of such a path
cg                         with an arbitrary plane or any general quadric
cg                         surface, or to find the initial position and velocity
cg                         and constant acceleration, given two or three points
cg                         at two or three times, see command "accelerate".

cg    paraboloid         A paraboloid is a surface in 3-D space described by an
cg                         implicit quadric equation.
cg
cg                         The standard forms of the implicit equations for
cg                         paraboloids are as follows (each coefficient must
cg                         have the preceding sign):
cg
cg                         Parabolic cylinder:               -QZ*z + x^2 = 0
cg                         Hyperbolic paraboloid (a saddle point surface):
cg                                            - |QZ|*z + x^2 - |QYY|*y^2 = 0
cg                         Elliptic paraboloid:
cg                                            - |QZ|*z + x^2 +  QYY *y^2 = 0
cg                         Circular paraboloid:
cg                                            - |QZ|*z + x^2 +       y^2 = 0
cg
cg                         In standard form, the z coordinate of the focus
cg                         is given by:
cg                           ZFOC = - QZ        (at y = 0)
cg                           ZFOC = - QZ / QYY  (at x = 0)
cg
cg                         To find the distance from a point to a paraboloid,
cg                         use command "distance", "proximal" or "side".
cg
cg                         To find the distance or intersection between a plane
cg                         and a paraboloid, use command "distance".
cg
cg                         To find the direction vectors of the embedded
cg                         straight lines at a particular point in a specified
cg                         hyperbolic paraboloid (a ruled surface), use
cg                         command "arc".

cc    parallel           An option in command "plane", to create a family of
cc                         parallel planes by repeated translations of a base
cc                         plane by a specified increment in the normal
cc                         direction.
cc                         Also see commands "axisym", "cone", "cylinder",
cc                         "ellipsoid", "point", "quadric", "sphere",
cc                         option "move".
cc
cc                         This command option can be replaced, using a do loop.
cc                         See "do loop use", commands "do", "enddo".

cg    parallel           Two lines or planes are parallel if the difference
cg                         between their direction vectors or normal vectors,
cg                         respectively, is no more than the estimated error
cg                         in calculating that differences, based on the
cg                         numerical tolerance limit TOL.
cg                         Also see command "plane", option "parallel", and
cg                         commands "axisym", "cone", "cylinder", "ellipsoid",
cg                         "point", "quadric", "sphere", option "move".
cg
cg                         To see if a line is parallel to another line, a
cg                         plane, or the axis of a circular cylinder, use
cg                         command "distance".
cg
cg                         To see if a triangle or a plane is parallel to
cg                         another triangle or plane, use command "distance".
cg
cg                         To see if a plane is parallel to the axis of a
cg                         circular cylinder, use command "distance".
cg
cg                         To see if two circular cylinders are parallel, use
cg                         command "distance".
cg
cg                         To see if two quadric surfaces have parallel axes,
cg                         display them both using command "quadric", and see
cg                         if their rotation tensor operators are the same.

cg    parallel           See "parallel planes", "parallel scaling",
cg                         "parallel vectors".

cg    parallel planes    A pair of parallel planes may be represented by a
cg                         single quadric surface for which the standard
cg                         equation is:
cg                          - 1 + QXX*x^2 = 0,        (QXX > 0).
cg                         which may be factored into the two equations:
cg                          - 1 + sqrt (QXX) * x = 0 (a simple plane),
cg                            1 + sqrt (QXX) * x = 0 (a simple plane).
cg                         The distance between the two planes is 2 / sqrt(QXX).
cg
cg                         The general implicit quadric equation for parallel
cg                         planes is the product of the implicit equations for
cg                         two simple parallel planes:
cg                           (QC1 + QX * x + QY * y + QZ * z) *
cg                           (QC2 + QX * x + QY * y + QZ * z) = 0.
cg
cg                         This is a limit case of a hyperboloid of two sheets,
cg                         an ellipsoid, an elliptic cylinder or a hyperbolic
cg                         cylinder.
cg                         A pair of imaginary parallel planes is a quadric
cg                         surface for which the standard equation is:
cg                            1 + QXX*x^2 = 0,        (QXX > 0).
cg
cg                         See "non-simple planes".
cg                         See command "plane", option "parallel".
cg
cg                         To create such a quadric surface, use command
cg                         "quadric", option "plane".
cg
cg                         Given a plane with the implicit quadric equation
cg                           QC + QX * x + QY * y + QZ * z,
cg                         the equation of the parallel plane at distance D
cg                         (away from the origin), has QC replaced by
cg                         QC - D * |N|, where
cg                           |N| = sqrt (QX^2 + QY^2 + QZ^2).

cg    parallel scaling   To scale in a direction parallel to a specified axis,
cg                         use command "operator", option "linear", and command
cg                         "scale".

cg    parallel vectors   To find out if two vectors are parallel:
cg
cg                         vector VNAME cross VNAME1 VNAME2
cg
cg                           Create vector VNAME, the cross (outer) product of
cg                           vectors VNAME1 and VNAME2.
cg
cg                         If vector VNAME has zero length, then vectors VNAME1
cg                         and VNAME2 are parallel.

cc    param              A synonym for parameter.

cc    parameter          An option in command "debug", to display GEOM internal
cc                         integer and floating point size limits,
cc                         array size parameters and numerical constants:
cc                         bignum, ebase, euler, iamax, idmax, iemax, naliasm,
cc                         nbinm, nbitm, nbrickm, ncharm, nclustm, ncmdm,
cc                         ndiskm, nfilem, nlinem, nmarkm, noperm, npairm,
cc                         npdfm, npdfxm, nplcolm, nplinem, nplptm, npointm,
cc                         npolygm, npolyhm, nptmarkm, nquadm, nsymbm, nsynm,
cc                         ntempm, ntetm, ntrigm, nvarm, nvectm, nverpgm,
cc                         nwordm, nzonem, nzquadm, pi, rgold, small, tolmin,
cc                         tolz.
cc                         Synonyms:  [parameter, param].

cc    parameter          An option in command "debug", to display all GEOM
cc                         internal parameters (specified with FORTRAN parameter
cc                         statements).  These include integer and floating
cc                         point size limits, array size limits, and some
cc                         physical constants.
cc                         Synonyms:  [parameter, param].

cg    parameters         See "internal" for GEOM internal parameters.

cg    parentheses        Object names may include parenthesized explicit
cg                         integer subscripts.  See "subscripted names".

cg    parenthesis        The left or right parenthesis character, "(" or ")".

cg    parse              Input lines for GEOM are stored first as 80 characters,
cg                         then separated into words, based on the current
cg                         field delimiter.  Multiple statements are separated,
cg                         appended comments are removed, and comment lines
cg                         are recognized and ignored.  The words are then
cg                         interpreted as character strings, integers or
cg                         floating point, aliases are replaced, symbols are
cg                         replaced, and subscripts specified by variables are
cg                         replaced with their current numerical values.
cg                         All this is called parsing, and the GEOM internal
cg                         parameters and variables involved by be displayed
cg                         by using command "parse", to turn the parsing flag
cg                         on, then turning it off when desired.

cc    parse              An option in command "debug", to display GEOM internal
cc                         parameters and variables for parsing input lines:
cc                         afd, aparse, ncharm, achar, nwordm, nword, aword,
cc                         iword, fword, lword, lwords, mtype, nstats, nstat,
cc                         ncharm, istat, astat, nifnest.
cc                         This only displays the data for this one "debug"
cc                         command.  To display parsing data for other commands,
cc                         use command "parse".

cc    parse              Command to turn parsing flag on or off.  When on,
cc                         GEOM internal parameters and variables related to
cc                         parsing of input lines are displayed for each input
cc                         line/
cc
cc                         Forms include:
cc
ccin                       help parse
ccin                       parse [help]
cc
cc                           Display the command options.
cc
ccin                       parse on
cc
cc                           Turn the parsing flag on.  Display the GEOM
cc                           internal parameters and variables for parsing each
cc                           input line, until the flag is turned off.
cc                           Compare with command "debug parse", which only
cc                           displays the parsing data for itself.
cc
ccin                       parse off
cc
cc                           Turn the parsing flag off.
cc
cc                         Synonyms:  [help, h].
cc

cg    partial            See "partial sum".

cg    partial sum        The n'th partial sum of a continued fraction is the
cg                         value of the ratio of integers obtained by evaluating
cg                         only the first n coefficients of the continued
cg                         fraction.
cg
cg                         To find the n'th convergent P(n) / Q(n), and the
cg                         n'th partial sum x(n), given the coefficients
cg                         C(n), n = 1, N:
cg                           P(0) = 1,            P(1) = C(1),
cg                           P(n) = C(n) * P(n-1) + P(n-2), n = 2, N,
cg                           Q(0) = 0,            Q(1) = 1,
cg                           Q(n) = C(n) * Q(n-1) + Q(n-2), n = 2, N,
cg                           x(n) = P(n) / Q(n), n = 2, N.
cg
cg                         These values are found and displayed when command
cg                         "contfr" is used.

cg    partition          To partition a triangle into two parts of equal area
cg                         and equal perimeter with straight line cuts, see
cg                         "triangle cutting", commands "cuts" and "cute".
cg                         The partitioning cutting lines will be displayed
cg                         whenever a triangle is created or displayed.

cc    path               An option in command "accelerate", to use specified
cc                         values of the initial position and velocity, and
cc                         a constant acceleration, to find points on the
cc                         resulting parabolic trajectory at the vertex, and at
cc                         specified times, or at the intersection or proximal
cc                         point with a point or a plane, or at any intersection
cc                         with a quadric surface.

cg    path               The path of a point undergoing a random walk process
cg                         is the complete path from the initial position,
cg                         along each step of the random walk, to the final
cg                         position.  The path length is the sum of all of the
cg                         individual displacements for each step.

cg    path               See "path length", "parabolic path", "total path".

cg    path length        In a display of data resulting from command
cg                         "accelerate", the distance along the parabolic
cg                         trajectory of a uniformly accelerated particle,
cg                         measured from time zero.

cc    PATHMAX            In command "walk", the maximum path length a point is
cc                         to be randomly walked, if it does not reach distance
cc                         DISTMAX from its initial position first.
cc                         May be integer, floating point or a variable.

cc    PAXIS              The name of a point on an axis.
cc                         Used in commands "cone", "cylinder", "plane",
cc                         "twist", "volume".

cc    PCEN               The name of a point at the center of an annular disk,
cc                         quadric surface, regular polygon or regular
cc                         polyhedron.
cc                         Used in commands "axisym", "disk", "ellipsoid",
cc                         "polygon", "polyhedron", "project", "sphere".

cc    PCOP               The name of a point in the plane of a regular polygon,
cc                         but not collinear with points PCEN and PVER.
cc                         Used in command "polygon".
cc                         The name of a point in the plane of the center and
cc                         two adjacent vertices of a regular polyhedron.
cc                         Used in command "polyhedron".

cg    pdf                A pdf is a probability distribution function,
cg                         consisting of a set of probability bins, each
cg                         specifying the integral probability of a discrete
cg                         event or object, or a discrete value of a random
cg                         variable, or specifying the differential
cg                         probabilities of a range of values of a random
cg                         variable.  One or more of those bins may represent
cg                         another pdf, making it possible to use one pdf as a
cg                         part of any number of other pdfs.
cg                         The maximum number of pdfs is now 1000.
cg                         The maximum number of probability bins assigned to
cg                         pdfs is now 1000, counting bins assigned to more
cg                         than one pdf.
cg
cg                         The following commands relate to pdfs:
cg                         bin, copy, debug, delete, help, last, list, pdf,
cg                         rename, repack, sample, search, sort, symbol,
cg                         synonym, tables.
cg
cg                         See "distribution".

cc    pdf                An option in command "debug", to display GEOM internal
cc                         parameters and variables for probability distribution
cc                         functions: npdfm, npdfs, apdf, prpdf, prpdft,
cc                         vexpdf, nbinpdf, apdfs, lapdfs, npdfxm, npdfxs,
cc                         apdfx, abinx.

cc    pdf                Command to display one or more pdfs or to create a
cc                         probability distribution function (pdf).
cc                         See commands "bin", "sample".
cc                         See "do loop use".
cc
cc                         Command "pdf" relates to objects:  bin, pdf, symbol.
cc
cc                         Forms include:
cc
ccin                       help pdf
ccin                       pdf [help]
cc
cc                           Display the command options.
cc
ccin                       pdf [all,list]
cc
cc                           Display all pdfs.
cc
ccin                       pdf list PDFNAME1 PDFNAME2 PDFNAME3 ...
cc
cc                           Display pdfs PDFNAME1, PDFNAME2, PDFNAME3, ...,
cc                           with or without subscripts.

cc
ccin                       pdf PDFNAME
cc
cc                           Display pdf PDFNAME.
cc
ccin                       pdf PDFNAME all
cc
cc                           Create new pdf PDFNAME, consisting of all
cc                           probability bins.
cc
cc                         In the following commands, use "+" or "-" instead of
cc                           PDFNAME to form the name by incrementing or
cc                           decrementing the base name for pdfs.
cc                           See command "last", "increment names".
cc
ccin                       pdf PDFNAME = BINAME1 ... BINAME2 [thru] BINAME3 ...
cc
cc                           Create new pdf PDFNAME, consisting of the
cc                           probability bins BINAME1, ..., BINAME2 [, through]
cc                           BINAME3, ...
cc
ccin                       pdf PDFNAME & BINAME1 ... BINAME2 [thru] BINAME3 ...
cc
cc                           Add to existing pdf PDFNAME (do not use "+" or
cc                           "-"), probability bins BINAME1, ..., BINAME2
cc                           [, through] BINAME3, ...  If option "thru" is used,
cc                           the adjacent probability bins need not exist.
cc
ccin                       pdf PDFNAME - BINAME1 ... BINAME2 [thru] BINAME3 ...
cc
cc                           Remove from existing pdf PDFNAME (do not use "+"
cc                           or "-"), probability bins BINAME1, ...  BINAME2
cc                           [, through] BINAME3, ...  If option "thru" is used,
cc                           the adjacent points need not exist.
cc                           A warning message is displayed if the pdf PDFNAME
cc                           does not already exist.
cc                           WARNING:  an error in one word does not
cc                           cancel the entire command.  For a backup, copy the
cc                           pdf before modifying.
cc
cc                         Synonyms:  [help, h].

cc    pdf                An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more probability
cc                         distribution functions.

cg    pdf arrays         Pdfs may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cc    PDFNAME            The name of a probability distribution function (pdf).
cc                         May have up to 24 characters, including any
cc                         subscripts, and may be ASCII, integer or floating
cc                         point.
cc                         No pdf name may be "+", "-", "all", "h", "help",
cc                         "list" or "thru", begin with "!" or contain ";".
cc                         Specified with command "pdf".
cc
cc                         Probability distribution functions may be used in
cc                         commands:
cc                         copy, delete, rename, sample.
cc                         Also see commands:  last, list, repack, search, sort.

cg    percent            The percent character, "%".

cg    perim              Abbreviation for perimeter.

cg    period             The period or dot character, ".".

cc    perm               An option in command "icalc", to find the number of
cc                         permutations of M things taken N at a time.
cc
cc                         icalc M perm N

cc    perm               An option in command "sudoku", to display all digits
cc                         permitted in each Sudoku cell.

cg    permutations       The number of permutations of M things taken N at a
cg                         time is P(M,N) = M! / (M - N)!.
cg                         Use command "icalc M perm N" or
cg                         "big BIGNAME = INT1 perm INT2".
cg                         See "combinations".

cg    perpendicular      See "orthogonal".

cg    perpendicular      See "perpendicular scaling".

cg    perpendicular scaling
cg                         To scale in a direction perpendicular to a specified
cg                         axis, use command "operator", option "radial", and
cg                         command "scale".

cc    pers               A synonym for perspective.

cc    perspective        To make a perspective projection of a point, a cluster
cc                         of points or all points, use one of the following
cc                         forms of command "project":
cc
cc                         project point   PNAME  perspective PVIEW PFOC PTL PTR
cc                         project cluster CLNAME perspective PVIEW PFOC PTL PTR
cc
cc                         alias prpa = "project point all"
cc
cc                           Create an alias for the following command, if
cc                           desired, to reduce typing.
cc
cc                         project point   all    perspective PVIEW PFOC PTL PTR
cc
cc                           Project point PNAME, cluster CLNAME or all points,
cc                           toward point PVIEW, onto a focal plane through
cc                           point PFOC and perpendicular to the line from point
cc                           PVIEW to point PFOC, then rotate and translate the
cc                           focal plane onto the plane z = 0, with point PFOC
cc                           at the origin, and with the line from point PTL to
cc                           point PTR parallel to the x axis, from left to
cc                           right.  Points behind point PVIEW will not be
cc                           projected.
cc
cc                         The viewing point is point PVIEW.
cc                         The line of sight is toward point PFOC.
cc                         The field of view is within 90 degrees of the line
cc                         of sight, and does not include points behind PVIEW.
cc                         The projection plane is through point PFOC,
cc                         perpendicular to the line of sight.
cc                         The horizon is parallel to the line from point PTL
cc                         on the left, to point PTR on the right.
cc
cc                         Also see command "quadric", option "ortho".

cc    perspective        An option in command "project", to make a perspective
cc                         projection.  See command "quadric", option "ortho".
cc                         Synonyms:  [perspective, pers].

cc    PFOC               In command "project", the name of a point to be used
cc                         as a focal point.

cc    pg                 A synonym for polygon.

cc    PGNAME             The name of a regular polygon.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.  No polygon name may be "+", "-", "all",
cc                         "h", "help", "list" or "thru", begin with "!" or
cc                         contain ";".
cc                         Also referred to as PGNAME1, PGNAME2, PGNAME3, ...
cc                         Specified with command "polygon".
cc
cc                         Regular polygons may appear in commands:
cc                         copy, delete, point, polygon, rename.
cc                         Also see commands:  last, list, repack, search, sort.

cc    ph                 A synonym for polyhedron.

cc    PHI                The polar angle phi of a point, in a spherical
cc                         coordinate system.  The angle from the z axis,
cc                         ranging from 0 to 180 degrees (0 to pi radians).
cc                         May be integer, floating point or a variable.
cc                         Used in commands "point", "vector", "plane" and
cc                         "brick".
cc                         PHI = arcsin (RCYL / RSPH)
cc                         PHI = arccos (Z / RSPH)
cc
cc                         See "phi cone", "azimuth", "theta".

cc    phi                An option in command "brick", to make surfaces of
cc                         constant phi the shared coordinate surfaces between
cc                         adjacent bricks.

cg    phi                The polar angle, measured from the positive z axis,
cg                         and ranging from 0 to 180 degrees (0 to pi radians).
cg                         Used in a spherical coordinate system.
cg                         See "azimuth", "theta".

cg    phi                See "phi cone".

cg    phi cone           To create a cone at a constant value of PHI degrees in
cg                         a spherical (rsph, theta, phi) orthogonal coordinate
cg                         system:
cg                         point pcen = 0 0 0
cg                         vector vaxis = 0 0 1
cg                         cone cone1 angle pcen vaxis PHI
cg
cg                         To create a family of nested cones with the same
cg                         vertex and axis, and with equally spaced vertex
cg                         half-angles, use command "cone" with option "nest".

cc    PHNAME             The name of a regular polyhedron.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.  No polyhedron name may be "+", "-", "all",
cc                         "h", "help", "list" or "thru", begin with "!" or
cc                         contain ";".
cc                         Also referred to as PHNAME1, PHNAME2, PHNAME3, ...
cc                         Specified with command "polyhedron".
cc
cc                         Regular polyhedrons may appear in commands:
cc                         copy, delete, point, polyhedron, rename.
cc                         Also see commands:  last, list, repack, search, sort.

cc    pi                 An option in command "symbol", argument SYMBNAME,
cc                         to create symbol pi with value acos(-1) =
cc                         3.14159265358979323846264338327950288419716939937510
cc                         Store as a big integer with command "input big.mac",
cc                         with file big.mac in the same directory as GEOM.

cg    pi                 Numerical constant = 3.141592653589793...
cg                         Generated as a symbol at the beginning of a
cg                         GEOM run.  Note:  pi radians is 180 degrees.
cg                         pi = acos (-1) =
cg                         3.14159265358979323846264338327950288419716939937510

cg    piano              See "piano tuning".

cg    piano tuning       A piano has 88 keys.  Each key produces a fundamental
cg                         musical note separated from that of an adjacent key
cg                         by a semitone.  The first or lowest note is A0,
cg                         with a frequency of 27.5 Hz.  The last or highest
cg                         note is C8, with a frequency of 4186 Hz.
cg                         Using an equally-tempered scale, the n'th note has a
cg                         frequency f(n) given by:
cg                           f(n) = 27.5 * 2^((n-1)/12),
cg                         and the nearest note n for a specified frequency
cg                         f is given by:
cg                           n = 1 + 12 * log (f/27.5) / log (2).
cg                         Any difference between n and an integer, multiplied
cg                         by 100, is called "cents".
cg
cg                         A difference of 12 semitones is an octave or a
cg                         frequency ratio of 2.  The difference between any
cg                         two frequencies f1 and f2, on a logarithmic scale,
cg                         may be measured in cents as follows:
cg                           cents = 1200 * log (f2/f1) / log (2).
cg                         With an equally-tempered scale, an octave is 1200
cg                         cents, and each semitone is 100 cents.
cg
cg                         In America, Concert Standard tuning is A4 = 440 Hz.
cg
cg                         See "equally-tempered", "harmony", "harmonics",
cg                         input file "scale.mac".

cc    PINIT              In command "accelerate", the name of the point at time
cc                         zero on the parabolic trajectory of a uniformly
cc                         accelerated particle.

cc    PINV               In commands "invert", "reflect", "rotate", "scale",
cc                         "point", "vector", "sphere", "cylinder", "cone",
cc                         "ellipsoid", "axisym" and "quadric", the invariant
cc                         point of the specified symmetry operation.
cc                         The object operated upon is translated to put point
cc                         PINV at the origin, operated on by an operator
cc                         created with its invariant point at the origin, and
cc                         the object is then translated to move the origin to
cc                         point PINV.
cc                         If not specified in the command, point PINV always
cc                         defaults to the origin.

cg    pipe               The pipe symbol, "|".

cc    PITCH              The distance along a twist axis for one complete
cc                         rotation around the axis, or the radial distance
cc                         from a specified radius for one complete rotation
cc                         around the twist axis.  Used in command "twist".
cc                         Positive for clockwise rotation, negative for
cc                         counterclockwise rotation.

cg    pitch              For musical pitches, see "equally-tempered",
cg                         "piano tuning".

cc    pl                 A synonym for plane.

cg    place holder       A place holder is an argument shown in upper case in a
cg                         command, indicating that it is to be provided by the
cg                         user.

cc    planar             An option in command "operator", to create a tensor
cc                         operator that performs a rotation of a plane
cc                         specified by two vectors, to be parallel to another
cc                         plane specified by two vectors, around an axis
cc                         parallel to both planes, and through the origin.
cc                         This is only one of an infinite number of ways to
cc                         align the planes, since either plane may be rotated
cc                         around any axis perpendicular to it before or after
cc                         the rotation.
cc                         See "axial", "serial", "triple", "vector".

cc    Planck             A synonym for planck.

cg    Planck             See "Planck spectrum".

cc    planck             An option in command "bin", to create a probability bin
cc                         with a Planck probability distribution.
cc                         Synonyms:  [planck, Planck].

cg    Planck spectrum    A Planck spectrum is a type of probability bin or
cg                         probability distribution function (pdf), which is an
cg                         equilibrium radiation spectrum, for which the pdf is
cg                         p(x) = (15 / pi^4) * x^3 / (exp (x) - 1),
cg                         where x = XNU / BTEMP, and XNU is a photon frequency,
cg                         and BTEMP is the black-body temperature of the
cg                         spectrum.  The expectation value of x is 3.83223...,
cg                         with a standard deviation of 2.  A random sample of
cg                         100,000 values ranged from 0.07 to 20.1.
cg                         1 Hz = 4.1356692E-15 eV (+/- 0.3 ppm).
cg                         See commands "bin", "pdf", "sample".
cg                         Also see "Wien".

cg    Plane              In the output display for a quadric surface, indicates
cg                         the quadric surface is a plane.

cg    plane              A plane is a flat quadric surface in 3-D space,
cg                         specified by a point and a normal vector, by three
cg                         points, by a triangle or by an annular disk.
cg                         NOTE!  A plane is not dependent on the subsequent
cg                         movement of any point(s) used to create it.
cg                         At any point in the plane, any straight line drawn
cg                         perpendicular to the normal vector lies entirely
cg                         within the plane.  Along with cylinders, cones,
cg                         hyperbolic paraboloids and hyperboloids of one sheet,
cg                         a plane is a ruled surface.
cg
cg                         The standard forms of the implicit equations for
cg                         planes are as follows (each coefficient must have the
cg                         preceding sign):
cg                         Simple plane:                                x = 0
cg                         Coincident planes:                         x^2 = 0
cg                         Real parallel planes:            - 1 + QXX*x^2 = 0
cg                         Real intersecting planes:      x^2 - |QYY|*y^2 = 0
cg                         Imaginary parallel planes:         1 + QXX*x^2 = 0
cg                         Imaginary intersecting planes: x^2 +  QYY *y^2 = 0
cg
cg                         The general form of the implicit quadric equation for
cg                         a plane is as follows:
cg                           F(x,y,z) = QC + QX * x + QY * y + QZ * z = 0
cg
cg                         A plane has the same normal vector everywhere:
cg                           N    = (QX, QY, QZ)
cg                           N^2 = QX^2 + QY^2 + QZ^2
cg
cg                         The point in the plane nearest the origin is given
cg                         by:  XCEN = -(QC / N^2) QX
cg                              YCEN = -(QC / N^2) QY
cg                              ZCEN = -(QC / N^2) QZ
cg
cg                         The absolute distance of this point from the origin
cg                         is |QC|/|N|.
cg
cg                         To find the equation of a parallel plane at distance
cg                         D (away from the origin) from the plane above,
cg                         replace QC with QC + D * |N|.
cg
cg                         The word "plane" in a command means a simple plane.
cg                         Otherwise, use "quadric".
cg
cg                         To find the distance from a plane to a point, use
cg                         command "side" or "distance".  To find the distance
cg                         and proximal point, and/or the intersections
cg                         of a plane and a line, a triangle, another plane,
cg                         a sphere or a circular cylinder, use command
cg                         "distance".  To find the distance or intersection
cg                         of a plane with any quadric surface, use command
cg                         "distance".
cg
cg                         To find any 3-way intersection between three planes,
cg                         use command "triple".
cg
cg                         The distance from the point P1 = (X1, Y1, Z1) to the
cg                         plane QC + QX * x + QY * y + QZ * z = 0 is
cg                           -(QC + QX * X1 + QY * Y1 + QZ * Z1) / |N|,
cg                         where |N| = sqrt (QX^2 + QY^2 + QZ^2).
cg
cg                         To find the intersection between the parabolic
cg                         trajectory of a uniformly accelerated particle and a
cg                         point, a plane or any general quadric surface,
cg                         use command "accelerate".
cg
cg                         The following commands relate to planes:
cg                         accelerate, arc, axisym, copy, debug, delete,
cg                         distance, extrema, help, invert, last, list, move,
cg                         operator, plane, point, project, proximal, quadric,
cg                         reflect, rename, repack, rotate, scale, search, side,
cg                         slice, sort, symbol, synonym, tables, track, triple,
cg                         vector, zone.
cg
cg                         Also see "plane of refl", "theta plane", "x plane",
cg                         "y plane", "z plane".

cc    plane              Command to display or create one or more planes.
cc                         Creating a plane replaces any existing quadric
cc                         surface having the same name.  The display for a
cc                         plane includes the normal vector and the point on the
cc                         plane nearest to the origin.
cc                         A plane is a quadric surface, and must have a unique
cc                         name for that object type.  Once a plane has been
cc                         specified, it no longer depends on any point used in
cc                         its specification.
cc                         See "do loop use".
cc
cc                         Command "plane" relates to objects:  axisym, disk,
cc                         operator, plane, point, quadric, symbol, triangle,
cc                         variable, vector.
cc
cc                         Forms include:
cc
ccin                       help plane
ccin                       plane [help]
cc
cc                           Display the command options.
cc
ccin                       plane [all,list]
cc
cc                           Display all planes.
cc
ccin                       plane list PLNAME1 PLNAME2 PLNAME3 ...
cc
cc                           Display planes PLNAME1, PLNAME2, PLNAME3, ...,
cc                           with or without subscripts.
cc
ccin                       plane PLNAME
cc
cc                           Display plane PLNAME.
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         PLNAME or PL(2) to form the name by incrementing or
cc                         decrementing the base name for planes.
cc                         See command "last", "increment names".
cc
ccin                       plane PLNAME = U V W DU DV DW
cc
cc                           Create plane PLNAME, through point (U, V, W) with
cc                           normal vector (DU, DV, DW).
cc
ccin                       plane PLNAME vector PNAME VNORM
cc
cc                           Create plane PLNAME, through point PNAME with
cc                           normal vector VNORM.
cc
ccin                       plane PLNAME fit PNAME1 PNAME2 PNAME3
cc
cc                           Create plane PLNAME, passing through points PNAME1,
cc                           PNAME2 and PNAME3, which must not be collinear or
cc                           coincident.
cc
ccin                       plane PLNAME triangle TRNAME
cc
cc                           Create plane PLNAME, coplanar with the triangle
cc                           TRNAME.
cc
ccin                       plane PLNAME disk DKNAME
cc
cc                           Create plane PLNAME, coplanar with the annular disk
cc                           DKNAME.
cc
ccin                       plane PLNAME tangent PNAME QNAME
cc
cc                           Create plane PLNAME, tangent at point PNAME to one
cc                           of the family of quadric surfaces represented by
cc                           quadric surface QNAME.
cc
ccin                       plane PLNAME outline PNAME QNAME
cc
cc                           Create any unique real plane PLNAME, containing the
cc                           outline of quadric surface QNAME, as seen from
cc                           point PNAME.
cc                           Display the "central point" of the outline plane,
cc                           nearest the center of QNAME.  The outline curve may
cc                           be hidden or may be imaginary.
cc
ccin                       plane PLNAME project VNAME QNAME
cc
cc                           Create any unique plane PLNAME, containing the
cc                           outline of quadric surface QNAME, as seen from an
cc                           infinite distance in the direction of vector VNAME.
cc                           Plane PLNAME will contain the curve on quadric
cc                           surface QNAME where the normal vector is
cc                           perpendicular to vector VNAME.
cc                           Display the "central point" of the outline plane,
cc                           nearest the center of QNAME.
cc                           See command "quadric", option "project".
cc
ccin                       plane PLNAME parallel NUMPL INC PNAME1 PNAME2
cc
cc                           Create a family of NUMPL parallel planes PLNAME,
cc                           ..., with names incremented by INC characters,
cc                           equally spaced along the normal axis from point
cc                           PNAME1 to point PNAME2.  See "increment names".
cc
cc                           This command can be replaced, using a do loop.
cc                           See "do loop use", commands "do", "enddo".
cc
ccin                       plane PL(2) move NUMPL INC PL(1) VMOVE
cc
cc                           Create a family of NUMPL parallel planes PL(2),
cc                           ..., with names incremented by INC characters,
cc                           translated by vector VMOVE from the preceding
cc                           plane, starting from plane PL(1).
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       plane PL(2) rotate NUMPL INC PL(1) PAXIS VAXIS DANGLE
cc
cc                           Create a family of NUMPL planes PL(2), ..., with
cc                           names incremented by INC characters, rotated around
cc                           an axis through point PAXIS in the direction of
cc                           vector VAXIS, by equal angular increments DANGLE
cc                           (counterclockwise, with the axis pointed at the
cc                           observer), starting from plane PL(1).
cc                           Such a family of planes is sometimes called a fan.
cc                           See "quadric", "rotated planes".
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
ccin                       plane PL(2) scale NUMPL INC PL(1) OPNAME PINV
cc
cc                           Create a family of NUMPL planes PL(2), ..., with
cc                           names incremented by INC characters, by scaling the
cc                           preceding plane with operator OPNAME and invariant
cc                           point PINV, starting from plane PL(1).
cc                           See "increment names".
cc                           See command "quadric" for an equivalent set of
cc                           commands using subscripted names and a do loop.
cc
cc                         Synonyms:  [disk, disc, dk], [help, h],
cc                         [move, mv, trans, translate], [parallel, par],
cc                         [plane, pl], [project, proj], [rotate, rot],
cc                         [tangent, tan], [triangle, tri], [vector, v, vect].

cc    plane              An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more planes.

cc    plane              An option in command "project", to project onto a
cc                         plane.

cg    plane              See "plane of refl".

cg    plane arrays       Planes may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    plane of refl      A reflection operator OPNAME, specified with command
cg                         "operator", has associated with it the normal
cg                         vector of a plane of reflection, which may be
cg                         displayed with command "operator OPNAME".
cg                         The specific plane is determined by the invariant
cg                         point PINV specified in command "reflect" that
cg                         uses OPNAME.

cg    planegen           A type of plane, not perpendicular to a major axis.

cg    planes             A family of planes may be created with command "plane",
cg                         options "parallel", "rotate", "move" or "scale" or
cg                         as follows:
cg
cg                         p pinv (options)
cg
cg                           Create an invariant point, if needed.
cg
cg                         pl PL(1) (options)
cg
cg                           Create a base plane.
cg
cg                         op opr (options)
cg
cg                           Create a tensor operator, to reflect, rotate,
cg                           invert or scale, if needed.
cg
cg                         v vmove (options)
cg
cg                           Create a vector for use as a translation operator,
cg                           if needed.
cg
cg                         cp PL(1) PL(2)
cg
cg                           Copy plane PL(1) to the first plane.
cg
cg                         Repeat the following block of commands as many times
cg                         as needed to create the rest of the family of planes.
cg                         See "shortcuts".
cg
cg                         mv pl PL(1) vmove
cg
cg                           Move the base plane by amount vmove, if needed.
cg
cg                         [invert, reflect, rotate, scale] pl PL(1) opr [pinv]
cg
cg                           Invert, reflect, rotate or scale the base plane
cg                           with tensor operator opr, if needed.
cg
cg                         cp pl PL(1) +
cg
cg                           Copy the base plane to the next plane.

cg    planetary          See "planetary orbits".

cg    planetary orbits   A planetary orbit is an ellipse, parabola or hyperbola,
cg                         with a focal point at the center of gravity of the
cg                         attracting body and the orbiting body.
cg
cg                         A general equation of such an orbit, if its axis is
cg                         known, is 1 / RSPH = A + B cos (PHI), where RSPH is
cg                         the distance from the focal point, PHI is the angle
cg                         angle from the axis, and A > 0.  The orbit type is
cg                         determined by:
cg                         A + B > 0:  elliptic, with the center on the axis
cg                                     at distance A / (A^2 - B^2) from the
cg                                     focal point.
cg                         A + B = 0:  parabolic, with the vertex at distance
cg                                     1 / (A - B) from the focal point.
cg                         A + B < 0:  hyperbolic, with the symmetry point on
cg                                     the axis, at distance A / (A^2 - B^2)
cg                                     from the focal point.
cg
cg                         The intersections of the orbit with the axis are at
cg                         the radii R = 1 / (A + B) and R = 1 / (A - B).
cg
cg                         The implicit quadric equation of the surface produced
cg                         by rotating such an orbit around the axis, with the
cg                         focal point at the origin and the axis in the
cg                         direction of the positive z axis, is:
cg                         - 1 + QZ*z + QXX^2*(x^2 + y^2) + QZZ*z^2 = 0,
cg                         where QZ = 2 * B, QXX = A^2, QZZ = A^2 - B^2,
cg                         and A = sqrt (AXX), B = 0.5 * QZ.
cg                         See command "axisym AXNAME focus ...".

cg    planex             A type of plane, with a constant value of x.

cg    planey             A type of plane, with a constant value of y.

cg    planez             A type of plane, with a constant value of z.

cc    PLNAME             The name of a plane.  May have up to 24 characters,
cc                         including any subscripts, and may be ASCII, integer,
cc                         or floating point.  Must not be the same as any other
cc                         quadric surface name QNAME.
cc                         No plane name may be "+", "-", "all", "h", "help",
cc                         "list" or "thru", begin with "!" or contain ";".
cc                         Also referred to as PLNAME1, PLNAME2, PLNAME3, ...,
cc                         PL(1), PL(2), ...
cc                         Specified with command "plane", "axisym" or
cc                         "quadric".
cc
cc                         Planes may appear in commands:
cc                         accelerate, arc, copy, delete, distance, extrema,
cc                         help, invert, last, list, move, operator, plane,
cc                         point, project, proximal, quadric, reflect, rename,
cc                         repack, rotate, scale, search, side, slice, sort,
cc                         track, triple, vector.

cg    plot               A plot of point coordinates may be made by using
cg                         command "plot" to specify a 2-D plot array of single
cg                         characters, representing the coordinates of plot
cg                         points along two axes.  If the plot status is "on",
cg                         all points displayed or created, including named
cg                         points and points of intersection or proximity,
cg                         will be stored as plot points, and may be mapped onto
cg                         the 2-D plot array and displayed with command
cg                         "plot array".
cg
cg                         The following commands relate to the plot:  debug,
cg                         delete, help, marker, plot, point, symbol, synonym,
cg                         tables, variable.

cc    plot               An option in command "debug", to display GEOM internal
cc                         parameters and parameters and variables for plots:
cc                         aplstat, aplmarkt, amark, amarkt, nplcen,
cc                         aplmarkh, amark, amarkt, axish, nplimh, splhcen,
cc                         splhmax, splhmin, spthmax, spthmin, zoomplh,
cc                         aplmarkv, amark, amarkt, axisv, nplimv, splvcen,
cc                         splvmax, splvmin, sptvmax, sptvmin, zoomplv,
cc                         nptmarkm, nptmarks, aptmark, amark, amarkt,
cc                         aptmarks, amark, amarkt,
cc                         nplptm, nplpts, aplotm, xplot, yplot, zplot,
cc                         nplcolm, nplcols, nplinem, nplines, aplbuf.

cc    plot               Command to define parameters for plotting points, and
cc                         to plot points.
cc                         See "aliases", for suggestions for short commands.
cc
cc                         Command "plot" relates to objects:  marker, plot,
cc                         point, symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help plot
ccin                       plot [help]
cc
cc                           Display the command options.
cc
ccin                       plot status
cc
cc                           Display the plot status (on or off), the plot
cc                           title, the plot axis labels and coordinate limits,
cc                           the plot point marker, and the size of the plot
cc                           array.  Also see command "status".
cc
ccin                       plot size [NCOLUMNS NLINES]
cc
cc                           Display [or specify] the number of plot array
cc                           columns [NCOLUMNS (<= 164)] and lines
cc                           [NLINES (<= 102)].  The pair 65 by 41 is almost
cc                           square, and fits nicely in a standard 80 by 47
cc                           window.  Other almost square sizes are 81 by 51,
cc                           101 by 61, 129 by 81 and 161 by 101.
cc
ccin                       plot off
cc
cc                           Stop storing plot points.  This is the initial
cc                           default.
cc
ccin                       plot on
cc
cc                           Resume storing plot points.
cc
ccin                       plot array
cc
cc                           Map the plot points to the plot array and display
cc                           it, along with all of the plot parameters.
cc
ccin                       plot point [all]
cc
cc                           Display all plot points.
cc
ccin                       plot point index N1 ... N2 [thru] N3 ...
cc
cc                           Display the plot points with indices N1, ..., N2,
cc                           [, through] N3, ...
cc
ccin                       plot point m AMNAME1 ... AMNAME2 [thru] AMNAME3 ...
cc
cc                           Display plot points with marks AMNAME1, ...,
cc                           AMNAME2, [, through] AMNAME3, ...
cc
ccin                       plot delete [all]
cc
cc                           Delete all plot points.
cc
ccin                       plot delete index N1 ... N2 [thru] N3 ...
cc
cc                           Delete the plot points with indices N1, ..., N2
cc                           [, through] N3, ...
cc
ccin                       plot delete m AMNAME1 ... AMNAME2 [thru] AMNAME3 ...
cc
cc                           Delete plot points with marks AMNAME1, ..., AMNAME2
cc                           [, through] AMNAME3, ...
cc
ccin                       plot title [AMNAME]
cc
cc                           Display [or specify the marker AMNAME whose text
cc                           is] the plot title.
cc
ccin                       plot axis [[h,v] COORD AMNAME]
cc
cc                           Display the plot axis coordinates and labels [or
cc                           specify the horizontal or vertical axis coordinate
cc                           COORD and the marker AMNAME whose text is the
cc                           horizontal or vertical axis label].  Argument
cc                           COORD must be x, y, z, rcyl, rsph, theta or phi.
cc                           See commands "coordinate", "angles", "marker".
cc
ccin                       plot marker [AMNAME]
cc
cc                           Display [or specify AMNAME to be] the marker to be
cc                           the plot array character.
cc
ccin                       plot marker all
cc
cc                           Display all plot point markers.
cc
ccin                       plot limits -
cc
cc                           Specify that the plot axis limits will include all
cc                           plot points (default).
cc
ccin                       plot limits [[h,v] SMIN SMAX]
cc
cc                           Display the plot axis limits [or specify the
cc                           horizontal or vertical plot axis limits to be SMIN
cc                           and SMAX].
cc
ccin                       plot center -
cc
cc                           Default the plot center to the center
cc
ccin                       plot center [SCENH SCENV]
cc
cc                           Display [or specify SCENH and SCENV to be the
cc                           horizontal and vertical coordinates of the plot
cc                           center.
cc
ccin                       plot zoom [[all,h,v] ZOOMULT]
cc
cc                           Display [or multiply by ZOOMULT] the zoom factors
cc                           [for both, the horizontal or vertical axis].  Use
cc                           ZOOMULT = 0 to make the zoom factor 1 (no zoom).
cc
cc                         Synonyms:  [help, h], [limits, lim],
cc                         [marker, m, mark],
cc                         [point, p, pnt, pt], [status, stat].

cg    plot               See "aliases", "plot aliases".

cg    plot aliases       See "aliases".

cg    plus               The plus sign, "+".

cc    PMESH              The root name (without the subscripts k, l and m) of
cc                         a mesh point.  The full name, including subscripts,
cc                         parentheses and commas, may not exceed 24 characters.
cc                         The naming restrictions for PNAME apply.

cc    PNAME              The name of a point.  May have up to 24 characters,
cc                         including any subscripts, and may be ASCII, integer,
cc                         or floating point.
cc                         No point name may be "+", "-", "all", "cl",
cc                         "cluster", "h", "help", "list" or "thru", begin with
cc                         "!" or "-", or contain ";".
cc                         Specified with command "point".
cc                         A point released from the mesh will have its first
cc                         character replaced by a minus sign, and must be
cc                         renamed before being reassigned to the mesh.
cc
cc                         Also referred to as PMESH(K,L,M).
cc                         Also referred to as PNAME1, PNAME2, PNAME3, ...
cc                         Also referred to as PNAME(1), PNAME(2), PNAME(3), ...
cc                         Also referred to as PAXIS, PCEN, PCOP, PFOC, PINIT,
cc                         PINV, PMID, PTL, PTR, PVER, PVIEW, ...
cc
cc                         Used to assign mesh indices with command "mesh".
cc                         Used to release mesh indices with command "mesh".
cc                         Used to create clusters with command "cluster".
cc                         Used to create lines with command "line".
cc                         Used to create vectors with command "vector".
cc                         Used to create triangles with command "triangle".
cc                         Used to create regular polygons with command
cc                         "polygon".
cc                         Used to create general polygons in command "area".
cc                         Used to create annular disks with command "disk".
cc                         Used to create regular polyhedrons with command
cc                         "polyhedron".
cc                         Used to create tetrahedrons with command
cc                         "tetrahedron".
cc                         Used to create planes with command "plane".
cc                         Used to create quadrics with commands "quadric",
cc                         "sphere", "cylinder", "cone", "ellipsoid" or
cc                         "axisym".
cc                         Used to find a circle through three points.
cc
cc                         Points may appear in commands:
cc                         accelerate, arc, area, axisym, bisect, circle,
cc                         cluster, cone, copy, cylinder, delete, disk,
cc                         distance, ellipsoid, help, hyperb, invert, last,
cc                         line, list, mesh, move, operator, plane, plot, point,
cc                         polygon, polyhedron, project, proximal, quadric,
cc                         reflect, rename, repack, rotate, scale, search, side,
cc                         search, sort, sphere, tetrahedron, track, triangle,
cc                         triple, twist, vector, volume, walk, where.

cc    pnt                A synonym for point.

cg    Point              In the output display for a quadric surface, indicates
cg                         that the quadric surface is confined to an extremely
cg                         small spherical volume of space.  Sometimes a false
cg                         point is indicated when the quadric surface is very
cg                         close to a limiting case, for example, when an
cg                         ellipsoid has very sharp tips.

cg    point              A point is a geometric object having a name PNAME, and
cg                         a position in 3-D space specified by three
cg                         coordinates in the current coordinate system, which
cg                         may be rectangular (x, y, z), cylindrical
cg                         (rcyl, theta, z) or spherical (rsph, theta, phi).
cg                         A point is created with command "point".
cg                         See "coordinate system".
cg                         If a point is also a mesh point, it is also specified
cg                         by the logical mesh indices k, l and m, assigned with
cg                         command "mesh".
cg                         The maximum number of points is now 1000.
cg
cg                         No point name may be "+", "-", "all", "cl",
cg                         "cluster", "h", "help", "list" or "thru", begin with
cg                         "!" or contain ";".
cg
cg                         A point may be in any number of clusters of points.
cg                         Many commands operate on either points or clusters.
cg
cg                         One point may be used to specify the bound point of
cg                         a vector, a point on an axis, or the center of
cg                         a circle, a sphere or an annular disk.
cg                         Two points may be used to create a line.
cg                         Two points may be used to specify the direction and
cg                         (optionally) the magnitude of a vector.
cg                         Three points may be used to create a triangle, a
cg                         plane, a regular polygon or a regular polyhedron.
cg                         From 3 to 24 points may be used to create a general
cg                         polygon with command "area" (temporarily).
cg                         Four points may be used to create a tetrahedron.
cg                         Points may also be used to create a quadric surface.
cg
cg                         A point may be moved with commands "move", "scale",
cg                         "invert", "reflect" or "rotate".
cg                         If a point is moved, all lines, triangles, regular
cg                         polygons, regular polyhedrons and tetrahedrons
cg                         specified by that point will change their geometry.
cg                         A point may not be deleted if a line, triangle,
cg                         regular polygon, regular polyhedron or tetrahedron is
cg                         specified by that point.  All such objects must be
cg                         deleted first.
cg                         If the name of a point is changed, the change also
cg                         applies to any clusters, lines, triangles, regular
cg                         polygons, annular disks, regular polyhedrons and
cg                         tetrahedrons specified by the point.
cg
cg                         To find points where a major plane intersects a
cg                         quadric surface, use command "slice".
cg
cg                         To find a proximal point, use command "distance",
cg                         "proximal" or "side",
cg
cg                         The following commands relate to points:
cg                         accelerate, arc, area, axisym, bisect, circle,
cg                         cluster, cone, copy, cylinder, debug, delete, disk,
cg                         distance, ellipsoid, help, hyperb, invert, last,
cg                         line, list, mesh, move, plane, plot, point, polygon,
cg                         polyhedron, project, proximal, quadric, random,
cg                         reflect, rename, repack, rotate, scale, search, side,
cg                         sort, sphere, symbol, synonym, tables, tetrahedron,
cg                         track, triangle, triple, twist, vector, volume, walk,
cg                         where.

cc    point              An option in command "debug", to display GEOM internal
cc                         parameters and variables for points:  npointm,
cc                         npoints, apoint, lapoint, x, y, z, apoints, lapoints.
cc                         Synonyms:  [point, p, pnt, pt].

cc    point              Command to display or create one or more points.
cc                         Creating a point replaces the coordinates of any
cc                         existing point having the same name, but does not
cc                         change any assigned mesh indices of such a point.
cc                         See "do loop use", and commands "mesh point ...".
cc
cc                         Command "point" relates to objects:  axisym, brick,
cc                         cluster, cone, cylinder, disk, ellipsoid, hyperb,
cc                         mesh, operator, plane, point, polygon, polyhedron,
cc                         quadric, sphere, symbol, tetrahedron, triangle,
cc                         variable, vector, zone.
cc
cc                         Forms include:
cc
ccin                       help point
ccin                       point [help]
cc
cc                           Display the command options.
cc
ccin                       point [all,list]
cc
cc                           Display all points (short display).
cc
ccin                       point list PNAME1 PNAME2 PNAME3 ...
cc
cc                           Display points PNAME1, PNAME2, PNAME3, ..., with
cc                           or without subscripts (short display).
cc
ccin                       point PNAME
cc
cc                           Display point PNAME (long display).
cc
cc                         See command "cluster CLNAME nomesh" to display
cc                         non-mesh points.
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         the first point name to form the name by incrementing
cc                         or decrementing the base name for points.
cc                         See command "last", "increment names".
cc
ccin                       point PNAME [=] U V W
cc
cc                           Create point PNAME, with coordinates (U, V, W),
cc                           which may be (X, Y, Z), (RCYL, THETA, Z) or (RSPH,
cc                           THETA, PHI), depending on the current coordinate
cc                           system, which may be rectangular, cylindrical or
cc                           spherical, respectively.  See command "coordinate".
cc                           The "=" is optional if argument U is integer or
cc                           floating point or a symbolic word which, after
cc                           symbol replacement, is integer or floating point.
cc
ccin                       point PNAME triangle TRNAME W1 W2 W3
cc
cc                           Create point PNAME, at local coordinates or vertex
cc                           weights W1, W2 and W3 in triangle TRNAME.
cc                           For the midpoint of an edge, use weights of 0.5
cc                           for each of the bounding vertices.
cc                           Use weights of 0.5, 0.5, 0.5 for the centroid of
cc                           the triangle.
cc
ccin                       point PNAME tetrahedron TETNAME W1 W2 W3 W4
cc
cc                           Create point PNAME, at local coordinates or vertex
cc                           weights W1, W2, W3 and W4 in tetrahedron TETNAME.
cc                           For the midpoint of an edge, use weights of 0.5
cc                           for each of the bounding vertices.
cc                           Use weights of 0.5, 0.5, 0.5, 0.5 for the centroid
cc                           of the tetrahedron.
cc
ccin                       point PNAME brick BRNAME FU FV FW
cc
cc                           Create point PNAME, in brick BRNAME, at fractional
cc                           volumes FU, FV and FW between the minimum and
cc                           maximum u, v and w coordinates, respectively, of
cc                           the brick.  Use 0.5, 0.5, 0.5 to locate a point at
cc                           the centroid of the brick.
cc
ccin                       point PNAME(1) polygon PGNAME
cc
cc                           Create points PNAME(1), PNAME(2), PNAME(3), ..., at
cc                           the vertices of regular polygon PGNAME.  Make sure
cc                           PNAME(1) can be incremented enough for all of the
cc                           vertices.  See "increment names".
cc
ccin                       point PNAME(1) polyhedron PHNAME
cc
cc                           Create points PNAME(1), PNAME(2), PNAME(3), ..., at
cc                           the vertices of regular polyhedron PHNAME.  Make
cc                           sure PNAME(1) can be incremented enough for all of
cc                           the vertices (4 to 20).
cc                           See "increment names".
cc
ccin                       point PNAME QTYPE QNAME
cc
cc                           Create point PNAME, at the center of symmetry of
cc                           the quadric surface of type QTYPE (plane, sphere,
cc                           cylinder, cone, hyperb, ellipsoid, axisym,
cc                           quadric), with name QNAME.
cc
ccin                       point PNAME proximal
cc
cc                           Create point PNAME, at the last proximal point
cc                           found with command "accelerate", "distance",
cc                           "proximal", "side", "track" or "triple".
cc                           If no proximal point was found, will be
cc                           (-10^99,-10^99,-10^99).
cc                           See the particular command for details.
cc
ccin                       point PNAME random [RADIUS|1]
cc
cc                           Create point PNAME, sampled randomly in a sphere of
cc                           radius RADIUS centered at the origin.  If not
cc                           specified, RADIUS defaults to 1.  Also see command
cc                           "bin".
cc
ccin                       point PNAME brick BRNAME random
ccin                       point PNAME disk DKNAME random
ccin                       point PNAME sphere SPHNAME random
ccin                       point PNAME tetrahedron TETNAME random
ccin                       point PNAME triangle TRNAME random
cc
cc                           Create point PNAME, sampled randomly in
cc                           brick BRNAME, disk DKNAME, sphere SPHNAME,
cc                           tetrahedron TETNAME or triangle TRNAME,
cc                           respectively.
cc
ccin                       point PNAME zone ZNAME random BRNAME
cc
cc                           Create point PNAME, sampled randomly in zone ZNAME,
cc                           by trying up to 10000 points sampled randomly in
cc                           brick BRNAME.  Make sure brick BRNAME completely
cc                           encloses zone ZNAME (see command "mcvol").
cc                           WARNING:  this may fail, if the zone is much
cc                           smaller than the brick.  Also see command "bin".
cc
ccin                       point PMESH brick BRNAME mesh block [volume]
cc
cc                           Create a family of points PMESH(k,l,m), assigned to
cc                           the mesh, with indices k, l and m ranging over the
cc                           indices of the current mesh block, and representing
cc                           equal intervals [or volumes] between the minimum
cc                           and maximum u, v and w coordinates of the brick,
cc                           respectively, where u, v and w are in the
cc                           coordinate system used to create the brick.
cc                           See "briquette".
cc                           Points previously assigned to the mesh block will
cc                           be renamed by replacing the first character of
cc                           their names with "-".  That must not create any
cc                           duplicate names.
cc
ccin                       point PNAME(2) move NUMPT INC PNAME(1) VMOVE
cc
cc                           Create a family of NUMPT points PNAME(2), ..., with
cc                           names incremented by INC characters, and spaced at
cc                           intervals of vector VMOVE, starting from point
cc                           PNAME(1).   See "increment names".
cc
cc                           This command can be replaced, using a do loop.
cc                           See "do loop use", commands "do", "enddo".
cc
ccin                       point PNAME(2) rotate NUMPT INC PNAME(1) OPNAME PINV
cc
cc                           Create a set of NUMPT points PNAME(2), ..., with
cc                           names incremented by INC characters, by rotating
cc                           the preceding point with operator OPNAME and
cc                           invariant point PINV, starting from point PNAME(1).
cc                           See "increment names".
cc
cc                           This command can be replaced, using a do loop.
cc                           See "do loop use", commands "do", "enddo".
cc
ccin                       point PNAME(2) scale NUMPT INC PNAME(1) OPNAME PINV
cc
cc                           Create a family of NUMPT points PNAME(2), ..., with
cc                           names incremented by INC characters, by scaling the
cc                           preceding point with operator OPNAME and invariant
cc                           point PINV, starting from point PNAME(1).  See
cc                           "increment names".
cc
cc                           This command can be replaced, using a do loop.
cc                           See "do loop use", commands "do", "enddo".
cc
cc                         To assign a triply subscripted point name
cc                         PMESH(K,L,M) to a logical mesh, at indices K, L, M,
cc                         use the following command:
cc
cc                           "mesh point PMESH(K,L,M)"
cc
cc                         For ranges of values of K, L and M, bracket the
cc                         preceding command with "do" and "enddo" commands.
cc
cc                         Also see commands "mesh point PMESH(K,L,M)",
cc                         "mesh point PNAME K L M", "mesh point PMESH block".
cc
cc                         Synonyms:  [brick, br], [cylinder, cyl],
cc                         [ellipsoid, el, ell, ellipse], [help, h],
cc                         [hyperb, saddle], [move, mv, trans, translate],
cc                         [plane, pl], [point, p, pnt, pt], [proximal, prox],
cc                         [quadric, q, quad], [random, ?], [rotate, rot],
cc                         [sphere, sph], [tetrahedron, tet, tetra],
cc                         [triangle, tri], [volume, vol], [zone, z, zn].

cc    point              An option in command "accelerate", to find the points
cc                         on the parabolic trajectory of a uniformly
cc                         accelerated particle that intersect or are proximal
cc                         to a point.

cc    point              An option in commands "area", "copy", "delete", "last",
cc                         "list", "rename", "repack" and "sort", to perform
cc                         the specified operation on one or more points.
cc                         Synonyms:  [point, p, pnt, pt].

cc    point              An option in command "plot", to display the stored plot
cc                         points, with their indices and markers.

cc    point              An option in command "sphere", to find the sphere
cc                         with its surface going through four specified points.
cc                         Synonyms:  [point, p, pnt, pt].

cc    point              An option in command "triple", to use a point as the
cc                         initial guess for the triple point.  Used only when
cc                         two or more of the quadric surfaces are non-planar.
cc                         Synonyms:  [point, p, pnt, pt].

cc    point              An option in command "vector", to specify the vector
cc                         by specifying the names of two points defining the
cc                         direction and optionally the length of the vector.

cg    point              See "point element", "proximal point", "saddle point".

cg    point arrays       Points may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    point element      A point element in the mesh is a mesh point, created
cg                         with command "point", and assigned a single specified
cg                         value of each of the mesh indices k, l and m.
cg                         See "line element", "surface element",
cg                         "volume element".

cg    points             To create a family of variables with values
cg                         forming an arithmetic or geometric series, for use in
cg                         specifying the coordinates of points, use command
cg                         "variable", option "series".
cg
cg                         To create a family of points at the vertices of a
cg                         regular polygon or polyhedron, use command "point",
cg                         options "polygon", "polyhedron".
cg
cg                         To create a set of randomly sampled points, PNAME(1),
cg                         PNAME(2), ..., using command "point", option
cg                         "random":
cg
cg                         point PNAME(1) ... random ...
cg                         alias k = 'point + ... random ...'
cg                         k<rtn>k<rtn>k<rtn>...
cg
cg                         To create a family of equally spaced points in a
cg                         straight line, use command "point", option "move".
cg                         To create a family of points with equal ratio spacing
cg                         in a straight line, use command "point", option
cg                         "scale".
cg                         To make a family of points equally spaced around a
cg                         circle, use command "point", option "rotate".

cg    polar              The polar angle, phi, of a point is the angle formed by
cg                         the positive z axis and a line from the origin to the
cg                         point, and may range from 0 to 180 degrees (0 to pi
cg                         radians).  See "azimuth", "theta".

cc    polar              An option in command "project", to project points
cc                         onto the surface of a sphere, in the direction
cc                         parallel to the polar (z) axis of the sphere.

cc    poly               A synonym for polynomial.

cc    polyg              A synonym for polygon.

cg    polygon            To create a zone partially bounded by a a family of
cg                         planes with parallel lines of intersection, and
cg                         enclosing a regular polygonal cylinder, use command
cg                         "plane" with option "rotate", and command "zone".

cg    polygon            A regular polygon is a geometric object in 3-D space,
cg                         specified by the number of vertices, which must be
cg                         three or more, and by three points:  its center, one
cg                         of its vertices, and another point in its plane, not
cg                         collinear with the first two points.
cg                         The maximum number of polygons is now 1000.
cg                         The maximum number of vertices is now 1000.
cg
cg                         A regular polygon has vertices lying in a plane,
cg                         equidistant from a center, and at equal angular
cg                         increments around the center.
cg
cg                         A specified regular polygon will change if any of the
cg                         points used to specify it are moved.  No point may be
cg                         deleted if it is being used to specify an existing
cg                         regular polygon.  However, a change in the name of
cg                         such a point will not change the regular polygon.
cg
cg                         The vertices of a regular polygon may be specified
cg                         as points, using command "point", and then
cg                         included in a cluster, using command "cluster".
cg
cg                         The plane containing the regular polygon may be
cg                         created using the same three points used to create
cg                         the polygon.
cg
cg                         A general polygon may have any number of vertices,
cg                         which need not be coplanar.  To find the projected
cg                         area of such a polygon, use command "area".
cg
cg                         The following commands relate to regular
cg                         polygons:
cg                         copy, debug, delete, help, last, list, point,
cg                         polygon, rename, repack, search, sort, symbol,
cg                         synonym, tables.

cc    polygon            An option in command "debug", to display GEOM internal
cc                         parameters and variables for regular polygons:
cc                         npolygm, npolygs, apolyg, nverpg, acenpg, acoppg,
cc                         averpg, apolygs, lapolygs,
cc                         nverpgm, nverpgs, xver, yver, zver.
cc                         Synonyms:  [polygon, pg, polyg].

cc    polygon            Command to display one or more regular polygons or to
cc                         create a regular polygon.  Creating a polygon
cc                         replaces any existing polygon having the same name.
cc                         The display for a regular polygon includes the vertex
cc                         angle, the edge length, the area, the radii of the
cc                         inscribed and circumscribed circles, the coordinates
cc                         of the vertices, and the normal vector.
cc
cc                         Command "polygon" relates to objects:  point,
cc                         polygon, symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help polygon
ccin                       polygon [help]
cc
cc                           Display the command options.
cc
ccin                       polygon [all,list]
cc
cc                           Display all regular polygons (short display).
cc
ccin                       polygon list PGNAME1 PGNAME2 PGNAME3 ...
cc
cc                           Display regular polygons PGNAME1, PGNAME2, PGNAME3,
cc                           ..., with or without subscripts (short display).
cc
ccin                       polygon PGNAME
cc
cc                           Display regular polygon PGNAME (long display).
cc
cc                         In the following command, use "+" or "-" instead of
cc                         PGNAME to form the name by incrementing or
cc                         decrementing the base name for polygons.
cc                         See command "last", "increment names".
cc
ccin                       polygon PGNAME fit NVER PCEN PVER PCOP
cc
cc                           Create regular polygon PGNAME, with NVER (from 3 to
cc                           nverpgm, now 1000) vertices, centered at point
cc                           PCEN, with one vertex at point PVER, and in the
cc                           same plane as point PCOP, which must not be
cc                           collinear with points PCEN and PVER.  Any later
cc                           change in the coordinates or names of points PCEN,
cc                           PVER or PCOP will change the definition of regular
cc                           polygon PGNAME.
cc
cc                         Synonyms:  [help, h], [polygon, pg, polyg].
cc
cc                         To find the area of a general polygon, use command
cc                         "area".
cc
cc                         To create points at the vertices of a regular
cc                         polygon, use command "point".

cc    polygon            An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more polygons.

cg    polygon            See "polygon equations".

cg    polygon arrays     Polygons may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    polygon equations
cg                       The equations for a regular polygon are as follows:
cg
cg                         N  = the number of vertices or the number of edges.
cg                         PI = pi radians or 180 degrees.
cg                         TC = the central angle of an edge
cg                            = 2 * PI / N
cg                         TV = the interior angle at a vertex
cg                            = PI * (1 - 2 / N)
cg                         TT = the sum of all of the interior angles
cg                            = N * TT = PI * (N - 2)
cg
cg                         RI = the radius of the inscribed circle, or the
cg                              distance from the center of the polygon to the
cg                              center of an edge
cg                            = RC * cos (PI / N)
cg                         RC = the radius of the circumscribed circle or the
cg                              distance from the center to a vertex
cg                            = RI * sec (PI / N)
cg
cg                         S  = the length of an edge or the distance from one
cg                              vertex to an adjacent vertex
cg                            = 2 * RC * sin (PI / N) = 2 * RI * tan (PI / N)
cg
cg                         C  = the circumference or the sum of all the edge
cg                              lengths
cg                            = 2 * RC * N * sin (PI / N)
cg                            = 2 * RI * N * tan (PI / N)
cg
cg                         A  = the area
cg                            = RC^2 * (N / 2) * sin (2 * PI / N)
cg                            = RI^2 * N * tan (PI / N)
cg                            = RI * N * S / 2

cc    polyh              A synonym for polyhedron.

cg    polyhedron         A regular polyhedron is a geometric object in 3-D
cg                         space, specified by the number of vertices, which is
cg                         4 (a tetrahedron with 6 edges and 4 faces),
cg                         6 (an octahedron with 12 edges and 8 faces),
cg                         8 (a cube with 12 edges and 6 faces),
cg                         12 (an icosahedron with 30 edges and 20 faces),
cg                         or 20 (a dodecahedron with 30 edges and 12 faces),
cg                         and by three points:  its center, one of its
cg                         vertices, and another point that is in the plane of
cg                         the center, the specified vertex, and an adjacent
cg                         vertex, but is not collinear with the center and
cg                         first vertex.  The maximum number of polyhedrons
cg                         is now 1000.
cg
cg                         A regular polyhedron has vertices lying on a
cg                         circumscribed sphere, with faces tangent to an
cg                         inscribed sphere, with equal dihedral angles between
cg                         faces, equal edge lengths, and equal central angles
cg                         for each edge.
cg
cg                         A regular polyhedron will change if any of the points
cg                         used to specify it are moved.  No point may be
cg                         deleted if it is being used to specify an existing
cg                         regular polyhedron.  However, a change in the name of
cg                         such a point will not change the regular polyhedron.
cg
cg                         The vertices of a regular polyhedron may be specified
cg                         as points, using command "point", and then
cg                         included in a cluster, using command "cluster".
cg
cg                         The following commands relate to regular
cg                         polyhedrons:
cg                         copy, debug, delete, help, last, list, point,
cg                         polyhedron, rename, repack, search, sort, symbol,
cg                         synonym, tables.

cc    polyhedron         An option in command "debug", to display GEOM internal
cc                         parameters and variables for regular polyhedrons:
cc                         npolyhm, npolyhs, apolyh, atypeh, nverph, nedgph,
cc                         acenph, averph, acopph, apolyhs, lapolyhs,
cc                         nverphs, nedgphs, xverh, yverh, zverh, nver1, nver2.
cc                         Synonyms:  [polyhedron, ph, polyh].

cc    polyhedron         Command to display one or more regular polyhedrons or
cc                         to create a regular polyhedron.
cc                         Creating a regular polyhedron replaces any existing
cc                         regular polyhedron having the same name.
cc                         The display for a regular polyhedron includes the
cc                         coordinates of the vertices, the indices of the
cc                         vertices for each edge, the individual and total
cc                         edge length, the individual and total face area,
cc                         the volume, the dihedral angle (between faces),
cc                         the central angle of each edge, and the radii of the
cc                         inscribed and circumscribed spheres, in addition to
cc                         the points initially used to create the polyhedron.
cc
cc                         Command "polyhedron" relates to objects:  point,
cc                         polyhedron, symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help polyhedron
ccin                       polyhedron [help]
cc
cc                           Display the command options.
cc
ccin                       polyhedron [all,list]
cc
cc                           Display all regular polyhedrons (short display).
cc
ccin                       polyhedron list PHNAME1 PHNAME2 PHNAME3 ...
cc
cc                           Display regular polyhedrons PHNAME1, PHNAME2,
cc                           PHNAME3, ..., with or without subscripts (short
cc                           display).
cc
ccin                       polyhedron PHNAME
cc
cc                           Display regular polyhedron PHNAME (long display).
cc
cc                         In the following command, use "+" or "-" instead of
cc                         PHNAME to form the name by incrementing or
cc                         decrementing the base name for polyhedrons.
cc                         See command "last", "increment names".
cc
ccin                       polyhedron PHNAME fit NVER PCEN PVER PCOP
cc
cc                           Create regular polyhedron PHNAME, with NVER
cc                           vertices (4 for a tetrahedron, 6 for an octahedron,
cc                           8 for a cube, 12 for an icosahedron or 20 for a
cc                           dodecahedron), centered at point PCEN, with one
cc                           vertex at point PVER, and with an adjacent vertex
cc                           in the same plane as points PCEN, PVER and PCOP.
cc                           Any later change in the coordinates or names of
cc                           points PCEN, PVER or PCOP will change the
cc                           definition of regular polyhedron PHNAME.
cc
cc                         Synonyms:  [help, h], [polyhedron, ph, polyh].
cc
cc                         To find the area of a general polyhedron, use
cc                         command "area".
cc
cc                         To create points at the vertices of a regular
cc                         polyhedron, use command "point".

cc    polyhedron         An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more polyhedrons.
cc                         Synonyms:  [polyhedron, ph, polyh].

cg    polyhedron arrays
cg                       Polyhedrons may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cc    polynomial         An option in command "variable", to create a variable
cc                         with the value of a polynomial function of a
cc                         specified argument.
cc                         Synonyms:  [polynomial, poly].

cg    polynomial         To find the real and/or complex roots of a quadratic,
cg                         cubic or quartic polynomial equation with specified
cg                         real coefficients, or to find the real coefficients
cg                         of a polynomial equation for from one to four
cg                         specified real and/or complex roots, use command
cg                         "roots".
cg
cg                         To search for real roots, extrema and inflection
cg                         points of a polynomial equation of higher order by
cg                         Newtonian iteration, or to find the real coefficients
cg                         of a polynomial equation of order N with N specified
cg                         real roots, use command "root".
cg
cg                         For the n'th order polynomial equation
cg                           P(z) = a0 + a1*z + a2*z^2 + ... + an*z^n = 0,
cg                         where z = x + i*y, i = sqrt (-1), and all
cg                         coefficients have been divided by the coefficient of
cg                         z^n, the n roots r1, r2, ..., rm, rn (m = n - 1)
cg                         satisfy the equations:
cg                           a0 =  (-1)^n * (product of all n roots)
cg                           a1 = -(-1)^n * (sum of all products of n-1 roots)
cg                           a2 =  (-1)^n * (sum of all products of n-2 roots)
cg                           a3 = -(-1)^n * (sum of all products of n-3 roots)
cg                           ...
cg                           al =  (sum of all products of two roots) (l = n-2)
cg                           am = -(sum of all of the roots) (m = n-1)
cg                           an = 1
cg
cg                         To find the value of a polynomial with specified
cg                         coefficients, and a specified argument, use command
cg                         "variable", option "polynomial" or command "roots",
cg                         option "?".

cg    position           In a display of data resulting from command
cg                         "accelerate", the x, y, z coordinates of a particle
cg                         on a parabolic trajectory resulting from uniform
cg                         acceleration.

cg    pound              The number or pound character "#".

cc    POW                Exponent of a power-law relative differential
cc                         probability in a probability bin specified with
cc                         command "bin".
cc
cc                         For relative differential probabilities PL at VRANL
cc                         and PR at VRANR,
cc                         POW = log (PR / PL) / log (VRANR / VRANL).

cc    pow                A synonym for power.

cg    power              The power, superscript or up arrow symbol, "^".

cg    power              To find powers of real or integer numbers, see
cg                         commands "variable", "icalc".

cc    power              An option in command "bin", to create a power-law
cc                         probability distribution function.
cc                         Synonyms:  [power, pow].

cg    power-law          A power-law probability distribution is a type of
cg                         probability bin or probability distribution function
cg                         (pdf) for which the relative differential probability
cg                         is a power-law function of the value of the random
cg                         variable, for a specified range of such values.
cg                         A power of zero is the same as a uniform probability.
cg                         A power of 1 is the same as a linear probability, and
cg                         is useful for sampling a radius in a circular
cg                         cylinder or an annulus of a circular cylinder.
cg                         A power of 2 is useful for sampling a radius in a
cg                         sphere or a spherical annulus.
cg                         See "discrete event", "discrete object",
cg                         "discrete value", "uniform", "linear", "exponential".
cg                         See commands "bin", "pdf", "sample".

cg    powers             To create a table of powers of a specified constant,
cg                         FMULT:
cg
cg                         variable VAR(1) = 1
cg
cg                           Create variable VAR(1) with value 1.
cg
cg                         variable VAR(2) series NUMVAR INC VAR(1) 0 FMULT
cg
cg                           Create a series of NUMVAR variables VAR(2), ...,
cg                           with names incremented by INC characters, for which
cg                           the n'th value is the n'th power of FMULT.
cg                           See "increment names".

cg    precision          The DEC version does floating point arithmetic with
cg                         about 15 to 16 significant figures, with exponents
cg                         from -308 to +308.
cg
cg                         Integer arithmetic, using command "icalc", uses
cg                         64 bits (about 18 significant figures) in the DEC
cg                         version.
cg
cg                         Big integer arithmetic, using command "big", uses
cg                         up to 1001 decimal digits.  Command "big" may be
cg                         used for precise calculations with numbers with
cg                         digits to the right of the decimal point by first
cg                         multiplying them by a sufficiently large power of
cg                         ten, then moving the decimal point in the final
cg                         result of the calculation.
cg
cg                         Control the precision of GEOM results with
cg                         variable TOL.  See command "tol".
cg                         See "error estimate".

cg    preset             Many command and option words have preset synonyms.
cg                         These may be displayed with command "synonym".

cg    primality          See "primality proof".

cg    primality proof    One simple proof of the primality of a number is
cg                         as follows:  if (N - 1)! + 1) / N = M, where N and
cg                         M are integers, and (N-1)! indicates the factorial
cg                         function 1*2*3*4* ... * (N-2)*(N-1), then N is
cg                         prime.  If N is not prime, then it can be replaced
cg                         by K*L, and it easy to show that (N-1)! contains
cg                         as factors both K and L. Then (N-1)! / (K*L) is also
cg                         an integer J.  Substituting in the first equation,
cg                         J + 1/N = M, clearly impossible.  So N is prime.
cg                         This is impractical for large primes, because the
cg                         factorial increases so rapidly with the size of n.
cg
cg                         A more practical primality proof is as follows:
cg                         For N>1, if for every prime factor Q of N - 1, there
cg                         is an integer A such that A^(N - 1) = 1 (mod N) and
cg                         A^((N - 1)/Q) is not 1 (mod N), then N is prime.
cg                         See "totient function", commands "factor",
cg                         "variable VARNAME = K ^ L mod M".

cc    prime              An option in command "variable", to create variables
cc                         with the values of the first NUMPRIME prime numbers.

cg    prime              See "prime factors", "prime integers", "prime number".

cg    prime factors      The prime factors of an integer are the prime numbers
cg                         which divide the integer with no remainder.
cg                         An integer with no prime factors other than 1 and
cg                         itself is prime.
cg                         See "prime number", "factorial", "common divisor",
cg                         commands "factor", "prime".

cg    prime integers     To create and store variables with the values of
cg                         a specified number of prime integers, beginning with
cg                         1, see command "variable VAR(1) prime NUMPRIME".
cg
cg                         File ~/work/geom/test/prime.10K.mac contains commands
cg                         to define variables equal to the first 1230 prime
cg                         integers, from 1 to 9973, with comment lines allowing
cg                         the variables to be defined in groups.
cg                         File ~/work/math/prime.mac contains the first 664580
cg                         primes, from 1 to 9999991.
cg                         The first 100 prime integers are as follows:
cg                         1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
cg                         43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,
cg                         103, 107, 109, 113, 127, 131, 137, 139, 149, 151,
cg                         157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
cg                         211, 223, 227, 229, 233, 239, 241, 251, 257, 263,
cg                         269, 271, 277, 281, 283, 293, 307, 311, 313, 317,
cg                         331, 337, 347, 349, 353, 359, 367, 373, 379, 383,
cg                         389, 397, 401, 409, 419, 421, 431, 433, 439, 443,
cg                         449, 457, 461, 463, 467, 479, 487, 491, 499, 503,
cg                         509, 521, 523.

cg    prime number       A prime number is a positive integer that has no
cg                         factors other than 1 and itself.
cg                         See "primality proof", "composite number".
cg                         See  command "variable VAR(1) prime NUMPRIME".

cg    prime products     A function of prime numbers analogous to the factorial
cg                         function is the product of all prime numbers up to
cg                         the n'th prime number.  Its value is approximately
cg                         exp (n - sqrt (n)).  See file prime_products.mac
cg                         in directory <g/g17/edwards/work/geom/test> on YANA.

cn    prime.10K.mac      A macro file containing commands to define variables
cn                         equal to the first 1230 prime integers, from
cn                         1 to 9973.
cn                         In ~/work/geom/test .

cg    principal          See "principal axis", "principal radius".

cg    principal axis     A principal axis is one of the major axes defining the
cg                         coordinate system, e.g., the x, y and z axes in a
cg                         Cartesian or rectangular coordinate system.
cg
cg                         The principal axis transformation, applied to a
cg                         quadric surface, translates the center of symmetry of
cg                         the quadric surface to the origin, and rotates the
cg                         three orthogonal axes of the quadric surface to the
cg                         x, y and z axes, with the z axis being the primary
cg                         axis.  This transformation makes simple and parallel
cg                         planes perpendicular to the x axis.

cg    principal radius   At a point on a quadric surface, the principal radii of
cg                         curvature are those for which the inverse (the
cg                         curvature) is zero or extreme.
cg                         A straight line has zero curvature.  All curves
cg                         through any point on a plane are straight lines.
cg                         One curve through any point on a cylinder is a
cg                         straight line, parallel to the axis of the cylinder.
cg                         Two curves through a saddle point on a hyperbolic
cg                         paraboloid or a hyperboloid of one sheet are straight
cg                         lines.  See "radius of curvature".
cg                         See "ruled surface".  See command "arc".

cc    print              A synonym for list in command "list".

cg    probability        The probability of something is a measure of its
cg                         likelyhood of existing or occurring, and may be
cg                         differential or integral, relative or absolute.
cg                         A differential probability p(x)dx of a value x of a
cg                         random variable must be integrated over a range of
cg                         values of x to find the integral probability.
cg                         A relative (unnormalized) probability must be divided
cg                         by the integral or sum of all of the relative
cg                         probabilities of possible events/objects/values to
cg                         obtain the absolute (normalized) probability.
cg                         See "bin", "distribution", "pdf", "random", "sample".

cg    probability        See "probability bin", "probability dist".

cg    probability bin    A probability bin specifies the relative probability
cg                         of a discrete event or object, a discrete value of a
cg                         random variable or a range of values of a random
cg                         variable.  In the latter case, the probability may be
cg                         a uniform, linear, power-law, exponential, normal,
cg                         relativistic Maxwellian, Planck or Wien function of
cg                         the random variable value.  A discrete object may be
cg                         any of the objects used in GEOM, including a
cg                         probability distribution function (pdf).
cg
cg                         A set of probability bins may be combined to form a
cg                         probability distribution function (pdf) over a set of
cg                         discrete events, discrete values of a random
cg                         variable or a continuous or discontinuous set of
cg                         ranges of values of a random variable.
cg                         See commands "bin", "pdf", "sample".

cg    probability dist   See "distribution".

cg    problem            Problem refers to the current GEOM run.
cg                         See "problem title".

cg    problem            See "Chinese Remainder".

cc    problem title      Any desired name or identification of the user and/or
cc                         the problem, specified with command "title".

cg    problems           See "test problems".

cg    product            For the inner (dot) or outer (cross) product of two
cg                         vectors, use command "dot" or "cross".
cg                         See "inner product", "outer product".
cg
cg                         For the scalar or vector triple product of three
cg                         vectors, see "triple product".

cg    profile            See "outline".

cg    prohibited         See "names", "prohibited names".

cg    prohibited names   Some names are disallowed for all or some object names,
cg                         because they make interpretation of a command
cg                         ambiguous.
cg                         No name may be "+", "-", "all", "h", "help",
cg                         "list" or "thru", begin with "!" or contain ";".
cg                         A point name must not be "cluster" or a synonym.
cg                         A cluster name must not be "point" or a synonym.
cg                         A quadric surface name must not be "types".

cc    proj               A synonym for project.

cc    project            Command to project a point, a cluster of points or all
cc                         points onto a plane, circular cylinder or sphere,
cc                         with various options.
cc                         See "aliases", for suggestions for short commands.
cc                         See option "project" in commands "plane" and
cc                         "quadric".  See option "ortho" in command "quadric".
cc
cc                         To find the projected area of a general polygon, use
cc                         command "area".
cc
cc                         To return points to their original coordinates before
cc                         projection, use command "undo".
cc
cc                         Command "project" relates to objects:  axisym,
cc                         cluster, cylinder, plane, point, quadric, sphere,
cc                         symbol, variable, vector.
cc
cc                         Forms include:
cc
ccin                       help project
ccin                       project [help]
cc
cc                           Display the command options.
cc
ccin                       project point   PNAME  plane PLNAME
ccin                       project cluster CLNAME plane PLNAME
ccin                       project point   all    plane PLNAME
cc
cc                           Project point PNAME, cluster CLNAME or all points,
cc                           respectively, onto plane PLNAME, in the direction
cc                           of its normal vector.
cc
ccin                       project point   PNAME  plane PCEN VNORM
ccin                       project cluster CLNAME plane PCEN VNORM
ccin                       project point   all    plane PCEN VNORM
cc
cc                           Project point PNAME, cluster CLNAME or all points,
cc                           respectively, onto the plane through point PCEN, in
cc                           the direction of its normal vector VNORM.
cc
ccin                       project point   PNAME  pers PVIEW PFOC PTL PTR
ccin                       project cluster CLNAME pers PVIEW PFOC PTL PTR
ccin                       project point   all    pers PVIEW PFOC PTL PTR
cc
cc                           Create a perspective view, by projecting point
cc                           PNAME, cluster CLNAME or all points, respectively,
cc                           toward point PVIEW, onto a focal plane through
cc                           point PFOC and perpendicular to the line from point
cc                           PVIEW to point PFOC, then rotate and translate the
cc                           focal plane onto the plane z = 0, with point PFOC
cc                           at the origin, and with the line from point PTL to
cc                           point PTR parallel to the x axis, from left to
cc                           right.  Note:  points behind point PVIEW will not
cc                           be projected.
cc
ccin                       project point   PNAME  viewfactor PVIEW VNORM
ccin                       project cluster CLNAME viewfactor PVIEW VNORM
ccin                       project point   all    viewfactor PVIEW VNORM
cc
cc                           Create a geometric view factor view, by projecting
cc                           point PNAME, cluster CLNAME or all points onto a
cc                           plane through point PVIEW, with normal vector
cc                           VNORM, by first projecting them onto a unit sphere
cc                           centered at PVIEW, then projecting them
cc                           perpendicularly onto the plane, then translating
cc                           and rotating the projected points into the plane
cc                           z = 0, with the image of point PVIEW at the origin.
cc                           The ratio of any area on the projection to the area
cc                           of the unit circle, pi, is the differential
cc                           geometric view factor of the area, as seen from the
cc                           differential surface element at point PVIEW with
cc                           normal vector VNORM.
cc
ccin                       project point   PNAME  cylinder RCYL PCEN VAXIS [cyl]
ccin                       project cluster CLNAME cylinder RCYL PCEN VAXIS [cyl]
ccin                       project point   all    cylinder RCYL PCEN VAXIS [cyl]
cc
cc                           Project point PNAME, cluster CLNAME or all points
cc                           onto the surface of a circular cylinder, toward the
cc                           axis of the cylinder.  The cylinder has radius
cc                           RCYL, and an axis through point PCEN in the
cc                           direction of vector VAXIS.  Point PCEN will not be
cc                           projected.  Points on the axis can not be
cc                           projected.
cc
ccin                       project point   PNAME  cylinder RCYL PCEN VAXIS sph
ccin                       project cluster CLNAME cylinder RCYL PCEN VAXIS sph
ccin                       project point   all    cylinder RCYL PCEN VAXIS sph
cc
cc                           Project point PNAME, cluster CLNAME or all points
cc                           onto the surface of a circular cylinder, toward
cc                           point PCEN on its axis.  The cylinder has radius
cc                           RCYL, and an axis through point PCEN in the
cc                           direction of vector VAXIS.  Point PCEN will not be
cc                           projected.  Points on the axis can not be
cc                           projected.
cc
ccin                       project point   PNAME  cylinder RCYL PCEN VAXIS cos
ccin                       project cluster CLNAME cylinder RCYL PCEN VAXIS cos
ccin                       project point   all    cylinder RCYL PCEN VAXIS cos
cc
cc                           Project point PNAME, cluster CLNAME or all points
cc                           onto the surface of a circular cylinder.  The
cc                           cylinder has radius RCYL, and an axis through point
cc                           PCEN in the direction of vector VAXIS.  Make the
cc                           axial distance of each point from point PCEN equal
cc                           to the cosine of the original polar angle phi of
cc                           the point, as measured relative to point PCEN, in
cc                           the positive direction along the axis.  For points
cc                           initially on a sphere centered at point PCEN, this
cc                           projection preserves relative areas.  Point PCEN
cc                           will not be projected.  Points on the axis cannot
cc                           be projected.
cc
ccin                       project point   PNAME  sphere RSPH PCEN [sph]
ccin                       project cluster CLNAME sphere RSPH PCEN [sph]
ccin                       project point   all    sphere RSPH PCEN [sph]
cc
cc                           Project point PNAME, cluster CLNAME or all points,
cc                           respectively, onto the surface of the sphere of
cc                           radius RSPH, centered at point PCEN, projecting
cc                           toward the center point PCEN.
cc
ccin                       project point   PNAME  sphere RSPH PCEN cyl
ccin                       project cluster CLNAME sphere RSPH PCEN cyl
ccin                       project point   all    sphere RSPH PCEN cyl
cc
cc                           Project point PNAME, cluster CLNAME or all points,
cc                           respectively, onto the surface of the sphere of
cc                           radius RSPH, centered at point PCEN, projecting
cc                           toward the polar (z) axis, which passes through
cc                           point PCEN in the direction of vector VAXIS.  Point
cc                           PCEN will not be projected.  Points on the polar
cc                           (z) axis can not be projected.  Points further than
cc                           RSPH axially from PCEN will be put on a pole.
cc
ccin                       project point   PNAME  sphere RSPH PCEN polar
ccin                       project cluster CLNAME sphere RSPH PCEN polar
ccin                       project point   all    sphere RSPH PCEN polar
cc
cc                           Project point PNAME, cluster CLNAME or all points,
cc                           respectively, onto the surface of the sphere of
cc                           radius RSPH, centered at point PCEN, projecting in
cc                           the direction parallel to the polar (z) axis,
cc                           which passes through point PCEN in the direction of
cc                           vector VAXIS.  Point PCEN will not be projected.
cc                           Points in the equatorial plane can not be
cc                           projected.  Points further than RSPH from the polar
cc                           (z) axis will be put on the equator.
cc
cc                         Synonyms:  [cluster, cl], [cosine, cos],
cc                         [cylinder, cyl], [cylindrical, cyl], [help, h],
cc                         [perspective, pers], [plane, pl],
cc                         [point, p, pnt, pt], [project, proj], [sphere, sph],
cc                         [spherical, sph], [viewfactor, view].

cc    project            An option in command "plane", to find any unique
cc                         plane containing the curve on a specified quadric
cc                         surface where the normal vector is perpendicular to
cc                         a specified vector.  That curve is the outline of the
cc                         quadric surface as seen from an infinite distance in
cc                         the direction of the specified vector.
cc                         An option in command "quadric", to find the
cc                         cylindrical quadric surface parallel to a specified
cc                         vector and tangent to a specified quadric surface.

cg    projection         Points may be projected onto a plane, circular cylinder
cg                         or sphere, with various options.
cg
cg                         Onto a plane:
cg                           normally (parallel projection);
cg                           toward a view point (perspective projection);
cg                           radially onto a unit sphere centered in the plane,
cg                           then normally onto the plane (geometric view
cg                           factor projection).
cg                         Onto a circular cylinder:
cg                           perpendicular to the axis of the cylinder;
cg                           toward a point on the axis of the cylinder
cg                           (Mercator projection);
cg                           radially onto a unit sphere centered on the
cg                           axis of the cylinder, then onto the cylinder
cg                           perpendicularly to the axis of the cylinder
cg                           (preserves relative areas, as viewed from the
cg                           center of the unit sphere).
cg                         Onto a sphere:
cg                           toward the center of the sphere;
cg                           toward the polar (z) axis of the sphere
cg                           parallel to the polar (z) axis of the sphere.
cg
cg                         Vectors may be projected onto a plane.  See "vector".
cg
cg                         See "perspective", "project", "view factor",
cg                         "outline", command "quadric", option "ortho".
cg
cg                         The outline of a quadric surface may be projected
cg                         onto an x, y or z plane by setting the x, y or z
cg                         component of its normal vector to zero, solving for
cg                         x, y or z, and substituting the result into the
cg                         equation of the quadric surface, respectively.
cg
cg                         For example, given the equation f for the aligned
cg                         ellipsoid centered at point (A, B, C), with semiaxes
cg                         SX, SY and SZ in the x, y and z directions:
cg
cg                           f(x,y,z) = - 1 + ((x - A)/SX)^2 +
cg                                            ((y - B)/SY)^2 +
cg                                            ((z - C)/SZ)^2 = 0
cg
cg                         To project onto the z plane, set the z component of
cg                         the normal vector N = (NX, NY, NZ) to 0:
cg
cg                           NZ = -2 * C / SZ + (2 / SZ) * z = 0
cg
cg                         Solve for z (z = C), then substitute in f to get:
cg
cg                           f(x,y,C) = - 1 + ((x - A)/SX)^2 +
cg                                            ((y - B)/SY)^2 = 0
cg
cg                         which is an elliptic cylinder centered on the line
cg                         through point (A, B, z), for any z, parallel to the
cg                         z axis, with semiaxes SX and SY, as expected.
cg                         This example is particularly simple because the
cg                         ellipsoid is initially aligned with the axes.
cg
cg                         The quadric curve of the intersection of a plane and
cg                         any quadric surface may be projected onto an x, y or
cg                         z plane by solving the implicit quadric equation of
cg                         the plane for x, y or z, and substituting the result
cg                         into the implicit quadric equation of the quadric
cg                         surface, to obtain an equation independent of x, y
cg                         or z, respectively.  See command "distance", option
cg                         "plane".
cg
cg                         The outline of any quadric surface may be projected
cg                         onto any plane with command "quadric", option
cg                         "project".  The quadric surface through the original
cg                         outline may be found with command "quadric", option
cg                         "ortho".

cg    prolate            See "prolate spheroid".

cg    prolate spheroid   A prolate spheroid is a circular ellipsoid for which
cg                         the symmetry axis is along the larger semiaxis.
cg                         See "circular ellipsoid".
cg
cg                         The surface area of a prolate spheroid with equal
cg                         semiaxes SAX = SAY, longer semiaxis SAZ, and
cg                         eccentricity e = sqrt (1 - (SAX / SAZ)^2), is
cg                         area = 2 * pi * SAX^2 * (1 + (SAZ / SAX) * f(e)),
cg                         where f(e) = arcsin (e) / e.

cg    prompt             A prompt is a message to the user's terminal, asking
cg                         for more input, when input is from the user's
cg                         terminal.  A long prompt or a short prompt may be
cg                         specified by the user.  See command "prompt'.

cc    prompt             Command to change the input prompt to the short form
cc                         if long, and to the long form if short, and to
cc                         specify the character string (initially "?") for the
cc                         short form.
cc
cc                         Forms include:
cc
ccin                       help prompt
ccin                       prompt [help]
cc
cc                           Display the command options.
cc
ccin                       prompt
cc
cc                           Switch between the long and short prompts.
cc                           values.
cc
ccin                       prompt [=] STRING
cc
cc                           Replace the short prompt string (initially "?")
cc                           with the ASCII string STRING, with up to 8
cc                           characters, not beginning with "!", not containing
cc                           ";', and not an integer or floating point number.
cc                           Note:  use of a command word may cause confusion.
cc                           Suggestions for STRING:  ?  >  :  Input?  Input:
cc                           Command:  Command?  Next:  Next?
cc
cc                         Synonyms:  [help, h].

cg    proof              See "primality proof".

cc    prox               A synonym for proximal.

cg    proximal           A proximal point is a point nearest to another
cg                         geometric object, such as the origin, a point, a
cg                         line, a linear track, a parabolic trajectory,
cg                         a triangle, a plane or a quadric surface.
cg                         See "accelerate", "distance", "proximal", "side",
cg                         "track".
cg                         The opposite of proximal is distal.
cg
cg                         Given a general quadric surface with the equation
cg                         F(x,y,z) = 0, and normal vector N = (NX, NY, NZ),
cg                         where NX = dF/dx, NY = dF/dy, NZ = dF/dz (partial
cg                         derivatives), and given any point P1 = (X1, Y1, Z1),
cg                         any proximal point P2 = (X2, Y2, Z2) on the quadric
cg                         surface must satisfy the equations:
cg                         (X2 - X1) / NX = (Y2 - Y1) / NY = (Z2 - Z1) / NZ =
cg                         D / |N| = K, where N is evaluated at P2, D is the
cg                         distance from P1 to P2, and K is to be determined.
cg                         This may be much easier if the quadric surface is
cg                         first translated and rotated to its standard form
cg                         at the origin, along with point P1.  Each singular
cg                         case that arises when X1, Y1 or Z1 is zero must be
cg                         considered separately.
cg
cg                         Given a general plane with the equation:
cg                           QC + QX * x + QY * y + QZ * z = 0
cg                         The normal vector is N = (QX, QY, QZ).  A point in
cg                         the plane is P = (XP, YP, ZP) = -(QC / N^2) N.
cg                         The nearest point in the plane to a point
cg                         P1 = (X1, Y1, Z1) is the point (X2, Y2, Z2), where
cg                         X2 = X1 + K * QX, Y2 = Y1 + K * QY, Z2 = Z1 + K * QZ,
cg                         and K = N dot (P - P1) / N^2.
cg                         The distance is N dot (P - P1) / |N|.
cg
cg                         Given a circular cylinder at the origin, aligned with
cg                         the z axis, R^2 = x^2 + y^2 = A^2,
cg                         and a point (X1, Y1, Z1), the nearest point on the
cg                         cylinder is (X2, Y2, Z1):
cg                         X2 = X1 * A / R1, Y2 = Y1 * A / R1,
cg                         where R1 = sqrt (X1^2 + Y1^2).
cg                         The distance is R1 - R.
cg
cg                         Given a sphere at the origin,
cg                         R^2 = x^2 + y^2 + z^2 = A^2,
cg                         and a point (X1, Y1, Z1), the nearest point on the
cg                         sphere is (X2, Y2, Z2):
cg                         X2 = X1 * A / R1, Y2 = Y1 * A / R1, Z2 = Z1 * A / R1,
cg                         where R1 = sqrt (X1^2 + Y1^2 + Z1^2).
cg                         The distance is R1 - R.
cg
cg                         Given a cone at the origin, aligned with the z axis,
cg                         R^2 - A^2 * z^2 = (x^2 + y^2) - A^2 * z^2 = 0,
cg                         and a point (X1, Y1, Z1) = (R1, Z1), the nearest
cg                         point on the cone is (X2, Y2, Z2) = (R2, Z2):
cg                         X2 = X1 * R2 / R1, Y2 = Y1 * R2 / R1
cg                              (both indeterminate if R1 = 0), where
cg                         R2 = A * (K * Z1 + A * R1) / (1 + A^2)
cg                         Z2 = (Z1 + K * A * R1) / (1 + A^2)
cg                              (same sign as Z1),
cg                         at distance:
cg                         D  = abs (K * A * Z1 - R1) / sqrt (1 + A^2)
cg                         Where K = 1 if Z1 is positive, otherwise K = -1.
cg
cg                         Command "proximal" will return proximal points for a
cg                         quadric surface.
cg
cg                         To find proximal points on a pair of quadric surfaces
cg                         consisting of any combination of planes, spheres or
cg                         circular cylinders, use command "distance".
cg
cg                         To find any proximal points between a plane and a
cg                         general quadric surface, use command "distance".
cg
cg                         To find any proximal points between two quadric
cg                         surfaces, use command "quadric".
cg
cg                         To create a point PNAME with the coordinates of the
cg                         last proximal point found with command "accelerate",
cg                         "distance", "proximal", "side" or "track", use
cg                         command "point PNAME proximal".

cc    proximal           An option in command "debug", to display GEOM internal
cc                         variables for proximal points:  pdist, proxptx,
cc                         proxpty, proxptz, proxvx, proxvy, proxvz.
cc                         Synonyms:  [proximal, prox].

cc    proximal           Command to try to find the point on a quadric surface
cc                         nearest a specified point, if command "distance" or
cc                         "side" fails to do so, or to check that the result
cc                         from "distance" or "side" is not false.  The best
cc                         point, the distance and vector from the specified
cc                         point to the best point, and the angle between
cc                         that vector and the normal vector at the best point
cc                         are displayed.
cc
cc                         The minimum distance found may be stored in variable
cc                         VARNAME with command:
cc                         "variable VARNAME distance".
cc
cc                         The proximal point found on the quadric surface may
cc                         be stored in point PNAME with command
cc                         "point PNAME proximal".
cc
cc                         The vector between the point and the proximal point
cc                         may be stored in vector VNAME with command
cc                         "vector VNAME proximal".
cc
cc                         Command "proximal" relates to objects:  axisym, cone,
cc                         cylinder, ellipsoid, hyperb, plane, point, quadric,
cc                         sphere, symbol, variable.
cc
cc                         Forms include:
cc
ccin                       help proximal
ccin                       proximal [help]
cc
cc                           Display the command options.
cc
ccin                       proximal PNAME QNAME [NUMDIR|100]
cc
cc                           Find the point nearest point PNAME, on quadric
cc                           surface QNAME.  If a solution is not found by the
cc                           standard method, try certain special directions
cc                           from point PNAME, then try NUMDIR random
cc                           directions, then try 10 * NUMDIR directions
cc                           randomly sampled from a cosine-power distribution
cc                           peaked toward the last best direction, with a power
cc                           geometrically increasing from 1.0 to 10^10.  Accept
cc                           convergence as a relative change in distance less
cc                           than TOL, and the cosine of the angle between the
cc                           direction to the nearest point and the vector
cc                           normal to the surface at that point having an
cc                           absolute value within TOL of 1.0.
cc                           Check the proximal point with command "side", to
cc                           insure against truncation error that can occur when
cc                           point PNAME is very close to a symmetry plane or
cc                           center of quadric surface QNAME.
cc
cc                         Synonyms:  [help, h], [proximal, prox].

cc    proximal           An option in command "point", to create a point equal
cc                         to the last proximal point found with command
cc                         "distance", "proximal", "side" or "track".
cc                         Synonyms:  [proximal, prox].

cg    proximal           See "proximal distance", "proximal point".

cg    proximal distance
cg                       The proximal distance between two geometric objects is
cg                         the minimum distance separating the two objects.
cg                         If the objects are points, lines, quadric curves,
cg                         triangles, planes or quadric surfaces, the straight
cg                         line connecting the two proximal points will be
cg                         perpendicular to the geometric objects at the
cg                         proximal points.
cg                         Distances from a point to a quadric surface are
cg                         found with a relative precision of TOL or better,
cg                         with commands "distance" or "proximal".
cg
cg                         To find proximal distances of a pair of quadric
cg                         surfaces consisting of any combination of planes,
cg                         spheres or circular cylinders, use command
cg                         "distance".
cg
cg                         To find the proximal distance between a plane and a
cg                         general quadric surface, use command "distance".

cg    proximal point     For an external point P1 = (X1, Y1, Z1), the proximal
cg                         point P2 = (X2, Y2, Z2) (nearest to point P1) on a
cg                         quadric surface F(x,y,z) = 0, with normal vector
cg                         N = grad F = (NX, NY, NX), satisfies the equations:
cg
cg                         F(X2,Y2,Z2) = 0
cg
cg                         (X2 - X1) / NX(X2,Y2,Z2) =
cg                         (Y2 - Y1) / NY(X2,Y2,Z2) =
cg                         (Z2 - Z1) / NZ(X2,Y2,Z2) =
cg                         (P2 - P1) dot N / N^2 = distance / |N|,
cg
cg                         where N = N(X2,Y2,Z2).
cg
cg                         In general, this is a set of sixth order equations.
cg                         Commands "distance", "proximal" and "side" usually
cg                         find accurate solutions to these equations.  If the
cg                         standard methods fail, approximate iterative methods
cg                         are used, including a method of extrema, a method of
cg                         tangent planes, and a method of centers of curvature.
cg                         Such iterative methods may also fail, but if they
cg                         converge, they find a proximal distance with a
cg                         relative precision of TOL or better, and the
cg                         coordinates of the proximal point with a relative
cg                         precision of sqrt(TOL) or better.
cg
cg                         To create a point PNAME equal to the last proximal
cg                         point found with command "accelerate", "distance",
cg                         "proximal", "side" or "track", use the command
cg                         "point PNAME proximal".

cc    PRTOT              A relative total probability of a discrete event or
cc                         object or a discrete value VRAN of a random
cc                         variable, in command "bin".
cc                         Must not be negative.
cc                         See "DPR", "DPRL", "DPRR".
cc
cc                         For an exponential probability distribution function,
cc                         with relative differential probabilities PL at VRANL
cc                         and PR at VRANR, DVAL = VRANR - VRANL,
cc                         SIGMA = log (PL / PR) / DVAL,
cc                         and PRTOT = (PL - PR) / SIGMA.
cc
cc                         For a power-law probability distribution function,
cc                         with relative differential probabilities PL at VRANL
cc                         and PR at VRANR,
cc                         POW = log (PR / PL) / log (VRANR / VRANL),
cc                         PRTOT = (VRANR*PR - VRANL*PL) / (POW + 1).
cc                         or if POW = -1:
cc                         PRTOT = PL * VRANL * log (PL / PR).

cc    pt                 A synonym for point.

cc    PTL                In command "project", the name of a point at the left
cc                         end of a horizon line.

cc    PTR                In command "project", the name of a point at the
cc                         right end of a horizon line.

cc    PVER               The name of a point at the vertex of a regular polygon.
cc                         Used in command "polygon".
cc                         The name of a point at the vertex of a regular
cc                         polyhedron.  Used in command "polyhedron".

cc    PVIEW              In command "project", the name of a view point for
cc                         a perspective projection.

cg    pyramid            To create a zone partially bounded by three or more
cg                         planes intersecting at a common point, use command
cg                         "plane" with option "rotate" and command "zone".

Q-------------------------------------------------------------------------------

cc    q                  A synonym for quadric.

cc    Q1                 See "Q1, Q2, Q3, ...".

cc    Q1, Q2, Q3, ...    The names of the quadric surfaces (including planes)
cc                         that bound a zone.  Specified with command "zone".

cc    Q2                 See "Q1, Q2, Q3, ...".

cc    Q3                 See "Q1, Q2, Q3, ...".

cn    qalign.mac         A macro file to find the center, central vector,
cn                         rotation matrix and standard form of a quadric
cn                         surface.  In ~/work/geom/test .

cg    qaligned           In a data display, indicates a quadric surface for
cg                         which the principal axes are aligned with the x, y
cg                         and z coordinate axes.

cc    QC                 The constant term in the general implicit equation of a
cc                         quadric surface, F(x,y,z) = 0.  The value of F at the
cc                         origin.

cc    QMULT1             In command "quadric", option "sum", the weight of the
cc                         first quadric to be added.  See "QMULT2".

cc    QMULT2             In command "quadric", option "sum", the weight of the
cc                         second quadric to be added.  See "QMULT1".

cc    QNAME              The name of a quadric surface.  May have up to 24
cc                         characters, and may be ASCII, integer or floating
cc                         point.  No quadric name may be "+", "-", "all",
cc                         "h", "help", "list" or "thru", begin with "!" or
cc                         contain ";".  Specified with command "quadric".
cc
cc                         Also referred to as QNAME1, QNAME2, QNAME3, ...,
cc                         QNAME(1), QNAME(2), QNAME(3), ...
cc
cc                         May be the name of a plane, a sphere, a circular
cc                         cylinder, a circular cone, an ellipsoid, an axially
cc                         symmetric quadric surface, a hyperbolic paraboloid,
cc                         or any general quadric surface.  No two quadric
cc                         surfaces may have the same name, regardless of what
cc                         type of quadric surface they are.
cc
cc                         Quadric surfaces may appear in commands:
cc                         accelerate, axisym, cone, copy, cylinder, delete,
cc                         distance, ellipsoid, extrema, hyperb, invert, move,
cc                         name, operator, plane, point, proximal, quadric,
cc                         reflect, rename, rotate, scale, side, slice, sphere,
cc                         surface, track, vector.
cc                         Also see commands:  last, list, repack, search, sort.

cc    QNAME(1)           In command "quadric", the name of a base quadric
cc                         surface, used to create a family of spheres.

cg    qnotalig           In a data display, indicates a quadric surface for
cg                         which the principal axes are not all aligned with the
cg                         x, y and z coordinate axes.

cc    QTYPE              In a command, a type of quadric surface:  plane,
cc                         sphere, cylinder, cone, ellipsoid, hyperb, axisym or
cc                         quadric.
cc                         Note that cylinders and cones must be circular to be
cc                         specified this way.
cc
cc                         Synonyms:  [plane, pl], [sphere, sph],
cc                         [cylinder, cyl], [ellipsoid, el, ell, ellipse],
cc                         [quadric, q, quad].

cc    quad               A synonym for quadric.

cg    quadalig           In a data display, indicates a quadric surface for
cg                         which the principal axes are aligned with the x, y
cg                         and z coordinate axes.  OBSOLETE.

cg    quadgen            In a data display, indicates a quadric surface for
cg                         which the principal axes are not all aligned with the
cg                         x, y and z coordinate axes.  OBSOLETE.

cg    quadrangle         A quadrangle is a figure with 4 edges and 4 vertices,
cg                         not necessarily all in one plane.  Such a figure,
cg                         with vertices 1, 2, 3 and 4, in that order, may be
cg                         represented by two triangles, in one of two ways:
cg                         triangle 123 and triangle 341, or triangle 234 and
cg                         triangle 412.  If not coplanar, these two
cg                         representations are different surfaces and have
cg                         different areas.  To find the projected area of a
cg                         quadrangle, use command "area".
cg
cg                         A quadric surface (a hyperbolic paraboloid) may be
cg                         created that contains the four vertices, the four
cg                         edges and the center point of a nonplanar quadrangle.
cg                         See command "hyperb".
cg
cg                         The surface elements of a mesh are quadrangles,
cg                         and may have non-linear edges.

cg    quadratic          A quadratic polynomial equation in z = x + i*y, where
cg                         z is real or complex, x and y are real, and
cg                         i = sqrt (-1):
cg                           P(z) = a0 + a1*z + a2*z^2 = 0,
cg                         with real coefficients a0, a1 and a2 = 1, has
cg                         two roots r1 and r2, of which zero or two are
cg                         real, and has one real extremum (maximum or minimum).
cg                         The roots satisfy P(z) = 0.  The extremum satisfies
cg                         P'(z) = 0.
cg                         Some interesting relationships between the
cg                         coefficients and the properties of the polynomial
cg                         are as follows:
cg                          -a0     = r1*r2
cg                          -a1 / 2 = (1/2)*(r1 + r2) = e1
cg                           a2     = 1
cg
cg                         Note that the extreme point e1 is at the average z
cg                         value of the roots.
cg
cg                         The roots r1 and r2 are most accurately found as
cg                         follows:
cg                           cc = a1^2 - 4 * a0
cg                           q  = sqrt (abs (cc))
cg
cg                         If cc < 0:  r1 = -a1 / 2 - i * q / 2
cg                                     r2 = -a1 / 2 + i * q / 2
cg                         If cc = 0:  r1 = -a1 / 2
cg                                     r2 = -a1 / 2 = r1
cg                         If cc > 0, and a1 < 0:
cg                                     r1 = 2 * a0 / (q - a1)
cg                                     r2 = (q - a1) / 2
cg                         If cc > 0, and a1 > 0:
cg                                     r1 = -2 * a0 / (q + a1)
cg                                     r2 = -(q + a1) / 2
cg                         If cc > 0, and a1 = 0:
cg                                     r1 =  q / 2 = -r2
cg                                     r2 = -q / 2 = -r1
cg
cg                         A continued fraction solution for a root is
cg                           r1 = CF {-a1/a2, a1/a0, -a1/a2, a1/a0, ...}
cg                         which is a standard continued fraction if the
cg                         coefficients are positive integers.
cg
cg                         See command "roots".

cg    quadric            See "quadric curve", "quadric family",
cg                         "quadric surface", "quadric types", "projection",
cg                         "characteristics".

cc    quadric            An option in command "debug", to display GEOM internal
cc                         parameters and variables for quadric surfaces:
cc                         aqtypes, nquadm, nquads, aquad, laquad,
cc                         qc, qx, qy, qz, qxy, qyz, qzx, qxx, qyy, qzz,
cc                         aqtype, aqform, asym, xcen, ycen, zcen,
cc                         aquads, laquads, acones, lacones, acyls, lacyls,
cc                         aellips, laellips, ahyperbs, lahyperbs, aplanes,
cc                         laplanes, aspheres, laspheres, axisyms, laxisyms,
cc                         rotq.
cc                         Synonyms:  [quadric, q, quad].

cc    quadric            Command to display or create one or more quadric
cc                         surfaces.  Creating a quadric surface replaces any
cc                         existing quadric surface having the same name.
cc                         See "do loop use".
cc
cc                         NOTE:  see "quadric types", "limit cases".
cc
cc                         The long display for a quadric surface includes the
cc                         surface type, the center (see commands "point" and
cc                         "vector"), the coefficients, any major axis extrema,
cc                         the principal axis rotation operator (see command
cc                         "operator", the coefficients of the aligned standard
cc                         form, axis intercepts, foci, radii and half-angles,
cc                         any simple planes included in the quadric surface,
cc                         the invariants of the quadric surface equation,
cc                         and other principal characteristics, as applicable.
cc                         See "characteristics".
cc
cc                         Command "quadric" relates to objects:  axisym, cone,
cc                         cylinder, ellipsoid, hyperb, operator, plane, point,
cc                         quadric, sphere, symbol, variable, vector.
cc
cc                         Forms include:
cc
ccin                       help quadric
ccin                       quadric [help]
cc
cc                           Display the command options.
cc
ccin                       quadric types
cc
cc                           Display the quadric surface types and the standard
cc                           forms of their implicit quadric equations.
cc
ccin                       quadric [all,list]
cc
cc                           Display (short) all quadric surfaces.
cc
ccin                       quadric list QNAME1 QNAME2 QNAME3 ...
cc
cc                           Display quadrics QNAME1, QNAME2, QNAME3, ..., with
cc                           or without subscripts (short display).
cc
ccin                       quadric QNAME
cc
cc                           Display (long) quadric surface QNAME.
cc
ccin                       quadric QNAME [* /] FACT
cc
cc                           [Multiply Divide] the coefficients of the implicit
cc                           equation for quadric surface QNAME by FACT.
cc                           This command has no effect on planes, for which
cc                           the implicit equation is always normalized to make
cc                           the normal vector a unit vector.  For nonplanar
cc                           quadric surfaces, this command is useful for giving
cc                           one of the invariants of the implicit equation a
cc                           specified value.  See "invariants of quadric".
cc
cc                         In the following commands, use "+" or "-" instead of
cc                         QNAME or QNAME(2) to form the name by incrementing or
cc                         decrementing the base name for general quadric
cc                         surfaces.  See command  "last", "increment names".
cc
ccin                       quadric QNAME = QC QX QY QZ [QXY QYZ QZX QXX QYY QZZ]
cc
cc                           Create general quadric surface QNAME, with
cc                           coefficients in the general implicit quadric
cc                           surface equation QC, QX, QY, QZ [, QXY, QYZ, QZX,
cc                           QXX, QYY and QZZ].
cc                           If necessary, create variables with short names for
cc                           the coefficients, to fit then all on one line, or
cc                           use the following command for long coefficients.
cc
cc                         quadric QNAME & CFNAME1 CFVAL1 [CFNAME2 CFVAL2 ...]
cc
cc                           For existing quadric QNAME, replace coefficient
cc                           CFNAME1 with new value CFVAL1, where CFNAME1 may
cc                           be any of the coefficient names QC, QX, QY, QZ,
cc                           QXY, QYZ, QZX, QXX, QYY, QZZ.  Likewise for
cc                           CFNAME2, CFVAL2, ...
cc
ccin                       quadric QNAME fit PNAME1 PNAME2 ... PNAME9
cc
cc                           Create quadric surface QNAME to fit the nine unique
cc                           and distinct points PNAME1 PNAME2 ... PNAME9, if
cc                           any such unique quadric surface exists.
cc
ccin                       q QNAME sum QNAME1 QNAME2 [[QMULT1|1] [[QMULT2|1]]]
cc
cc                           Create quadric surface QNAME, the weighted sum
cc                           QMULT1 * QNAME1 + QMULT2 * QNAME2,
cc                           where QMULT1 and QMULT2 default to 1.
cc
ccin                       quadric QNAME plane PLNAME1 PLNAME2
cc
cc                           Create quadric surface QNAME, representing the two
cc                           coincident, parallel or intersecting planes PLNAME1
cc                           and PLNAME2.
cc
ccin                       quadric QNAME slice QNAME2 [x,y,z|x] [VAL|0]
cc
cc                           Create quadric surface QNAME, the intersection of
cc                           quadric surface QNAME2 with the plane (default x)
cc                           defined by [x, y, z] = VAL (default zero).
cc                           The quadric surface QNAME will be an infinite
cc                           cylinder perpendicular to the specified major
cc                           plane.  Also display any minimum and maximum points
cc                           and intersection points of the quadric curve in the
cc                           slice plane.  Note:  if the quadric curve is a pair
cc                           of hyperbolas, the minimum of one hyperbola will be
cc                           larger than the maximum of the other hyperbola.
cc
ccin                       quadric QNAME intersect PLNAME QNAME2
cc
cc                           Create cylindrical quadric surface QNAME,
cc                           perpendicular to plane PLNAME, and through the
cc                           intersection of plane PLNAME and quadric surface
cc                           QNAME2.
cc
ccin                       quadric QNAME project VNAME QNAME2
cc
cc                           Create cylindrical quadric surface QNAME, parallel
cc                           to vector VNAME, and tangent to quadric surface
cc                           QNAME.  This is the locus of all projections of
cc                           quadric surface QNAME onto planes perpendicular to
cc                           vector VNAME.
cc                           See command "plane", option "project".
cc                           See command "quadric", option "ortho".
cc
ccin                       quadric QNAME ortho QNAME1 QNAME2
cc
cc                           Create quadric surface QNAME, the locus of all
cc                           points at which the two vector fields represented
cc                           by quadric surfaces QNAME1 and QNAME2 are
cc                           orthogonal.  If QNAME1 is a plane, cylinder or
cc                           sphere, respectively, than QNAME passes through the
cc                           outline of QNAME2 projected perpendicularly toward
cc                           the plane, the axis of the cylinder or the center
cc                           of the sphere, respectively.  Any members of the
cc                           families of quadric surfaces QNAME1 and QNAME2 are
cc                           orthogonal to each other on QNAME.
cc                           See "perspective", command "project".
cc
ccin                       quadric QNAME(2) move NUMQ INC QNAME(1) VMOVE
cc
cc                           Create a family of NUMQ quadric surfaces QNAME(2),
cc                           ..., with names incremented by INC characters,
cc                           spaced at intervals of vector VMOVE, starting from
cc                           quadric surface QNAME(1).  See command "plane",
cc                           option "parallel".  See "increment names".
cc
cc                           This command can be replaced by the following
cc                           commands, using subscripted names and a do loop:
cc
ccin                         copy quadric QNAME(N1) QNAMETEMP
ccin                         variable NQ = N1
ccin                         do NL 2 NUMQ
ccin                           variable NQ = NQ + INC
ccin                           move quadric QNAMETEMP VMOVE
ccin                           copy quadric QNAMETEMP QNAME(NQ)
ccin                         enddo
cc
ccin                       quadric QNAME(2) rotate NUMQ INC QNAME(1) OPNAME PINV
cc
cc                           Create a family of NUMQ quadric surfaces QNAME(2),
cc                           ..., with names incremented by INC characters, by
cc                           rotating the preceding quadric surface with
cc                           operator OPNAME and invariant point PINV, starting
cc                           from quadric surface QNAME(1).
cc                           See "increment names".
cc
cc                           This command can be replaced by the following
cc                           commands, using subscripted names and a do loop:
cc
ccin                         copy quadric QNAME(N1) QNAMETEMP
ccin                         variable NQ = N1
ccin                         do NL 2 NUMQ
ccin                           variable NQ = NQ + INC
ccin                           rotate quadric QNAMETEMP OPNAME PINV
ccin                           copy quadric QNAMETEMP QNAME(NQ)
ccin                         enddo
cc
ccin                       quadric QNAME(2) scale NUMQ INC QNAME(1) OPNAME PINV
cc
cc                           Create a family of NUMQ quadric surfaces QNAME(2),
cc                           ..., with names incremented by INC characters, by
cc                           scaling the preceding quadric surface with operator
cc                           OPNAME and invariant point PINV, starting from
cc                           quadric surface QNAME(1).  See "increment names".
cc
cc                           This command can be replaced by the following
cc                           commands, using subscripted names and a do loop:
cc
ccin                         copy quadric QNAME(N1) QNAMETEMP
ccin                         variable NQ = N1
ccin                         do NL 2 NUMQ
ccin                           variable NQ = NQ + INC
ccin                           scale quadric QNAMETEMP VMOVE
ccin                           copy quadric QNAMETEMP QNAME(NQ)
ccin                         enddo
cc
cc                         Synonyms:  [help, h], [intersect, int, inter],
cc                         [move, mv, trans, translate],
cc                         [quadric, q, quad], [rotate, rot], [slice, cut],
cc                         [sum, add], [x, X]. [y, Y], [z, Z].
cc
cc                         See commands:  plane, sphere, ellipsoid,
cc                         cylinder, cone, axisym, slice.

cc    quadric            An option in commands "copy", "delete", "last", "list",
cc                         "rename", "repack" and "sort", to perform the
cc                         specified operation on one or more quadric surfaces.
cc                         Synonyms:  [quadric, q, quad].

cg    quadric            See "quadric commands", "quadric curve",
cg                         "quadric families", "quadric family",
cg                         "quadric surface", "quadric types",
cg                         "quadric properties".

cg    quadric arrays     Quadrics may be created as arrays with subscripted
cg                         names.  See "subscript", "subscripted names",
cg                         commands "do" and "enddo".

cg    quadric commands   The command and key words used for quadric surfaces are
cg                         as follows:
cg
cg                         Command
cg                         or option   Name      Description
cg
cg                         axisym      AXNAME    Axisymmetric quadric surface
cg                         cone        CNAME     Circular cone (type 12)
cg                         cylinder    CYLNAME   Circular cylinder (type 7)
cg                         ellipsoid   ELNAME    Ellipsoid (type 15)
cg                         hyperb      HPNAME    Hyperbolic paraboloid (type 8)
cg                         plane       PLNAME    Simple plane (type 0)
cg                         quadric     QNAME     General quadric surface
cg                         sphere      SPHNAME   Sphere (type 16)
cg
cg                         See "quadric surfaces", "quadric types".

cg    quadric curve      A quadric curve is a planar curve representing the
cg                         intersection of a plane with a quadric surface.
cg                         The types of quadric curves, and their standard
cg                         forms (centered at the origin, and aligned with the
cg                         x and y axes), are as follows (each coefficient must
cg                         have the preceding sign):
cg
cg                         Real quadric curves:
cg                           Simple straight line:                        x = 0
cg                           Coincident straight lines:                 x^2 = 0
cg                           Real parallel lines:             - 1 + QXX*x^2 = 0
cg                           Real intersecting lines:       x^2 - |QYY|*y^2 = 0
cg                           Parabola:                       - |QY|*y + x^2 = 0
cg                           Hyperbola (2 curves):    1+QXX*x^2 - |QYY|*y^2 = 0
cg                           Real ellipse:          - 1 + QXX*x^2 + QYY*y^2 = 0
cg                           Real circle:           - 1 + QXX * (x^2 + y^2) = 0
cg
cg                         Imaginary and degenerate quadric curves:
cg                           Imaginary parallel lines:          1 + QXX*x^2 = 0
cg                           Imaginary intersecting lines:
cg                             (a point)                      x^2 + QYY*y^2 = 0
cg                           Imaginary ellipse:       1 + QXX*x^2 + QYY*y^2 = 0
cg                           Imaginary circle:        1 + QXX * (x^2 + y^2) = 0
cg
cg                         Quadric curves in standard form have the following
cg                         defining properties, derivable from their implicit
cg                         equations:
cg
cg                         Line:  a point on the line and a direction vector.
cg                         Parabola:  a center and a focus.
cg                         Hyperbola:  a center, a half-angle, and two axis
cg                         intercepts.
cg                         Ellipse:  a center, two foci and two semi-axes.
cg                         Circle:  a center and a radius.
cg
cg                         A quadric curve may also represent a cylindrical
cg                         quadric surface with infinite extent in the direction
cg                         perpendicular to the plane of the quadric curve.
cg
cg                         To find the quadric curve resulting from slicing a
cg                         quadric surface with a major plane, use command
cg                         "slice".
cg
cg                         To find the quadric curve resulting from slicing a
cg                         quadric surface with any plane, use command
cg                         "intersection" or "distance".

cg    quadric families   A family of quadric surfaces may be created with
cg                         command "quadric" with option "move", "rotate" or
cg                         "scale" or as follows (note use of synonyms):
cg
cg                         p pinv (options)
cg
cg                           Create an invariant point, if needed.
cg
cg                         quadric QNAME(1) (options)
cg
cg                           Create a base quadric surface.
cg
cg                         op opr (options)
cg
cg                           Create a tensor operator, to reflect, rotate,
cg                           invert or scale, if needed.
cg
cg                         v vmove (options)
cg
cg                           Create a vector for use as a translation operator,
cg                           if needed.
cg
cg                         cp QNAME(1) QNAME(2)
cg
cg                           Copy quadric surface QNAME(1) to the first quadric
cg                           surface.
cg
cg                         Repeat the following block of commands as many times
cg                         as needed to create the rest of the family of quadric
cg                         surfaces.  See "shortcuts".
cg
cg                         mv q QNAME(1) vmove
cg
cg                           Move the base quadric surface by amount vmove,
cg                           if needed.
cg
cg                         [invert,reflect,rotate,scale] q QNAME(1) opr [pinv]
cg
cg                           Invert, reflect, rotate or scale the base quadric
cg                           surface with tensor operator opr, if needed.
cg
cg                         cp q QNAME(1) +
cg
cg                           Copy the base quadric surface to the next quadric
cg                           surface.

cg    quadric family     A quadric surface is a member of a quadric surface
cg                         family, with implicit equations differing only in
cg                         the value of the constant term QC.
cg                         All members of the family have the same 3-D vector
cg                         field of normal vectors, given by the gradient of the
cg                         implicit equation.  See "family", "quadric surface".

cg    quadric properties
cg                       Certain properties of quadric surfaces may be used to
cg                         classify them.
cg
cg                         Define the matrices M1 and E:
cg
cg                              |QXX   QXY/2 QZX/2|
cg                         M1 = |QXY/2 QYY   QYZ/2|
cg                              |QZX/2 QYZ/2 QZZ  |,
cg
cg                              |QXX   QXY/2 QZX/2  QX/2|
cg                              |QXY/2 QYY   QYZ/2  QY/2|
cg                         M2 = |QZX/2 QYZ/2 QZZ    QZ/2|
cg                              |QX/2  QY/2  QZ/2   QC  |,
cg
cg                         r3 = rank M1,
cg                         r4 = rank M2,
cg                         D  = det  M2,
cg                         S  = sign (D)
cg
cg                         and k1, k2 and k3 are the roots of
cg
cg                             |QXX - x   QXY/2     QZX/2  |
cg                             |QXY/2     QYY - x   QYZ/2  | = 0.
cg                             |QZX/2     QYZ/2     QZZ - x|
cg
cg                         Also,
cg
cg                         K = 1 if the signs of nonzero k's are the same,
cg                         K = 0 if the signs of nonzero k's are different.
cg
cg                         The properties of the 24 type of quadric surfaces
cg                         listed under "quadric types" are as follows:
cg
cg                         Type                               r3  r4   S     K
cg                         ---------------------------------------------------
cg                          0 Simple plane                    0    0
cg                          1 Coincident planes               1    1
cg                          2 Real parallel planes            1    2
cg                          3 Real intersecting planes        2    2         0
cg                          4 Parabolic cylinder              1    3
cg                          5 Hyperbolic cylinder             2    3         0
cg                          6 Real elliptic cylinder          2    3         1
cg                          7 Real circular cylinder          2    3         1
cg                          8 Hyperbolic paraboloid           2    4    +    0
cg                          9 Elliptic paraboloid             2    4    -    1
cg                         10 Circular paraboloid             2    4    -    1
cg                         11 Real elliptic cone              3    3         0
cg                         12 Real circular cone              3    3         0
cg                         13 Hyperboloid of one sheet        3    4    +    0
cg                         14 Hyperboloid of two sheets       3    4    -    0
cg                         15 Real ellipsoid                  3    4    -    1
cg                         16 Real sphere                     3    4    -    1
cg                         17 Imaginary parallel planes       1    2
cg                         18 Imaginary intersecting planes   2    2         1
cg                         19 Imaginary elliptic cylinder     2    3         1
cg                         20 Imaginary circular cylinder     2    3         1
cg                         21 Imaginary elliptic cone         3    3         1
cg                         22 Imaginary circular cone         3    3         1
cg                         23 Imaginary ellipsoid             3    4    +    1
cg                         24 Imaginary sphere
cg
cg                         See "invariants of quadric".

cg    quadric surface    A quadric surface is a surface in 3-D space described
cg                         by an implicit equation in (x, y, z) coordinates,
cg                         of the form:
cg                           F(x,y,z) =    QC          +
cg                           QX  * x     + QY  * y     + QZ  * z     +
cg                           QXY * x * y + QYZ * y * z + QZX * z * x +
cg                           QXX * x^2   + QYY * y^2   + QZZ * z^2  = 0
cg
cg                         At any point (x, y, z) in the surface, the surface
cg                         has the normal vector N = (NX, NY, NZ), where:
cg                           NX = QX + 2 * QXX * x +     QXY * y +     QZX * z
cg                           NY = QY +     QXY * x + 2 * QYY * y +     QYZ * z
cg                           NZ = QZ +     QZX * x +     QYZ * y + 2 * QZZ * z
cg
cg                         At any point (x. y, z) in space, the vector N is
cg                         normal to one of a family of quadric surfaces having
cg                         implicit equations differing only in the value of the
cg                         constant term QC.  N(x,y,z) is a linear vector field.
cg                         See command "vector", option "normal".
cg
cg                         The positive side of a quadric surface is in the
cg                         direction of the normal vector.
cg                         See "invariants of quadric".
cg
cg                         The data displayed for a quadric surface includes;
cg                         its name, type, coefficients, and its central point;
cg                         any extreme points (minima, maxima, saddle points);
cg                         its standard form, when moved to the origin and
cg                         rotated to be aligned with the x, y and z axes;
cg                         its intercepts on the major axes in both the original
cg                         and standard form; its foci, in both the original and
cg                         standard form, if one of the types:
cg                         parabolic cylinder, hyperbolic paraboloid,
cg                         elliptic paraboloid or circular paraboloid;
cg                         its half-angle, if one of the types: intersecting
cg