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.
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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".
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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 planes, hyperbolic cylinder, hyperbolic paraboloid,
cg cone or hyperboloid of one or two sheets.
cg
cg The quadric surface may be specified in many ways.
cg NOTE! A quadric surface is NOT dependent on the
cg subsequent movement of any point(s) used to create
cg it!
cg
cg The maximum number of quadric sufaces is now 1000.
cg
cg Quadric surfaces may be deleted, copied or renamed.
cg
cg If the name of a quadric surface is changed, the
cg change also applies to any zones bounded by the
cg quadric surface.
cg
cg In a command the word "quadric" includes planes,
cg spheres, circular cylinders, circular cones,
cg hyperbolic paraboloids, ellipsoids, axially symmetric
cg quadric surfaces, and all other types of quadric
cg surfaces.
cg
cg Quadric surfaces may be translated, rotated, scaled,
cg or inverted. See "center", "align".
cg
cg To find the distance along a specified track from a
cg point to a quadric surface, use command "track".
cg
cg To find which side of a quadric surface a point is
cg on, and the distance from the point to the surface,
cg use command "distance", "proximal" or "side".
cg
cg To find the intersection of a quadric surface with a
cg major plane, and to find points on the resulting
cg quadric curve in that plane, or to find x given y and
cg z, or y given x and z, or z given x and y, use
cg command "slice".
cg
cg To find the distance or intersection of two quadric
cg surfaces consisting of any combination of plane,
cg sphere or circular cylinder, see command "distance".
cg
cg To find the distance or intersection of any quadric
cg surface with a line, point or plane, see command
cg distance.
cg
cg To find any extreme points of a quadric surface in a
cg specified direction, use command "extrema".
cg
cg Quadric surfaces may be displayed and specified by
cg commands: plane, sphere, cylinder, cone,
cg ellipsoid, hyperb, axisym, quadric.
cg
cg The following commands relate to quadric
cg surfaces:
cg accelerate, arc, axisym, cone, copy, cylinder,
cg debug, delete, distance, ellipsoid, extrema, help,
cg hyperb, invert, last, list, move, plane, point,
cg project, proximal, quadric, reflect, rename, repack,
cg search, scale, side, slice, sort, sphere, symbol,
cg synonym, tables, track, triple, vector, zone.
cg
cg See "family", "quadric types", "radius of curvature".
cg quadric types Types of quadric surfaces include planes, spheres,
cg circular cylinders, circular cones, hyperbolic
cg paraboloids, ellipsoids, axially symmetric quadric
cg surfaces and general quadric surfaces. Axially
cg symmetric quadric surfaces include planes, spheres,
cg circular cylinders, circular cones, circular
cg paraboloids, and circular hyperboloids.
cg quadric types The types of quadric surfaces, and their standard
cg forms (centered at the origin, and aligned with the
cg x, y and z axes), are as follows (each coefficient
cg must have the preceding sign):
cg
cg Real quadric surface types:
cg
cg 0 Simple plane: x = 0
cg 1 Coincident planes: x^2 = 0
cg 2 Real parallel planes: - 1 + QXX*x^2 = 0
cg 3 Real intersecting planes: x^2 - |QYY|*y^2 = 0
cg 4 Parabolic cylinder: - |QY|*y + x^2 = 0
cg 5 Hyperbolic cylinder:
cg 1 + QXX*x^2 - |QYY|*y^2 = 0
cg 6 Real elliptic cylinder:
cg - 1 + QXX*x^2 + QYY*y^2 = 0
cg 7 Real circular cylinder:
cg - 1 + QXX * (x^2 + y^2) = 0
cg 8 Hyperbolic paraboloid:
cg - |QZ|*z + x^2 - |QYY|*y^2 = 0
cg 9 Elliptic paraboloid:
cg - |QZ|*z + x^2 + QYY*y^2 = 0
cg 10 Circular paraboloid:
cg - |QZ|*z + x^2 + y^2 = 0
cg 11 Real elliptic cone:
cg x^2 + QYY*y^2 - |QZZ|*z^2 = 0
cg 12 Real circular cone: x^2 + y^2 - |QZZ|*z^2 = 0
cg 13 Hyperboloid of 1 sheet (elliptic or circular):
cg - 1 + QXX*x^2 + QYY*y^2 - |QZZ|*z^2 = 0
cg 14 Hyperboloid of 2 sheets (elliptic or circular):
cg 1 + QXX*x^2 + QYY*y^2 - |QZZ|*z^2 = 0
cg 15 Real ellipsoid (general or circular):
cg - 1 + QXX*x^2 + QYY*y^2 + QZZ*z^2 = 0
cg 16 Real sphere:
cg - 1 + QXX * (x^2 + y^2 + z^2) = 0
cg
cg Imaginary and degenerate quadric surface types:
cg
cg 17 Imaginary parallel planes: 1 + QXX*x^2 = 0
cg 18 Imaginary intersecting planes:
cg (a straight line) x^2 + QYY*y^2 = 0
cg 19 Imaginary elliptic cylinder:
cg 1 + QXX*x^2 + QYY*y^2 = 0
cg 20 Imaginary circular cylinder:
cg 1 + QXX * (x^2 + y^2) = 0
cg 21 Imaginary elliptic cone:
cg (a point) x^2 + QYY*y^2 + QZZ*z^2 = 0
cg 22 Imaginary circular cone:
cg (a point) x^2 + y^2 + QZZ*z^2 = 0
cg 23 Imaginary ellipsoid:
cg 1 + QXX*x^2 + QYY*y^2 + QZZ*z^2 = 0
cg 24 Imaginary sphere:
cg 1 + QXX * (x^2 + y^2 + z^2) = 0
cg
cg WARNING: if one or more coefficients are extremely
cg small or large compared with others or almost equal
cg to others, the type of quadric surface may be
cg numerically indeterminate. See "limit cases".
cg
cg See "plane", "sphere", "cylinder", "cone",
cg "ellipsoid", "axisymmetric", "quadric properties".
cg quadrilateral See "quadrangle".
cg quadrpl In output from command "roots", indicates a root of
cg order 4 of a polynomial equation or order 4.
cg I.e., the equation is an expansion of (x - r)^4.
cg quartic A quartic 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 + a3*z^3 + a4*z^4 = 0,
cg with real coefficients a0, a1, a2, a3 and a4 = 1,
cg has four real/and or complex roots r1, r2, r3, r4,
cg has three extrema (maxima and minima) e1, e2, e3, of
cg which either one or three are real, has two inversion
cg points v1 and v2, of which either zero or two are
cg real, and one real symmetry point s1.
cg The roots satisfy P(z) = 0. The extrema satisfy
cg P'(z) = 0. The inversion points satisfy P''(z) = 0.
cg The symmetry point satisfies P'''(z) = 0.
cg P', P'', P''' are the first, second and third
cg derivatives, respectively, of 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*r4
cg -a1 / 4 = (1/4)*(r1*r2*r3 + r2*r3*r4 +
cg r3*r4*r1 + r4*r1*r2)
cg = e1*e2*e3
cg a2 / 6 = (1/6)*(r1*r2 + r2*r3 + r3*r4 +
cg r4*r1 + r1*r3 + r2*r4)
cg = (1/3)*(e1*e2 + e2*e3 + e3*e1)
cg = v1*v2
cg -a3 / 4 = (1/4)*(r1 + r2 + r3 + r4)
cg = (1/3)*(e1 + e2 + e3)
cg = (1/2)*(v1 + v2)
cg = s1
cg a4 = 1
cg
cg Note that the symmetry point s1 is at the average z
cg value of the roots, the extrema, and the inversion
cg points, and that the average value of the products of
cg pairs of roots, extrema and inversion points are the
cg same, as are the average value of the products of
cg triplets of roots and extrema.
cg
cg If all of the roots are very close together, it may
cg increase accuracy to shift the origin to a point z0
cg near the roots. For u = z - z0,
cg P(u+z0) = b0 + b1*u + b2*u^2 + b3*u^3 + b4*u^4
cg where b0 = P(z0), b1 = P'(z0), b2 = P''(z0) / 2,
cg b3 = P'''(z0) / 6 = a3 + 4*a4*z0,
cg b4 = P''''(z0) / 24 = a4.
cg
cg See command "roots".
cg question mark The question mark, "?".
cc quit A synonym for end.
cg quote mark The single (') or double (") quote mark.
cc quoted string An input argument bracketed by single or double quotes.
cc See commands "alias", "symbol", "marker",
cc "variable", option "tag", "big", option "tag".
cc QX The coefficient of x in the general implicit equation
cc of a quadric surface, F(x,y,z) = 0.
cc of a quadric surface, F(x,y,z) = 0.
cc QXX The coefficient of x^2 in the general implicit
cc equation of a quadric surface, F(x,y,z) = 0.
cc QXY The coefficient of x*y in the general implicit equation
cc QY The coefficient of y in the general implicit equation
cc of a quadric surface, F(x,y,z) = 0.
cc QYY The coefficient of y^2 in the general implicit
cc equation of a quadric surface, F(x,y,z) = 0.
cc QYZ The coefficient of y*z in the general implicit equation
cc of a quadric surface, F(x,y,z) = 0.
cc QZ The coefficient of z in the general implicit equation
cc of a quadric surface, F(x,y,z) = 0.
cc QZX The coefficient of z*x in the general implicit equation
cc of a quadric surface, F(x,y,z) = 0.
cc QZZ The coefficient of z^2 in the general implicit
cc equation of a quadric surface, F(x,y,z) = 0.
R-------------------------------------------------------------------------------
cc r A synonym for input.
cc R1 See "R1, R2, R3, ...".
cc R1, R2, R3, ... In command "redo", replacements for arguments 1, 2, 3,
cc ..., respectively, of the command being redone.
cc A "." means no replacement, use original argument.
cc The replacement arguments may be ASCII, integer or
cc floating point, and do not have to be the same mode
cc as the arguments being replaced, which may also be
cc ASCII, integer or floating point.
cc R2 See "R1, R2, R3, ...".
cc R3 See "R1, R2, R3, ...".
cc rad A synonym for radians.
cc rad/deg An option in command "symbol", argument SYMBNAME,
cc to create symbol rad/deg with value pi / 180 =
cc 0.01745329251994329576923690768488612713442871888541
cg rad/deg Symbol, radians per degree. Generated as a
cg symbol at the beginning of the run.
cg rad/deg = pi / 180.
cg rad/deg = 0.01745329251994330 radians per degree.
cg deg/rad = 57.29577951308232 degrees per radian.
cg pi = 3.141592653589793... = acos (-1).
cg
cg symbol SYMBNAME rad/deg
cg
cg Set symbol SYMBNAME to
cg pi / 180 = 0.01745329251994330.
cg
cg symbol SYMBNAME deg/rad
cg
cg Set symbol SYMBNAME to
cg 180 / pi = 57.29577951308232.
cc RAD1 See "RAD1, RAD2", "RAD1, RAD2, RAD3", "RAD1-RAD4".
cc RAD1, RAD2 In commands "sphere" and "cylinder", the inner and
cc outer radii of a family of concentric surfaces.
cc RAD1, RAD2, RAD3 In command "kiss", the radii of three mutually tangent
cc circles. A radius with a negative sign means that
cc circle is outside of the other two circles. A radius
cc with a value much larger than the other two
cc approximates a straight line.
cc RAD1-RAD4 In command "kiss", the radii of four mutually tangent
cc spheres. A radius with a negative sign means that
cc sphere is outside of the other spheres. A radius
cc with a value much larger than the other two
cc approximates a plane.
cc RAD2 See "RAD1, RAD2", "RAD1, RAD2, RAD3", "RAD1-RAD4".
cc RAD3 See "RAD1, RAD2, RAD3", "RAD1-RAD4".
cc RAD4 See "RAD1, RAD2, RAD3", "RAD1-RAD4".
cc radial An option in command "operator", to create a tensor
cc operator that performs a radial scaling relative to a
cc specified axis in 3-D space, by factor RATIO, with
cc the axis passing through the origin. All distances
cc from the axis will be multiplied by RATIO.
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 "linear", "scale", "uniform".
cc radial An option in command "twist", to specify a rotation of
cc one or more points around an axis in proportion to
cc the radial distance of each point from a fixed radial
cc distance from the axis.
cg radial See "radial scaling".
cg radial scaling To scale in a direction perpendicular to a specified
cg axis, use command "operator", option "linear" or
cg command "scale".
cc radian A synonym for radians.
cc radians An option in command "angles", to specify that angles
cc are to be input and displayed in radians.
cg radians 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 RADIUS Radius of a circle, circular cylinder or sphere.
cg radius Radius of a circle, circular cylinder or sphere.
cg See "radius of curvature".
cg radius of curvature
cg The radius of curvature rcurve, of a 2-D curve in the
cg x-y plane, y = f(x) is given by:
cg rcurve = (1 + (dy/dx)^2)^3/2) / d(dy/dx)/dx or
cg rcurve = (1 + (y')^2)^3/2 / y'',
cg where y' and y'' are the first and second derivatives
cg of y with respect to x.
cg See "principal radius".
cg
cg Given a quadric surface Q in 3-D space, defined by:
cg QC + 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 and a point P = (x, y, z) on the quadric surface, at
cg which the vector N normal to the quadric surface is:
cg N = (NX, NY, NZ)
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 and an arbitrary unit vector U perpendicular to N
cg (tangent to the quadric surface):
cg U = (UX, UY, UZ), UX^2 + UY^2 + UZ^2 = 1,
cg UX*NX + UY*NY + UZ*NZ = 0,
cg the radius of curvature rcurve of quadric surface Q
cg in the direction of U is given by:
cg rcurve = - |N| / (dN/ds dot U)
cg where s is the distance in direction U:
cg dP = s*dU,
cg and:
cg |N| = sqrt (NX^2 + NY^2 + NZ^2)
cg dN/ds = (dNX/ds, dNY/ds, dNZ/ds)
cg dNX/ds = 2*QXX*UX + QXY*UY + QZX*UZ
cg dNY/ds = QXY*UX + 2*QYY*UY + QYZ*UZ
cg dNZ/ds = QZX*UX + QYZ*UY + 2*QZZ*UZ
cg dN/ds dot U =
cg 2*QXX*UX^2 + QXY*UX*UY + QZX*UZ*UX
cg + QXY*UX*UY + 2*QYY*UY^2 + QYZ*UY*UZ
cg + QZX*UZ*UX + QYZ*UY*UZ + 2*QZZ*UZ^2
cg
cg The principal radii of curvature at point P are those
cg for which the inverse (the curvature) is zero or
cg extreme.
cg
cg The center of curvature for a given rcurve is at
cg the point C = (cx, cy, cz) obtained by moving a
cg distance rcurve along the normal vector N from point
cg P (I do not know yet how to find the sign of the
cg direction):
cg C = P + N / (dN/ds dot U) or
cg C = P - N / (dN/ds dot U).
cg
cg In general, at any point P on the quadric surface,
cg there are minimum and maximum values of dN/ds dot U.
cg To find these values, rotate the quadric surface to
cg make the normal vector N at point P point in the z
cg direction, then set vector U = (cosa, sina, 0), where
cg cosa = cos (theta), sina = sin (theta), theta is an
cg arbitrary angle around the z axis. This results in
cg dN/ds dot U =
cg 2*(QXX*cosa^2 + QXY*cosa*sina + QYY*sina^2).
cg Set the derivative of this equation (with respect to
cg theta) to zero and solve:
cg (QYY - QXX) * cos(2*theta) + QXY * sin(2*theta) = 0
cg The two primary values of theta that satisfy this
cg equation, and the two corresponding values of
cg dN/ds dot U, provide the minimum and maximum values
cg of dN/ds dot U at point P.
cg If dN/ds dot U is zero at any value of theta, the
cg radius of curvature is infinite there. This will
cg occur if there are real solutions to the equation
cg QXX + QXY * tan (theta) + QYY * tan (theta)^2 = 0.
cg Applying the inverse rotation to the vectors U
cg corresponding to the values of theta found as shown
cg above will provide the tangent vectors in the
cg original coordinates.
cc RADIUS1 See "RADIUS1, RADIUS2".
cc RADIUS1 See "RADIUS1, RADIUS2".
cc RADIUS1, RADIUS2 In command "intcirc", the radii of two circles.
cc RADIUS2 See "RADIUS1, RADIUS2".
cc RADKIN In command "disk", the inner radius of an annular
cc disk. See "RADKOUT".
cc RADKOUT In command "disk", the outer radius of an annular
cc disk. See "RADKIN".
cc ran An option in command "big", to create a big integer
cc with a specified number of digits, each chosen
cc randomly from 0 to 9.
cc ran An option in command "icalc", to randomly sample an
cc integer from a range of values.
cc M ran N means random integer between M and N.
cc ran An option in command "variable", to randomly sample a
cc floating point number from a range of values.
cc A ran B means random real number between A and B.
cg random See "random numbers", "random points",
cg "random variables", "random vectors",
cg commands "big", "bin", "icalc", "last", "pdf",
cg "point", "sample", "spin", "variable", "walk".
cc random Command to display a help message for Monte Carlo
cc commands.
cc
cc Forms include:
cc
ccin help random
ccin random [help]
cc
cc Display a help message for Monte Carlo commands.
cc
cc Synonyms: [help, h], [random, ?].
cc random An option in command "walk", in place of a vector name
cc VNAME, to indicate that the initial direction vector
cc is to be randomly sampled from an isotropic
cc distribution.
cc random An option in command "help", to display help messages
cc for Monte Carlo commands.
cc Synonyms: [random, ?].
cc random An option in commands "last", "point", "variable" and
cc "vector", to select certain values at random.
cc random An option in command "spin", to set the random number
cc seed to a new value based on the date and clock.
cg random See "random numbers", "random order", "random points",
cg "random reordering", "random sampling",
cg "random vectors", "random walk".
cg random numbers To generate random numbers:
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 in the random
cg number sequence.
cg
cg variable VARNAME random
cg
cg Give variable VARNAME a random value uniformly
cg distributed between 0 and 1.
cg
cg variable VARNAME random VMIN VMAX
cg
cg Give variable VARNAME a random value uniformly
cg distributed between VMIN and VMAX.
cg
cg variable VARNAME random VMIN VMAX
cg variable VARNAME = 1 nint VARNAME
cg
cg Give variable VARNAME a random integer value
cg uniformly distributed between VMIN and VMAX.
cg
cg variable VARNAME = A ran B
cg
cg Give variable VARNAME a random value uniformly
cg distributed between A and B.
cg
cg icalc M ran N
cg
cg Randomly sample an integer in the range from M to
cg N, and display.
cg
cg big BIGNAME ran N
cg
cg Create big integer BIGNAME with N digits randomly
cg sampled from 0 through 9.
cg
cg You may also create probability bins, and then create
cg probability distribution functions (pdfs), each
cg consisting of a set of probability bins, and sample
cg from the pdfs, either with or without replacement.
cg See "bin", "pdf", "sample".
cg
cg To generate a large number of random numbers, repeat
cg one of the above commands many times, then edit
cg the output file(s) to collect the results.
cg
cg See command "spin".
cg random order See "order", "random reordering".
cg random points To randomly sample points, uniformly distributed over
cg an area or in a volume:
cg
cg point PNAME random [RADIUS|1]
cg
cg Create point PNAME, sampled randomly in a sphere
cg of radius RADIUS centered at the origin. If not
cg specified, RADIUS defaults to 1.
cg
cg point PNAME triangle TRNAME random
cg
cg Create point PNAME, sampled randomly in triangle
cg TRNAME.
cg
cg point PNAME tetrahedron TETNAME random
cg
cg Create point PNAME, sampled randomly in tetrahedron
cg TETNAME.
cg
cg point PNAME sphere SPHNAME random
cg
cg Create point PNAME, sampled randomly in sphere
cg SPHNAME.
cg
cg point PNAME disk DKNAME random
cg
cg Create point PNAME, sampled randomly in annular
cg disk DKNAME.
cg
cg point PNAME brick BRNAME random
cg
cg Create point PNAME, sampled randomly in brick
cg BRNAME.
cg
cg point PNAME zone ZNAME random BRNAME
cg
cg Create point PNAME, sampled randomly in the part of
cg zone ZNAME included in brick BRNAME.
cg See command "mcvol".
cg
cg To generate a large number of random points, repeat
cg one of the above commands many times, then edit
cg the output file(s) to collect the results.
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 Synonyms: [alias, al, a], [disk, disc, dk],
cg [point, p, pnt, pt], [sphere, sph],
cg [tetrahedron, tet, tetra], [triangle, tri].
cg
cg Also see commands "bin", "pdf", "sample".
cg random reordering
cg If a probability distribution (pdf), consisting of a
cg set of N equally probable probability bins, is
cg randomly sampled N times without replacement, the
cg resulting sequence of probability bins will be
cg randomly reordered from its initial sequence.
cg random sampling For random sampling of points in certain simple
cg surfaces and volumes, see "random points".
cg
cg To randomly sample points in the ellipsoid QNAME:
cg (1) Find the standard form of the implicit equation
cg for the ellipsoid, using command "quadric QNAME":
cg QC + QXX * x^2 + QYY * y^2 + QZZ * z^2 = 0
cg (QC < 0, QXX => QYY => QZZ > 0).
cg The ellipsoid has semiaxes sx = sqrt (-QC / QXX),
cg sy = sqrt (-QC / QYY), sz = sqrt (-QC / QZZ).
cg (2) Create the brick BRNAME with coordinates
cg -sx < x < sx, -sy < y < sy, -sz < z < sz, using
cg command "brick BRNAME = ...".
cg (3) Randomly sample points in brick BRNAME.
cg (4) Test each point with command "side", using the
cg ellipsoid in standard position. Reject points with
cg a positive result (outside of the ellipsoid).
cg (5) Rotate and translate each accepted point, using
cg the inverse of the translation and rotation of the
cg ellipsoid to its standard position.
cg random vectors Vectors may be randomly sampled from an isotropic
cg distribution; at a fixed angle from a specified
cg axis; and from a cosine-power distribution relative
cg to a specified axis:
cg
cg vector VNAME random
cg
cg Create unit vector VNAME, randomly sampled from an
cg isotropic distribution.
cg
cg vector VNAME random VAXIS cosine [POWER|1]
cg
cg Create unit vector VNAME, randomly sampled from a
cg cosine-power distribution with power POWER, around
cg the axis vector VAXIS.
cg
cg vector VNAME random VAXIS angle ANGLE
cg
cg Create unit vector VNAME, randomly sampled from a
cg uniform azimuthal distribution at angle ANGLE from
cg the axis vector VAXIS.
cg
cg Also see commands "bin", "pdf", "sample".
cg random walk A random walk is a series of displacements of a point
cg in space, each with a direction randomly sampled from
cg an isotropic distribution and a distance randomly
cg sampled from an exponential distribution whose
cg average value is the mean free path.
cg
cg The random walk produced with command "walk" is a
cg random walk with mean free path DMEAN, limited to a
cg maximum total path length PATHMAX, and to a specified
cg maximum distance DISTMAX from the initial position of
cg the point being walked (within a sphere with radius
cg DISTMAX centered on the initial coordinates of the
cg point.
cg
cg The expected values of the total path length <path>,
cg the net distance the point moves <dist>, and the
cg number of steps <nstep>, and their standard
cg deviations, may be estimated from diffusion theory
cg for the diffusion limit DMEAN << DISTMAX,
cg DMEAN << PATHMAX, with approximate corrections for
cg real transport that give correct results for <path>,
cg <dist> and <nstep> in the transport limit
cg DMEAN >> DISTMAX, DMEAN >> PATHMAX.
cg The standard deviations in <path> and <dist> given
cg below do not apply near the transport limit.
cg
cg If PATHMAX is reached first,
cg <path> = PATHMAX (with no standard deviation),
cg <dist> = FTRANR * sqrt (FACTP * DMEAN * PATHMAX),
cg with standard deviation
cg sigdist = sqrt ((2 - FACTP) * DMEAN * PATHMAX),
cg where FACTP = 16 / (3 * pi),
cg and FTRANR = 1 / sqrt (1 + FACTP * DMEAN / PATHMAX)
cg is an approximate correction for real transport, and
cg <nstep> = <path> / DMEAN,
cg with standard deviation
cg sigstep = sqrt (<n>).
cg From numerical experiments, the expected ratio of
cg maximum to final distance is about 1.3, with
cg a skewed standard deviation of about -0.3, +0.6,
cg and occurs at about 0.8 * PATHMAX, with a standard
cg deviation of about 0.2 * PATHMAX.
cg
cg If DISTMAX is reached first,
cg <dist> = DISTMAX (with no standard deviation),
cg <path> = FTRAND * 0.5 * DISTMAX^2 / DMEAN,
cg with standard deviation
cg sigpath = sqrt (0.1) * DISTMAX^2 / DMEAN,
cg where FTRAND = 1 + 2 * DMEAN / DISTMAX is an
cg approximate correction for real transport, and
cg <n> = <path>,
cg with standard deviation
cg sigstep = sqrt (<n>) + sigpath /DMEAN
cc rat A synonym for ratio.
cc RATIO In command "vector", a multiplier of the length of the
cc vector.
cc RATIO In command "operator", a scale factor for uniform,
cc radial or linear scaling. May be integer, floating
cc point or a variable.
cc If the following conditions occur, use of a scaling
cc operator may cause truncation errors, and it may not
cc be possible to get the original object back with
cc the inverse operator:
cc (1) the scaling operation is not going to use a
cc fixed point at the origin, a fixed axis on one of the
cc major axes (x, y or z), or a fixed plane on one of
cc the major planes (x, y or z), and
cc (2) the scaling operation will be on points, vectors
cc or planes, with a scale factor RATIO less than TOL,
cc or greater than 1 / TOL, or
cc (3) the scaling operation will be on a nonplanar
cc quadric surface, with the square of the scale factor,
cc RATIO^2, less than TOL or greater than 1 / TOL.
cc
cc See command "ratio".
cc RATIO In command "bin", when creating a relativistic
cc Maxwellian distribution, the ratio of particle
cc temperature (in energy units) to particle rest mass
cc (in energy units). RATIO is dimensionless.
cc 1 K = 8.617385E-5 eV (+/- 8.5 ppm),
cc 1 amu = 9.3149432E8 eV (+/- 0.3 ppm).
cc ratio Command to find the ratio between successive terms for
cc a geometric series with a specified first term,
cc a specified sum, and a specified number of terms.
cc
cc Command "ratio" relates to objects: symbol,
cc variable.
cc
cc Forms include:
cc
ccin help ratio
ccin ratio [help]
cc
cc Display the command options.
cc
ccin ratio TERM1 SUM NUMT
cc
cc Display the ratio between consecutive terms for a
cc geometric series that begins with the term TERM1,
cc and having a sum SUM for the first NUMT terms.
cc TERM1 must be positive. SUM must be greater than
cc TERM1. NUMT must be greater than 1.
cc
cc Synonyms: [help, h], [ratio, rat].
cg ratio See "ratio zoning".
cg ratio zoning Planes or other quadric surfaces may be created in a
cg family for which each surface is scaled by a constant
cg ratio from the preceding surface. See command
cg "ratio".
cg Points may be created in a family in which each point
cg is scaled by a constant ratio from the preceding
cg point. The scaling may be uniform, linear parallel
cg to an axis, or radial perpendicular to an axis.
cg See option "scale" of commands "point", "plane",
cg "sphere", "cylinder", "cone", "axisym", "quadric",
cg and see command "operator", options "scale",
cg "uniform", "linear", "radial".
cg See "equal zoning".
cg rational See "rational fraction".
cg rational fraction
cg To find a series of increasingly accurate rational
cg fraction approximations for a non-integer decimal
cg number VALUE, use command "contfr = VALUE".
cg ray tracing Ray tracing includes following the path of a light ray
cg and finding its intersections with surfaces, and may
cg include attenuation and reflection. See commands
cg "distance", "intersection", "proximal", "side",
cg "track", "where".
cc RCYL The radial coordinate rcyl of a point, in a cylindrical
cc coordinate system. Distance from the z axis. May be
cc integer, floating point or a variable.
cc Specified with command "point", "vector" or "plane".
cc RCYL^2 = X^2 + Y^2
cc RCYL = RSPH * sin (PHI)
cc See "rcyl cylinder".
cc rcyl An option in command "brick", to make surfaces of
cc constant rcyl the shared coordinate surfaces between
cc adjacent bricks.
cg rcyl See "rcyl cylinder".
cg rcyl cylinder To create a circular cylinder at a constant value RCYL
cg of rcyl in an (rcyl, theta, z) coordinate system:
cg point pcen = 0 0 0
cg vector vaxis = 0 0 1
cg cylinder cyl1 = pcen vaxis RCYL
cg
cg To create a family of concentric circular cylinders
cg with equally spaced radii, use command "cylinder"
cg with option "concentric".
cc rd A synonym for input.
cc read A synonym for input.
cg read See "input", "input file".
cg real See "real data", "real variable".
cg real data Real data is data which has been assigned type
cg real in GEOM. On execution, all such data is filled
cg with zeros. See "data types", "floating point".
cg real variable See "floating point".
cg reciprocal To find the reciprocal of a number to high accuracy,
cg get an integer by moving the decimal point to the
cg right end, say by multiplying by 10^M, and using
cg command "big", divide a large power of ten, say 10^N
cg (with at least twice as many digits) by the integer,
cg and then multiply the quotient by 10^(M-N).
cg reciprocal See "reciprocal series".
cg reciprocal series
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". To evaluate the decimal part of
cg a reciprocal series, use command "variable" with
cg option "/".
cg record See "record of session".
cg record of session
cg To record a continuous record of the screen contents
cg for a terminal session, execute the shell command:
cg
cg script MYSCREEN
cg
cg Record the screen contents in file MYSCREEN
cg (defaults to "typescript") until the shell command
cg "exit" is executed.
cg recover To recover from errors, use command "undo", "redo"
cg or "indo", and see "input file".
cc rect A synonym for cartesian.
cc rectangular A synonym for cartesian.
cg rectangular A 3-D coordinate system using the coordinates
cg (x, y, z). A Cartesian coordinate system.
cg Specified with command "coordinate".
cg To create a volume element bounded by surfaces of a
cg rectangular coordinate system, use command "brick".
cg Synonyms: rectangular, rect, xyz
cg
cg To create arrays of points on surfaces of a
cg rectangular coordinate system, see command
cg "point PMESH brick BRNAME mesh block [volume]".
cg recursion Recursive calculations may be made by using commands
cg "do" and "enddo", bracketing commands "variable", to
cg do the recursions. Conditions for ending the
cg recursion may be specified by using commands "if"
cg and "endif", bracketing a command to change the loop
cg variable to its final value.
cg
cg To generate a series with a single-term linear
cg recursion rule, use command "variable", option
cg "series".
cg
cg For a recursive method of evaluating continued
cg fractions, see "partial sum", "convergent".
cg redirect See "redirect input", "redirect output".
cg redirect input On the GEOM execution line, input may be redirected
cg from the standard input, the user's terminal, to
cg an input file IN_FILE, with the execution line
cg "geom IN_FILE". Do NOT use the UNIX execution line
cg "geom < IN_FILE", as messages requesting input
cg by the user would not then be seen on the user's
cg terminal, and the run would hang up.
cg redirect output On the GEOM execution line, output that would otherwise
cg go to file geom_hsp may be redirected to a file
cg OUT_FILE, with the execution line
cg "geom IN_FILE OUT_FILE" or the command "output".
cg File OUT_FILE must not be geom_cmd or tty.
cg The standard output to the user's terminal will not
cg be affected.
cg Do not use the UNIX execution line "geom > OUT_FILE"
cg or "geom IN_FILE > OUT_FILE" or no input file will
cg be specified, and prompts for input data will not be
cg displayed at the user's terminal.
cg
cg To redirect the standard output to a file STDOUT,
cg when an input file (not the user's terminal) is
cg specified, use the command
cg "geom IN_FILE OUT_FILE > STDOUT", or
cg "geom IN_FILE OUT_FILE >! STDOUT".
cg Use the latter form to write over an existing file
cg STDOUT.
cc redo Command to repeat an input line saved in memory,
cc optionally with one or more arguments replaced.
cc This is useful to correct an error in a preceding
cc command or to generate a series of commands with
cc only minor changes. Also see command "symbol" for
cc replacing erroneous input data.
cc The modified command may not exceed 80 characters.
cc See commands "undo", "indo".
cc To display all input lines saved in memory, use
cc command "input command ?".
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 Command "redo" relates to objects: symbol, variable.
cc
cc Forms include:
cc
ccin help redo
ccin redo help
cc
cc Display the command options.
cc
ccin redo [$]
cc
cc Redo the preceding input line with no replacements.
cc Same as commands "." and "indo [$]".
cc
ccin redo $ [R1 R2 R3 ...]
cc
cc Redo the preceding input line with arguments 1, 2,
cc 3, ..., replaced by R1, R2 R3, ...
cc
ccin redo -N [R1 R2 R3 ...]
cc
cc Redo the Nth preceding input line with arguments
cc 1, 2, 3, ..., replaced by R1, R2 R3, ...
cc
ccin redo LINE [R1 R2 R3 ...]
cc
cc Redo line LINE in the input lines saved in memory,
cc with arguments 1, 2, 3, ..., replaced by
cc R1, R2, R3, ... LINE must be an integer, not an
cc integer variable, and must not be less than the
cc index of the lowest input line saved in memory not
cc written over by a new command if the number of
cc input lines saved in memory exceeds the storage
cc space (see command "tables").
cc
cc
ccin redo STRING [R1 R2 R3 ...]
cc
cc Redo the last input line saved in memory that
cc begins with STRING, with arguments 1, 2, 3, ...,
cc replaced by R1, R2, R3, ...
cc
cc Note: for no replacement, use "." for R1, R2, R3,
cc ... (all trailing "." may be omitted).
cc
cc Synonyms: [help, h].
cg reduce See "reduce order", "reduce significant",
cg "reduce typing".
cg reduce order If one real root x = r of a polynomial equation P(x)
cg is known, the order of the equation may be reduced by
cg factoring out (x - r), using command "rootf".
cg If the initial set of coefficients is A(n), n = 0, N,
cg the coefficients of the reduced equation are
cg D(n), n = 0, N - 1, where:
cg D(n) = A(n+1) + A(n+2)*r + ... + A(N)*r^(N-n-1), or
cg D(n) = -[A(0) + A(1)*r + ... + A(n)*r^n] / r^(n+1).
cg One of these two forms may be numerically more
cg accurate to calculate than the other.
cg
cg If two or more real roots are known, the most
cg accurately known root (usually the one with the
cg largest absolute magnitude, but not near an extremum)
cg should be used first to reduce the order.
cg
cg See commands "root", "roots".
cg reduce significant
cg See "significant".
cg reduce typing See "shortcuts".
cg Refl In an output display, indicates a reflected vector,
cg as from a plane or quadric surface.
cc refl A synonym for reflect.
cg reflect 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 reflected with command
cg "reflect", after creating a reflection operator with
cg command "operator" (vectors may be reflected using a
cg simple normal vector).
cg Lines, triangles, regular polygons, regular
cg polyhedrons, and tetrahedrons may be reflected by
cg reflecting the points used to create them. This may
cg be easier if the points are put into clusters.
cg Zones may be reflected by reflecting the quadric
cg surfaces used to create them.
cg Annular disks may be reflected by redefining them,
cg after reflecting the center point and normal vector.
cg
cg Command "operator OPNAME" may be used to display
cg a reflection operator OPNAME.
cc reflect Command to reflect one or more points, clusters,
cc vectors or quadric surfaces (including planes,
cc spheres, circular cylinders, circular cones,
cc hyperbolic paraboloids, ellipsoids, axially symmetric
cc quadric surfaces and general quadric surfaces), using
cc a reflection operator previously created with command
cc "operator". The reflection plane passes through the
cc invariant point PINV, which defaults to the origin if
cc not specified. See commands "undo", "redo", "indo".
cc
cc Command "reflect" relates to objects: axisym,
cc cluster, cone, cylinder, ellipsoid, hyperb, operator,
cc plane, point, sphere, symbol, vector.
cc
cc Forms include:
cc
ccin help reflect
ccin reflect [help]
cc
cc Display the command options.
cc
ccin reflect point PNAME OPNAME [PINV]
cc
cc Reflect point PNAME with operator OPNAME, and
cc invariant point PINV.
cc
ccin reflect point all OPNAME [PINV]
cc
cc Reflect all points with operator OPNAME, and
cc invariant point PINV.
cc
ccin reflect cluster CLNAME OPNAME [PINV]
cc
cc Reflect cluster CLNAME with operator OPNAME, and
cc invariant point PINV.
cc
ccin reflect vector VNAME normal VNORM
cc
cc Reflect vector VNORM as if reflected at its bound
cc point by a surface with normal vector VNORM.
cc
ccin reflect vector VNAME OPNAME [PINV]
cc
cc Reflect vector VNAME with operator OPNAME, and
cc invariant point PINV.
cc
ccin reflect vector all OPNAME [PINV]
cc
cc Reflect all vectors with operator OPNAME, and
cc invariant point PINV.
cc
ccin reflect QTYPE all OPNAME [PINV]
cc
cc Reflect all quadric surfaces of type QTYPE (plane,
cc sphere, cylinder, cone, hyperb, ellipsoid, axisym,
cc quadric), with operator OPNAME, with invariant
cc point PINV.
cc
ccin reflect QTYPE QNAME OPNAME [PINV]
cc
cc Reflect the quadric surface of type QTYPE (plane,
cc sphere, cylinder, cone, hyperb, ellipsoid, axisym,
cc quadric) with name QNAME, with operator OPNAME,
cc with invariant point PINV.
cc
cc Synonyms: [cylinder, cyl],
cc [ellipsoid, el, ell, ellipse], [help, h],
cc [hyperb, saddle], [normal, norm], [plane, pl],
cc [point, p, pnt, pt], [quadric, q, quad],
cc [reflect, refl], [sphere, sph], [vector, v, vect].
cc reflect An option in command "operator", to create a tensor
cc operator that performs a reflection in a plane
cc through the origin, parallel to the specified plane
cc PLNAME. Equivalent to linear scaling with a scale
cc factor of -1.
cg reflect See "reflect the mesh".
cg reflect the mesh See "transform the mesh".
cg reflection See "normal", "reflect", "plane of reflection",
cg "track reflection".
cg reflection See "plane of reflection", "reflection surface".
cg reflection plane See "plane of reflection".
cg reflection surface
cg See "normal", "track reflection".
cg reflector A reflector surface in a telescope or a beamed light
cg source usually is a circular paraboloid.
cg regular See "polygon", "polyhedron",
cg "regular polygon", "regular polyhedron".
cg regular polygon See "polygon".
cg regular polyhedron
cg See "polyhedron", "tetrahedron", "cube", "octahedron",
cg "dodecahedron", "icosahedron".
cc rel An option in command "vector". If the vector is
cc specified by two points, and the option "rel" or "*"
cc is used, or if the option "abs" is not used, the
cc vector will have a length equal to RELLEN times
cc the distance between the two points. The value of
cc RELLEN defaults to 1.0 if not specified. Each of the
cc vector components will be multiplied by RELLEN.
cc See "RELLEN", "abs", "absolute", "reverse".
cc
cc The length, and each of the components of an existing
cc vector VNAME may be multiplied by the factor RATIO by
cc using one of the two equivalent commands:
cc
cc vector VNAME rel RATIO
cc vector VNAME * RATIO
cc
cc Use RATIO = -1 to reverse the vector direction.
cg relative A vector specified by two points is relative if key
cg word "abs" or "unit" is not used. If key word "rel"
cg or "*" is used, the vector will have a length equal
cg to RELLEN (defaults to 1) times the distance between
cg the two points. Once a vector has been created, it
cg no longer depends on any points used to create it.
cg See "RELLEN", "abs", "absolute", "vector".
cg relativistic See "Maxwellian".
cc relax An option in command "mesh", to relax a part of the
cc mesh, by moving each relaxed mesh point to the
cc average coordinates of its neighbors.
cc The relaxed points may be interior to the mesh block
cc or interior to the mesh, but outside the mesh block.
cg relax The coordinates of a mesh point may be relaxed in an
cg iterative process, by moving the point to the
cg average coordinates of its nearest neighbors in the
cg k, l and m index directions of the mesh.
cg In each index direction, there must be zero or two
cg neighbors, and neighbors with "vacuum" coordinates
cg (-10^99) are ignored. The point being relaxed may
cg not be an exterior point of the mesh or the mesh
cg block. See command "mesh relax ...".
cg RELEASED In the display for a point, indicates the point is
cg no longer a mesh point, having released its mesh
cg indices, by changing them to zero.
cc RELLEN Relative length of a "relative" vector. Defaults to 1.
cc May be integer, floating point or a variable.
cc A vector specified by two points, using key word
cc "rel" or "*", will have a length equal to RELLEN
cc times the distance between the two points.
cc Specified with command "vector".
cc RELTYPE An option in command "if", to indicate a relationship
cc between values A and B, each of which may be
cc explicitly numerical, a variable, or a symbolic word
cc which, after symbol replacement, is integer or
cc floating point.
cc "<" for A less than B,
cc "<=" for A less than or equal to B,
cc "=" for A equal to B,
cc ">=" for A greater than or equal to B,
cc ">" for A greater than B.
cc
cc Synonyms: [<, .lt.], [<=, =<, .le.], [=, .eq.],
cc [=>, >=, .ge.], [>, .gt.].
cg Remainder See "Chinese Remainder", command "crt".
cg remainder See "Chinese Remainder", command "crt".
cg remainder problem
cg See "Chinese Remainder", command "crt".
cg Remainder Theorem
cg See "Chinese Remainder", command "crt".
cc remove A synonym for delete.
cg rename See "rename objects", command "rename".
cc rename Command to rename 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 rename an object with the name of an
cc existing object of the same type.
cc
cc Note: no replacement of symbols will be done
cc in the commands "rename symbol ...", "rename alias".
cc
cc Command "rename" 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 rename
ccin rename [help]
cc
cc Display the command options.
cc
cc In the following command, if optioin "array" is not
cc used, you may use "+" or "-" instead of OBJNAME2 to
cc form the name by incrementing or decrementing the
cc base name for objects of type OBJTYPE (which may not
cc be OBJNAME). See command "last", "increment names".
cc
ccin rename OBJTYPE OBJNAME OBJNAME2 [array]
cc
cc Rename object type OBJTYPE (alias, axisym, big,
cc bin, brick, cluster, cone, cylinder, ellipsoid,
cc disk, hyperb, line, plane, marker, operator, pdf,
cc point, polygon, polyhedron, quadric, sphere,
cc symbol, tetrahedron, triangle, variable, vector,
cc zone) by replacing name OBJNAME with the new name
cc OBJNAME2.
cc If option "array" is used, only the stem names
cc will be changed, and not the appended subscripts.
cc Option "array" may not be used with big integers
cc or markers.
cc
cc If a point is renamed, it is also renamed wherever it
cc is used in the definition of a cluster, a line,
cc a triangle, a regular polygon, an annular disk, a
cc regular polyhedron or a tetrahedron.
cc A mesh point may not be renamed to a triply
cc subscripted name, unless the subscripts are the
cc k, l and m indices to which the point is assigned.
cc
cc If a quadric surface (plane, sphere, cylinder, cone,
cc ellipsoid, axially symmetric or general quadric) is
cc renamed, it is also renamed wherever it is used in
cc the definition of a zone.
cc
cc If a probability bin is renamed, it is also renamed
cc wherever it us used in the definition of a
cc probability distribution function (pdf).
cc
cc Synonyms: [alias, al, a], [brick, br],
cc [cluster, cl], [cylinder, cyl], [disk, disc, dk],
cc [ellipsoid, el, ell, ellipse], [help, h],
cc [hyperb, saddle], [line, l, ln], [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 [rename, sw, switch], [sphere, sph],
cc [symbol, sym, symb], [tetrahedron, tet, tetra],
cc [triangle, tri], [variable, s, set, var],
cc [vector, v, vect], [zone, z, zn].
cg rename objects See command "rename".
cg rename points See commands "mesh", "point" and "rename".
cg Points assigned to the mesh may be released from
cg the mesh and renamed by replacing the first character
cg of their name with "-". This must not create a
cg duplicate name. To avoid this, rename all such
cg points before releasing any more points.
cg reorder To sort objects into ASCII order, use command "sort".
cg To randomly reorder N objects:
cg create N probability bins, one for each object,
cg each with relative total probability = 1;
cg create a probability distribution function (pdf)
cg consisting of all of the probability bins;
cg randomly sample N times from the probability
cg distribution function (pdf), without replacement.
cg See commands "bin", "pdf", "sample".
cg repack The data arrays for a specified object type are
cg repacked by removing all deleted objects of that type
cg and shifting the remaining objects down in memory.
cg This is done when command "repack" is used, and is
cg also done automatically whenever commands "list",
cg "sort" or "table" are used.
cc repack Command to repack aliases, markers, variables, vectors,
cc tensor operators, points, clusters, meshes, lines,
cc triangles, regular polygons, annular disks, planes,
cc quadric surfaces, regular polyhedrons, tetrahedrons,
cc bricks, zones, probability bins, probability
cc distribution functions (pdfs) and symbols,
cc by shifting objects down into memory space previously
cc used by deleted objects. This operation is done
cc automatically whenever commands "list", "sort" or
cc "table" is used.
cc
cc Command "repack" relates to objects: alias, axisym,
cc bin, brick, cluster, cone, cylinder, disk, ellipsoid,
cc hyperb, line, marker, mesh, operator, pdf, plane,
cc point, polygon, polyhedron, quadric, sphere, symbol,
cc tetrahedron, triangle, variable, vector, zone.
cc
cc Forms include:
cc
ccin help repack
ccin repack [help]
cc
cc Display the command options.
cc
ccin repack all
cc
cc Repack all objects and mesh points.
cc
ccin repack OBJTYPE
cc
cc Repack objects of the type indicated by OBJTYPE.
cc OBJTYPE may be alias, marker, variable, vector,
cc operator, point, cluster, line, triangle, polygon,
cc disk, quadric, polyhedron, tetrahedron, brick,
cc zone, bin, pdf or symbol.
cc
ccin repack quadric
cc
cc Repack all quadric surfaces
cc (includes planes, spheres, circular cylinders,
cc circular cones, hyperbolic paraboloids, ellipsoids,
cc axially symmetric quadric surfaces, and general
cc quadric surfaces).
cc
cc repack QTYPE
cc
cc Repack all quadric surfaces of quadric type QTYPE
cc (plane, sphere, cylinder, cone, hyperb, ellipsoid,
cc axisym or quadric).
cc
ccin repack mesh
cc
cc Repack all mesh points.
cc
cc Synonyms: [alias, al, a], [brick, br],
cc [cluster, cl], [cylinder, cyl], [disk, disc, dk],
cc [ellipsoid, el, ell, ellipse], [help, h],
cc [hyperb, saddle], [line, l, ln], [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 [sphere, sph], [symbol, sym, symb],
cc [tetrahedron, tet, tetra], [triangle, tri],
cc [variable, s, set, var], [vector, v, vect],
cc [zone, z, zn].
cg repeat To repeat the preceding command, use command ".".
cg
cg To repeat a block of commands many times in many
cg different GEOM runs, put them into a separate file,
cg e.g. IN_FILE, followed by the command "return".
cg In a GEOM run, use command "input IN_FILE" as many
cg times as desired, or type "alias k = 'i IN_FILE'",
cg then type "k" as many times as desired.
cg
cg To repeat a block of commands previously used in the
cg same run, type "i c ?" to find what the input
cg line numbers of the commands are, e.g. LINE1 and
cg LINE2, then type "i c LINE1 LINE2" to repeat the
cg commands. To repeat then many times, type
cg "alias k = 'i c LINE1 LINE2'", then type "k" many
cg times.
cg
cg To repeat an entire run, perhaps with modifications,
cg copy the output file geom_cmd to a new input file
cg IN_FILE, make any needed modifications, then use it
cg in a GEOM run by typing "input IN_FILE".
cg
cg See "input file", "macro".
cc repl An option in command "sample", to sample with
cc replacement (the same probability bin may be sampled
cc more than once).
cg replace See "alias", "replace arguments", "replace name",
cg "replace object", "symbol".
cg replace See "replace arguments", "replace command",
cg "replace name", "replace object", "replace symbol".
cg replace arguments
cg See commands "redo", "indo", "alias", "symbol".
cg replace command See command "alias".
cg replace name See commands "rename", "symbol".
cg replace object An existing object may be replaced by creating it
cg again, using the same name, or by deleting it, then
cg creating a new object to replace it. In the latter
cg case, the same name or a different name may be used.
cg
cg To create objects, see commands "variable", "point",
cg "cluster", "line", "triangle", "polygon", "disk",
cg "vector", "operator", "plane", "sphere", "cylinder",
cg "cone", "axisym", "quadric", "polyhedron",
cg "tetrahedron", "brick", "zone", "bin", "pdf",
cg "symbol".
cg
cg Also see "copy", "delete", "rename".
cg replace symbol See "symbol", command "symbol".
cg replacement In a random sampling procedure without replacement, the
cg same probability bin can not be sampled more than
cg once, while doing NSET sets of NSAMP samples each.
cg See "random reordering".
cc REPLNAME In command "symbol", REPLNAME may be any delimited
cc ASCII character string that will fit on the input
cc line (limited to 80 characters). REPLNAME may be an
cc ASCII character string, an integer or floating point
cc value, a key word used as a command or command
cc option, the name of another symbol or the name of
cc any other object.
cc reset An option in command "sudoku", to restore the last set
cc of assigned digits saved by use of option "save".
cg restart To restart from an interrupted run, see "input file" to
cg find how to make a new input file to repeat the run
cg up to the point of interruption. End the new input
cg file with command "return" to return to the previous
cg input medium.
cc return You must type the <return> key at the end of every
cc input line typed at the user's terminal.
cc return Command to restore the input medium to the previous
cc input medium.
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 See command "input".
cc
cc Do not use the statement separator ";" on the same
cc line as this command.
cc
cc Forms include:
cc
ccin help return
ccin return help
cc
cc Display the command options.
cc
ccin return
cc
cc Read the previous input medium. If a file, start
cc at the line following the line previously read, and
cc use the same specification for the last line.
cc See "linked input", "trace".
cc
cc Synonyms: [help, h], [return, rtn].
cg reverse To reverse the effects of errors, use command "undo",
cg "redo" or "indo", and see "input file".
cg reverse To find the reciprocals of the roots of a polynomial
cg equation using command "root" or "roots", specify
cg the coefficients in reverse order. This sometimes
cg is more accurate, especially for roots of small
cg magnitude.
cg reverse To reverse the direction of a vector V1, use command
cg "vector V1 rel -1"
cg reversible In this document, reversible means capable of being
cg undone or returned to an original form, position or
cg state. Extreme scaling of a quadric surface may not
cg be reversible.
cg review To review commands used previously in the current
cg session of GEOM, use command "input", with options
cg "command" and "?".
cg revolution See "area of revolution", "volume of revolution",
cg "surface of revolution".
cc rgold An option in command "symbol", argument SYMBNAME,
cc to create symbol rgold with value (1 + sqrt(5))/2 =
cc 1.618033988749894848204586834365638117720309189
cg rgold The "golden ratio" 0.5 * (1.0 + sqrt (5.0)) =
cg 1.618033988749894848204586834365638117720309189...
cg Generated as a symbol at the beginning of a
cg GEOM run. The roots of 1 + x - x^2 = 0 are
cg rgold and 1 - rgold. Also rgold = 1 + 1 / rgold.
cg right See "right side".
cg right arrow The "greater than" or right arrow character, ">".
cg right bracket The right square bracket, "[" or the right curly
cg bracket, "{".
cg right side Aliases may be created, so that when the left side
cg (a delimited character string up to 24 characters
cg long, including any subscripts) is the first word of
cg an input line, it is replaced by the right side, as
cg long as the combined total number of characters does
cg not exceed 80. See command "alias".
cg
cg Markers may be created, so that when command
cg "marker AMNAME" is typed, the text assigned to
cg marker AMNAME is displayed.
cc RINV In command "twist", a radius from the twist axis.
cc rm A synonym for delete.
cg root To find the real square or cube root of a number, use
cg command "variable", with FUNCTION = "sqrt" or "curt".
cg
cg To find the real and/or complex roots of a quadratic,
cg cubic or quartic polynomial equation with real
cg coefficients, use command "roots".
cg
cg To search for real roots, extrema and inflection
cg points of a polynomial equation of higher order,
cg using Newtonian iteration, use command "root".
cg
cg To find the real coefficients of a polynomial
cg equation with from one to four 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 do a Newton's iteration of a value near a real or
cg complex root of a quadratic, cubic or quartic
cg polynomial equation with real coefficients, use
cg command "roots", option "?".
cg
cg To factor out a term (x - r) from a polynomial
cg equation in x with a root r, use command "rootf".
cc root Command to find the coefficients of an Nth-order
cc polynomial equation with N specified real roots.
cc Also, to search for real roots, extrema and
cc inflection points of a polynomial equation P(x) of
cc order 5 or higher (use command "roots" otherwise),
cc using Newtonian iteration, given a minimum, increment
cc and maximum of trial values of x, and a maximum
cc number of iterations.
cc
cc Note that a polynomial equation of order N may have
cc no more than N roots, N - 1 extrema, and N - 2
cc inflection points. If any of these limits are
cc exceeded, some adjacent pairs of points may represent
cc a single root, extremum or inflection point, with
cc slightly different approximate values.
cc
cc Two real roots must have an extremum between them.
cc Two extrema must have an inflection point between
cc them. If two maxima occur, there must be a minimum
cc between them, and vice versa.
cc If any of these conditions are violated, there are
cc missing roots, extrema or inflection points. Use
cc the procedure below to find them.
cc
cc To find all real roots, extrema and inflection
cc points:
cc (a) Make sure the search range includes the limits
cc displayed by the first use of command "root".
cc (b) Search smaller intervals between extrema.
cc (c) Allow more iterations (100 to 1000).
cc (d) Make TOL larger, to allow quicker convergence.
cc (e) If one or more real roots are found, use command
cc "rootf" to factor them out, starting with the
cc root of maximum magnitude, then try to find the
cc roots of the reduced equation.
cc (e) Specify the coefficients of the polynomial
cc equation P(x) in reverse order, and use the
cc reciprocals of any roots found. This is a better
cc and more accurate way to find very small roots.
cc
cc Also see command "rootf".
cc
cc Command "root" relates to objects: symbol, variable.
cc
cc Forms include:
cc
ccin help root
ccin root [help]
cc
cc Display the command options.
cc
ccin root = X(1) X(2) X(3) ... X(N)
cc
cc Find the coefficients a(n), n = 0, N, of the
cc polynomial equation of order N in x:
cc P(x) = sum(n = 0,N) {a(n) * x^n} = 0,
cc with the specified N real roots X(1), X(2), X(3),
cc ..., X(N). The displayed coefficients may be cut
cc and pasted to put them into memory.
cc
ccin root XMIN DX XMAX NITMAX A(0) A(1) A(2) A(3) ... A(N)
cc
cc Try to find any real roots, extrema and (if N is 3
cc or more) inflection points of the polynomial
cc equation in x:
cc P(x) = sum(n = 0,N) {A(n) * x^n} = 0,
cc first at x = 0, and then by Newtonian iteration,
cc with trial values of x from XMIN to XMAX, with
cc increment DX (0 defaults to XMAX - XMIN), using no
cc more than NITMAX (0 defaults to 100) iterations,
cc and using TOL (0 defaults to 10^(-11) to determine
cc convergence.
cc
cc As many coefficients as will fit on one input line
cc may be used (use short variable names to save
cc space).
cc
cc The most extreme values needed for XMIN and XMAX
cc for roots, extrema and inflection points will be
cc displayed,
cc
cc If the first n leading coefficients are zero, there
cc are n roots at x = 0.
cc
cc The values of x, P(x), the slope P'(x), and the
cc second derivative P''(x) at up to 32 equally spaced
cc trial values will be displayed, as well as any
cc trial values for which the sign of P, P' or P''
cc changes.
cc
cc Any roots, maximum and minimum points, and
cc inflection points found will be displayed, along
cc with the values of P, P' and P'' at those points.
cc Roots will be labeled "single", "dbl max",
cc "dbl min" or "triple", the latter including all
cc roots of degeneracy 3 or more. Extrema will be
cc labeled "maximum", "minimum" or "min/max", the
cc latter at an inflection point with zero slope.
cc Any remaining inflection points will be labeled
cc "inflectn".
cc
cc Synonyms: [help, h].
cc root An option in commands "big" and "variable", to find a
cc specified root of a specified big integer or
cc variable, respectively.
cg root name See "base name", "stem".
cc rootf Command to reduce the order of a polynomial equation
cc when one root is known, by factoring out a term
cc containing that root. See "reduce order".
cc
cc Command "rootf" relates to objects: symbol,
cc variable.
cc
cc Forms include:
cc
ccin help rootf
ccin rootf [help]
cc
cc Display the command options.
cc
ccin rootf XROOT A(0) A(1) A(2) A(3) A(4) ... A(N)
cc
cc If XROOT is a root of the polynomial equation in x,
cc P(x) = sum(n = 0,N) {A(n) * x^n} = 0, reduce the
cc order of the equation by factoring out the term
cc (x - XROOT), and display the resulting
cc coefficients. b(n), n = 0, N - 1, where b(n) =
cc A(n+1) + A(n+2)*r + ... + A(N)*r^(N-n-1) =
cc -[A(0) + A(1)*r + ... + A(n)*r^n] / r^(n+1)
cc (use the equation with the lesser error).
cc
cc If XROOT is not a root, a search is made to find
cc the nearest root by Newtonian iteration, and if
cc found, used instead of XROOT.
cc
cc Trailing zero coefficients are ignored. Each
cc leading zero coefficient is counted as a root at
cc x = 0, and removed.
cc
cc The displayed coefficients may be cut and pasted to
cc put them into memory.
cc
cc Synonyms: [help, h].
cg roots The roots of a polynomial equation in z, P(z), are
cg the real and/or complex values of z for which
cg P(z) = 0. To find the real and/or complex roots of a
cg quadratic, cubic or quartic polynomial equation in z,
cg use command "roots".
cg
cg To search for real roots, extrema and inflection
cg points of a higher order polynomial equation using
cg Newtonian iteration, use command "root".
cg
cg To factor a known root out of a polynomial equation,
cg use command "rootf".
cg
cg Any minima and/or maxima of the polynomial equation
cg in z,
cg P(z) = a0 + a1*z + a2*z^2 + a3*z^3 + a4*z^4 + ...
cg are at any real roots of the derivative of P(z),
cg P'(z) = a1 + 2*a2*z + 3*a3*z^2 + 4*a4*z^3 + ...
cg Any inflection points of P(z) are at any real roots
cg of the second derivative of P(z),
cg P''(z) = 2*a2 + 2*3*a3*z + 3*4*a4*z^2 + ...
cg The real roots of the third derivative of P(z),
cg P'''(z) = 2*3*a3 + 2*3*4*a4*z + ...
cg may also be of interest.
cg
cg To find the real coefficients of a polynomial
cg equation with from one to four specified real and/or
cg complex roots, use command "root" or "roots",
cg option "=".
cg
cg To improve an approximate real or complex root of a
cg polynomial equation up to fourth order by Newton's
cg iteration, use command "roots", option "value".
cg
cg To try to improve an approximate real root of any
cg polynomial equation, use command "root".
cg
cg If one real root r of a polynomial equation in x,
cg P(x) is known, the order of the equation may be
cg reduced by one, by using command "rootf" to factor
cg out (x - r). If the initial set of coefficients is
cg A(n), n = 0, N, the coefficients of the reduced
cg equation are D(n), n = 0, N - 1, where:
cg D(n) = A(n+1) + A(n+2)*r + ... + A(N)*r^(N-n-1) or
cg D(n) = -[A(0) + A(1)*r + ... + A(n)*r^n] / r^(n+1).
cg One of the two ways of calculating D(n) may be
cg numerically more accurate than the other.
cg
cg See "quadratic", "cubic", "quartic".
cc roots Command to find any real and/or complex roots, given
cc the real coefficients, or to find the real
cc coefficients, given the real and/or complex roots
cc (option "="), or to find the value for a given real
cc or complex argument (option "value"), of a quadratic,
cc cubic or quartic polynomial equation with real
cc coefficients:
cc P(z) = a0 + a1*z + a2*z^2 + a3*z^3 + a4*z^4.
cc The values of z at any extrema and inflection points
cc may also be found, as well as the value of P(z) at
cc those values of z.
cc The accuracy of an approximate root may also be
cc improved by Newton's iteration, using option "value".
cc
cc Any extrema (minima and/or maxima) of P(z) are at any
cc roots of the derivative of P(z),
cc P'(z) = a1 + 2*a2*z + 3*a3*z^2 + 4*a4*z^3 + ...
cc Any inflection points of P(z) are at any roots of the
cc second derivative of P(z),
cc P''(z) = 2*a2 + 6*a3*z + 12*a4*z^2 + ...
cc Also, P'''(z) = 6*a3 + 24*a4*z, P''''(z) = 24*a4.
cc
cc If TOL is zero, a temporary value of 10^(-11) will be
cc used. TOL is used to estimate error, and to test for
cc convergence when real roots are found by Newton's
cc method. If any of the roots are inaccurate, try
cc decreasing the value of TOL. Also try reversing the
cc order of the coefficients, and using the
cc reciprocal(s) of any resulting accurate root(s).
cc Some roots may be inaccurate if they differ by many
cc orders of magnitude from other roots, or if the
cc difference is small relative to the magnitude of the
cc root, or if the real and imaginary parts of a complex
cc root differ by many orders of magnitude.
cc
cc If all of the roots are very close together, it may
cc increase accuracy to shift the origin to a point z0
cc near the roots. For u = z - z0,
cc P(u+z0) = b0 + b1*u + b2*u^2 + b3*u^3 + b4*u^4
cc where b0 = P(z0), b1 = P'(z0), b2 = P''(z0) / 2,
cc b3 = P'''(z0) / 6 = a3 + 4*a4*z0,
cc b4 = P''''(z0) / 24 = a4.
cc
cc Command "roots" relates to objects: symbol,
cc variable.
cc
cc Forms include:
cc
ccin help roots
ccin roots [help]
cc
cc Display the command options.
cc
ccin roots A0 A1 [A2]
cc
cc Find the two real or complex roots r1 and r2 of the
cc quadratic equation A0 + A1 * z + A2 * z^2 = 0
cc (when A2 is nonzero). Also find the real extremum.
cc If Q = A1^2 - 4 * A0 * A2 is zero (or truncates to
cc zero, based on TOL), there is one double root. If
cc Q is positive, there are two real roots. If Q is
cc negative, there are two conjugate complex roots.
cc If A0 is zero, one root is zero, and the other root
cc is -A1 / A2. The roots r1 and r2 are related to
cc A0, A1 and A2 by:
cc A0 / A2 = r1*r2, A1 / A2 = -(r1 + r2).
cc
ccin roots A0 A1 A2 A3
cc
cc Find the real and any complex roots of the cubic
cc equation A0 + A1 * z + A2 * z^2 + A3 * z^3 = 0
cc (when A3 is nonzero). Also find any real extrema,
cc and the real inflection point. There may be a
cc triple real root, a single real root and a double
cc real root, a single real root and two conjugate
cc complex roots or three unique real roots.
cc If A0 = 0, one root is zero, and the other two
cc roots are the roots of the quadratic equation
cc A1 + A2 * x + A3 * x^2 = 0.
cc The roots r1, r2 and r3 are related to A0, A1, A2
cc and A3 by:
cc A0 / A3 = -r1*r2*r3,
cc A1 / A3 = r1*r2 + r2*r3 + r3*r1,
cc A2 / A3 = -(r1 + r2 + r3).
cc
ccin roots A0 A1 A2 A3 A4
cc
cc Find the four real and/or complex roots of the
cc quartic equation:
cc A0 + A1 * x + A2 * z^2 + A3 * z^3 + A4 * z^4 = 0
cc (when A4 is not zero). Also find the real extrema,
cc any real inflection points, and the real point
cc where the third derivative is zero. Any real roots
cc may be single, double, triple or quadruple. Any
cc complex roots occur in conjugate pairs.
cc
cc If A0 = 0, one root is zero, and the other three
cc roots are the roots of the cubic equation:
cc A1 + A2 * z + A3 * z^2 + A4 * z^3 = 0.
cc The roots r1, r2, r3 and r4 are related to A0, A1,
cc A2 and A3 by:
cc A0 / A4 = r1*r2*r3*r4,
cc A1 / A4 = -(r1*r2*r3 + r2*r3*r4 + r3*r4*r1 +
cc r4*r1*r2),
cc A2 / A4 = r1*r2 + r2*r3 + r3*r4 + r4*r1 +
cc r1*r3 + r2*r4,
cc A3 / A4 = -(r1 + r2 + r3 + r4).
cc If a pair of roots r1 and r2 are known, the other
cc two roots are the roots of the quadratic equation:
cc q(z) = B0 + B1 * z + z^2,
cc where B0 = (A0 / A4) / (r1 * r2)
cc and B1 = (A3 / A4) + r1 + r2,
cc and are either two real roots or a pair of
cc conjugate complex roots. The two real roots may be
cc the same.
cc
ccin roots = X1 [Y1 X2 Y2 [X3 Y3 [X4 Y4]]]
cc
cc Find the coefficients of the polynomial equation
cc with the from one to four real and/or complex roots
cc X1 + i*Y1, X2 + i*Y2, [X3 + i*Y3, [X4 + i*Y4]].
cc Specify any real roots first, with the imaginary
cc part zero, followed by any complex roots, in
cc conjugate pairs. The displayed coefficients may be
cc cut and pasted to put them into memory.
cc
ccin roots value X Y A0 A1 [[A2|0] [[A3|0] [[A4|0]]]]
cc
cc For the real or complex argument z = X + i*Y, find
cc the value of the polynomial
cc P(z) = A0 + A1*z + A2*z^2 + A3*z^3 + A4*z^4,
cc and use Newtonian iteration to try to find a root
cc of P(z) near z. If a complex root r1 is found, its
cc conjugate r2 is also a root. If any two roots of a
cc quartic equation are known, the equation for the
cc other two roots is B0 + B1 * z + z^2 = 0,
cc where B0 = (A0 / A4) / (r1 * r2)
cc and B1 = (A3 / A4) + r1 + r2.
cc This equation has either two real roots or a pair
cc of conjugate complex roots. The two real roots may
cc be the same.
cc
cc See command "variable", option "polynomial".
cc
cc Synonyms: [help, h].
cc rot A synonym for rotate.
cg rotate 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 rotated with command
cg "rotate", after specifying a rotation operator with
cg command "operator".
cg Lines, triangles, regular polygons, regular
cg polyhedrons, and tetrahedrons may be rotated by
cg rotating the points used to create them. This may
cg be easier if the points are put into clusters. Zones
cg may be rotated by rotating the planes and quadric
cg surfaces used to create them. Annular disks may be
cg rotated by redefining them, after rotating the center
cg point and normal vector.
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.
cg See command "plane" with option "rotate".
cg
cg A family of rotated quadric surface may be created by
cg rotating a base quadric surface around a specified
cg invariant point by a specified rotation operator.
cg See command "quadric" with option "rotate".
cg
cg Command "operator OPNAME" may be used to display
cg a rotation operator OPNAME and its inverse, and the
cg angle and axis of rotation.
cg
cg See "parallel", "move", "nest", "scale", "twist".
cc rotate Command to rotate one or more points, clusters,
cc vectors or quadric surfaces (including planes,
cc spheres, circular cylinders, circular cones,
cc hyperbolic paraboloids, ellipsoids, axially symmetric
cc quadric surfaces and general quadric surfaces), using
cc the original or inverse (option "-") of a rotation
cc operator previously created with command "operator",
cc based on an "axial", "planar", "serial", "triple" or
cc "vector" rotation.
cc A rotation operator rotates its three row vectors
cc onto the major axes, and the major axes onto its
cc three column vectors. Also see command "twist".
cc The rotation axis passes through the invariant point
cc PINV, which defaults to the origin if not specified.
cc See commands "undo", "redo", "indo".
cc
cc Command "rotate" relates to objects: axisym,
cc cluster, cone, cylinder, ellipsoid, hyperb, operator,
cc plane, point, sphere, symbol, vector.
cc
cc Forms include:
cc
ccin help rotate
ccin rotate [help]
cc
cc Display the command options.
cc
ccin rotate point PNAME [-] OPNAME [PINV]
cc
cc Rotate point PNAME with [the inverse of] operator
cc OPNAME, around the invariant point PINV.
cc
ccin rotate cluster CLNAME [-] OPNAME [PINV]
cc
cc Rotate cluster CLNAME with [the inverse of]
cc operator OPNAME, around the invariant point PINV.
cc
ccin rotate point all [-] OPNAME [PINV]
cc
cc Rotate all points with [the inverse of] operator
cc OPNAME, around the invariant point PINV.
cc
ccin rotate vector VNAME [-] OPNAME [PINV]
cc
cc Rotate vector VNAME with [the inverse of] operator
cc OPNAME, around the invariant point PINV.
cc
ccin rotate vector all [-] OPNAME [PINV]
cc Rotate all vectors with [the inverse of] operator
cc OPNAME, around the invariant point PINV.
cc
ccin rotate QTYPE all [-] OPNAME [PINV]
cc
cc Rotate all quadric surfaces of type QTYPE (plane,
cc sphere, cylinder, cone, ellipsoid, axisym,
cc quadric, hyperb), with [the inverse of] operator
cc OPNAME, around the invariant point PINV.
cc
ccin rotate QTYPE QNAME [-] OPNAME [PINV]
cc
cc Rotate the quadric surface of type QTYPE (plane,
cc sphere, cylinder, cone, hyperb, ellipsoid, axisym,
cc quadric) with name QNAME, with [the inverse of]
cc operator OPNAME, around the invariant point PINV.
cc If operator OPNAME has been specified with the
cc command: operator OPNAME QTYPE QNAME and the [-]
cc option is not used, the quadric surface will be
cc rotated to align it with the x, y and z axes.
cc See command "quadric", option "rotate".
cc
cc Synonyms: [cylinder, cyl],
cc [ellipsoid, el, ell, ellipse], [help, h],
cc [hyperb, saddle], [plane, pl], [point, p, pnt, pt],
cc [quadric, q, quad], [rotate, rot], [sphere, sph],
cc [vector, v, vect].
cc rotate An option in command "plane", to create a family of
cc intersecting planes by repeated rotations of a base
cc plane around a specified axis by a specified angular
cc increment.
cc See "concentric", "move", "nest", "parallel",
cc "scale".
cc
cc This command option can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc rotate An option in command "point", to create a family of
cc points by repeated rotations of a base point, using
cc a specified rotation operator and a specified
cc invariant point.
cc
cc This command option can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc rotate An option in commands "sphere", "cylinder", "cone",
cc "ellipsoid", "axisym", "quadric", to create a family
cc of quadric surfaces by repeated rotations of a base
cc quadric surface, using a specified rotation operator
cc and a specified invariant point.
cc See commands "operator", options "axial", "planar",
cc "QTYPE", 'serial", "triple", "vector".
cc See "concentric", "move", "nest", "parallel",
cc "scale".
cc
cc This command option can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc rotate An option in command "vector", to create a family of
cc vectors by repeated rotations of a base vector,
cc using a specified rotation operator and a specified
cc invariant point.
cc
cc This command option can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cg rotate See "rotate the mesh".
cg rotate the mesh See "transform the mesh".
cg rotated See "rotate".
cg rotated See "rotated planes", "rotated quadrics".
cg rotated planes A family of intersecting planes may be created by
cg rotating a base plane around a specified axis by a
cg specified angular increment.
cg See "parallel planes", "concentric", "nested cones".
cg
cg Create the rotation axis and the base plane:
cg
cg point PAXIS ...
cg
cg For options, see command "point".
cg Create the axial point PAXIS.
cg
cg vector VAXIS ...
cg
cg For options, see command "vector".
cg Create the axis vector VAXIS.
cg
cg plane PL(1) ...
cg
cg For options, see command "plane".
cg Create base plane PL(1).
cg
cg Check the axis and base plane to see if they will
cg produce the desired results:
cg
cg track PAXIS VAXIS plane PL(1)
cg
cg Find the distance to and coordinates of any point
cg of intersection of the track from point PAXIS in
cg the direction of vector VAXIS, with the plane
cg PL(1), or if the track is parallel to the plane,
cg the proximal distance.
cg
cg If the track is parallel to plane PL(1), at zero
cg distance, a family of radial planar vanes
cg intersecting at the axis will result. Each space
cg forms a wedge.
cg
cg If the track is parallel to plane PL(1), at a nonzero
cg distance, a family of planes intersecting at lines
cg parallel to and equidistant from the axis will
cg result. One space forms a polygonal cylinder.
cg
cg If the track is not parallel to plane PL(1), the
cg intersection point is where all of the lines of
cg intersection of the family of planes will meet.
cg Two spaces each form the apex of a pyramid.
cg
cg plane PL(2) rotate NUMPL INC PL(1) PAXIS VAXIS DANGLE
cg
cg Create a family of NUMPL planes PL(2), ..., with
cg names incremented by INC characters, rotated around
cg an axis through point PAXIS in the direction of
cg vector VAXIS, by equal angular increments DANGLE
cg (counterclockwise, with the axis pointed at the
cg observer), starting from plane PL(1).
cg See "increment names".
cg rotated quadrics A family of rotated quadric surfaces may be created by
cg rotating with a specified rotation operator around a
cg specified invariant point, starting from a base
cg quadric surface.
cg See commands "axisym", "cone", "cylinder",
cg "ellipsoid", "plane", "quadric", "sphere",
cg option "rotate".
cg rotation See "rotate", 'axis of rotation".
cg round-off 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.
cc row An option in command "sudoku", to assign up to 9
cc digits to a Sudoku row.
cg row See "family of objects", "move", "scale".
cc RSPH The radial coordinate rsph of a point, in a spherical
cc coordinate system. Distance from the origin. May be
cc integer, floating point or a variable.
cc Specified with command "point", "vector" or "plane".
cc RSPH^2 = X^2 + Y^2 + Z^2
cc See "RSHP sphere".
cc rsph An option in command "brick", to make surfaces of
cc constant rsph the shared coordinate surfaces between
cc adjacent bricks.
cg rsph See "rsph sphere".
cg rsph sphere To create a sphere at a constant value RSPH of rsph in
cg an (rsph, theta, phi) spherical coordinate system:
cg
cg point pcen = 0 0 0
cg sphere sph1 = pcen RSPH
cg
cg To create a family of concentric spheres with equally
cg spaced radii, use command "sphere", option
cg "concentric".
cg To create a family of concentric spheres with radii
cg and radial increments forming geometric series, use
cg command "sphere", option "scale".
cc rtn A synonym for return.
cg ruled surface A surface is a ruled surface if at any point in the
cg surface, one or more straight lines exist lying
cg entirely within the surface. Planes, cylinders
cg (circular, elliptic and hyperbolic), and cones
cg (circular and elliptic) are singly ruled surfaces.
cg Hyperbolic paraboloids and hyperboloids of one sheet
cg are doubly ruled surfaces.
cg
cg To see if a quadric surface is a ruled surface, use
cg command "arc" at any point in the surface, to find if
cg any of the surface curves through the point has zero
cg curvature, i.e. is a straight line.
cg
cg A quadric surface may be defined by:
cg F(x,y,z) = QC + 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,
cg with the normal vector:
cg N(x,y,z) = (NX, NY, NZ)
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 For a point P = (PX, PY, PZ) in the surface, and the
cg vector V = (VX, VY, VZ), the point P + f * V is also
cg in the surface for any value of f, only if
cg VX * NX + VY * NY + VZ * NZ = 0
cg and if
cg QXY * VX * VY + QYZ * VY * VZ + QZX * VZ * VX +
cg QXX * VX^2 + QYY * VY^2 + QZZ * VZ^2 = 0
cg Since the vector V has arbitrary magnitude, only two
cg cases need be considered: VZ = 0, and VZ = 1.
cg The equations above can then be solved for VX and VY.
cg The equation of the embedded line is P' = P + f * V.
cg run A single session of use of the GEOM code.
S-------------------------------------------------------------------------------
cc s A synonym for variable.
cg Saddle In the output display, indicates a saddle point in a
cg quadric surface, which must be a hyperbolic
cg paraboloid or a hyperboloid of one sheet.
cc saddle A synonym for hyperb.
cg saddle See "saddle points".
cg saddle points Saddle points are points on a surface at which the
cg curvature is negative, zero and positive, depending
cg on the direction in the surface.
cg The only quadric surfaces with saddle points are
cg hyperboloids of one sheet and hyperbolic paraboloids,
cg on both of which every point is a saddle point.
cg To find saddle points with normal vectors in the
cg x, y and z directions, use command "quadric".
cg To find saddle points with normal vectors in any
cg specified direction, use command "extrema".
cg To find the directions and magnitudes of zero and
cg extreme curvatures at a specified point on a surface,
cg use command "arc".
cg To fit the vertices, edges and center point of a
cg nonplanar quadrangle with a hyperbolic paraboloid,
cg use command "hyperb".
cg sample A sample is an event/object/value, randomly selected
cg from a probability distribution function (pdf).
cg See "random values", "random points",
cg "random vectors".
cg See commands "bin", "pdf", "spin".
cc sample Command to randomly sample one or more events and/or
cc objects and/or values from a probability distribution
cc function (pdf). See commands "bin", "pdf", "spin".
cc
cc Command "sample" relates to objects: pdf, symbol,
cc variable.
cc
cc Use command "spin" to spin the random number
cc sequence.
cc
cc Forms include:
cc
ccin help sample
ccin sample [help]
cc
cc Display the command options.
cc
ccin sample PDFNAME [NSAMP|1]
cc
cc Randomly sample from probability distribution
cc function (pdf) PDFNAME, to obtain NSAMP
cc events/objects/values. If specified, NSAMP must be
cc positive. Otherwise NSAMP defaults to 1.
cc Only the first 26 samples will be displayed on
cc the user's terminal. All will be displayed in
cc the current output file. If two or more values are
cc sampled, the total, the minimum and maximum, the
cc mean, the standard deviation and the expected total
cc are displayed.
cc
ccin sample PDFNAME NSAMP [norepl|repl]
cc
cc Randomly sample from probability distribution
cc function (pdf) PDFNAME, to obtain NSAMP
cc events/objects/values [without replacement].
cc NSAMP must be positive. If norepl is not
cc specified, sampling will be with replacement.
cc Only the first 26 samples will be displayed on
cc the user's terminal. All will be displayed in
cc the current output file. If two or more values are
cc sampled, the total, the minimum and maximum, the
cc mean, the standard deviation and the expected total
cc are displayed.
cc
ccin sample PDFNAME NSAMP [norepl,repl] [NSET|1]
cc
cc Randomly sample from probability distribution
cc function (pdf) PDFNAME, to obtain NSAMP
cc events/objects/values, for each of [NSET|1] sets of
cc samples, [without, with] replacement.
cc NSAMP must be positive. If specified, NSET must be
cc positive. If not specified, NSET defaults to 1.
cc Option "norepl" means sample without replacement
cc (no resampling of the same bin) while all NSET
cc sets of NSAMP samples are completed, and the total
cc number of samples, NSAMP * NSET, can not exceed the
cc number of probability bins in pdf PDFNAME. See
cc "reorder". Only the first 26 samples will be
cc displayed on the user's terminal. All
cc samples will be displayed in the current output
cc file. If two or more values are sampled, the
cc total, the minimum and maximum, the mean, the
cc standard deviation and the expected total are
cc displayed.
cc
cc Synonyms: [help, h].
cc save An option in command "sudoku" to save the current
cc assignments of digits to cells. They can be
cc restored later with option "reset".
cg save To save screen contents, see "record of session".
cg saved See "saved input lines".
cg saved See "saved input lines".
cg saved input lines
cg A history or list of all input lines in the current
cg session is saved, except that commands "call",
cg "goto", "i", "in", "indo", "input", "r", "rd",
cg "read", "redo", "return", "rtn", "u" and "undo
cg are converted to comments, preceded by "! ", and
cg limited to no more than 78 characters of the
cg original command. The latter commands are not
cg needed to reproduce the output from the problem.
cg The latter commands are preceded by the character
cg "<" in the current output file, initially geom_hsp.
cg
cg These input lines saved in memory may be
cg displayed or executed again with command:
cg
cg input command [options]
cg
cg The maximum storage space is now 2000 lines, so the
cg last 2000 input lines may be saved. Therefore,
cg input line saved in memory 2000 + N will write over
cg input line saved in memory N.
cg
cg Commands of type "input", "return", "indo", "redo",
cg and "undo" are saved only as comments in the command
cg summary file geom_cmd, because of the difficulty of
cg avoiding ambiguous or infinite input chains.
cg
cg However, any such command skipped because of being
cg in an "if-endif" block that was not executed is left
cg in the input lines saved in memory. This allows
cg blocks of saved input lines to be repeated until the
cg "if" condition is satisfied.
cg The output file provides a record of the source of
cg all input lines.
cg
cg See "command summary", "geom_cmd".
cc SAX The semiaxis of an aligned ellipsoid in the x
cc direction. Used in command "ellipsoid". May be
cc integer, floating point or a variable.
cc SAY The semiaxis of an aligned ellipsoid in the y
cc direction. Used in command "ellipsoid". May be
cc integer, floating point or a variable.
cc SAZ The semiaxis of an aligned ellipsoid in the z
cc direction. Used in command "ellipsoid". May be
cc integer, floating point or a variable.
cc sc A synonym for scale.
cg scalar A variable having only a name and a value.
cg See "symbol", "integer variable",
cg "floating point variable".
cg scalar product See "dot product".
cg scale See "equally-tempered", "musical scale".
cg scale 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 scaled with command
cg "scale", after creating a scaling operator using
cg command "operator".
cg Triangles, regular polygons, regular polyhedrons, and
cg tetrahedrons may be scaled by scaling the points used
cg to create them. This may be easier if the points
cg are put into clusters. Zones may be scaled by
cg scaling the quadric surfaces used to create them.
cg Annular disks may be scaled by redefining then, after
cg scaling the center point, normal vector, inner and
cg outer radius.
cg
cg Scaling may be uniform (in the direction of a
cg specified point, radial (perpendicular to a specified
cg axis), or linear ( parallel to a specified axis),
cg each with a specified scale factor.
cg See "scale factor".
cg Uniform scaling may be used to convert between
cg different length units, such as centimeters, inches,
cg feet and meters.
cg
cg Command "operator OPNAME" may be used to display
cg a scaling operator OPNAME and its inverse, the scale
cg factor, the scaling direction for linear scaling, and
cg the scaling axis for radial scaling.
cg
cg A family of quadric surfaces may be created by
cg starting from a base quadric surface, and repeatedly
cg scaling with a specified scaling operator and
cg invariant point. See command "quadric" with option
cg "scale".
cg
cg See "scale a vector".
cc scale Command to scale one or more points, clusters, vectors,
cc or quadric surfaces (including planes, spheres,
cc circular cylinders, circular cones, hyperbolic
cc paraboloids, ellipsoids, axially symmetric quadric
cc surfaces and general quadric surfaces), using the
cc original or inverse (option "-") of a scaling
cc operator previously created with command "operator"
cc with option "scale" = "uniform", "radial" or
cc "linear".
cc The scaling is relative to an invariant point PINV,
cc which defaults to the origin if not specified.
cc
cc Command "scale" relates to objects: axisym, cluster,
cc cone, cylinder, ellipsoid, hyperb, operator, plane,
cc point, sphere, symbol, vector.
cc
cc Forms include:
cc
ccin help scale
ccin scale [help]
cc
cc Display the command options.
cc
ccin scale point PNAME [-] OPNAME [PINV]
cc
cc Scale point PNAME with [inv] operator OPNAME,
cc and invariant point PINV.
cc
ccin scale cluster CLNAME [-] OPNAME [PINV]
cc
cc Scale cluster CLNAME with [inv] operator OPNAME,
cc and invariant point PINV.
cc
ccin scale point all [-] OPNAME [PINV]
cc
cc Scale all points with [inv] operator OPNAME,
cc and invariant point PINV.
cc
ccin scale vector VNAME [-] OPNAME [PINV]
cc
cc Scale vector VNAME with [inv] operator OPNAME,
cc and invariant point PINV.
cc
ccin scale vector all [-] OPNAME [PINV]
cc
cc Scale all vectors with [inv] operator OPNAME,
cc and invariant point PINV.
cc
cc WARNING: the following two commands may change an
cc axially symmetric quadric surface to an axially
cc unsymmetric quadric surface, if the scaling operator
cc is linear or radial, unless the scaling axis is
cc through the center of a sphere being scaled, or
cc parallel to the symmetry axis of any other axially
cc symmetric quadric surface being scaled.
cc
ccin scale QTYPE all [-] OPNAME [PINV]
cc
cc Scale all quadric surfaces of type QTYPE (plane,
cc sphere, cylinder, cone, hyperb, ellipsoid, axisym,
cc quadric), with [the inverse of] operator OPNAME,
cc with invariant point PINV.
cc
ccin scale QTYPE QNAME [-] OPNAME [PINV]
cc
cc Scale the quadric surface of type QTYPE (plane,
cc sphere, cylinder, cone, ellipsoid, hyperb, axisym,
cc quadric) with name QNAME, with [the inverse of]
cc operator OPNAME, with invariant point PINV.
cc
cc Synonyms: [cylinder, cyl],
cc [ellipsoid, el, ell, ellipse], [help, h],
cc [hyperb, saddle], [plane, pl], [point, p, pnt, pt],
cc [quadric, q, quad], [sphere, sph], [vector, v, vect].
cc scale An option in command "operator", to create a tensor
cc operator that performs uniform scaling in 3-D space,
cc by factor RATIO, with the invariant point at the
cc origin. See "RATIO", for a warning about truncation
cc error.
cc See "linear", "radial", "uniform".
cc Synonyms: [scale, uniform].
cc scale An option in command "point", to create a family of
cc points by repeated scaling of a base point, using a
cc specified scaling operator and a specified invariant
cc point.
cc
cc This command option can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc scale An option in command "vector", to create a family of
cc vectors by repeated scaling of a base vector, using a
cc specified scaling operator and a specified invariant
cc point.
cc
cc This command option can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc scale An option in commands "plane", "sphere", "cylinder",
cc "cone", "ellipsoid", "axisym", "quadric", to create a
cc family of quadric surfaces by repeated scaling of a
cc base quadric surface, using a specified scaling
cc operator and a specified invariant point.
cc See command "operator", options "axial", "linear",
cc "scale".
cc See "concentric", "move", "nest", "parallel",
cc "rotate".
cc
cc This command option can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc
cc WARNING: scaling may change an axially symmetric
cc quadric surface to an axially unsymmetric quadric
cc surface, if the scaling operator is linear or radial,
cc unless certain symmetry conditions are met.
cg scale See "scale a vector", "scale factor".
cg scale a vector To multiply the length of a vector VNAME by factor
cg RATIO, without changing its direction, use one of the
cg two equivalent commands:
cg
cg vector VNAME rel RATIO
cg vector VNAME * RATIO
cg scale factor A scaling operator OPNAME, specified with command
cg "operator", has a uniform, radial or linear scale
cg factor associated with it, which may be displayed
cg with command "operator OPNAME".
cg scale.mac A GEOM input file to generate and display the frequency
cg ratios for one octave, and absolute frequencies for
cg eight octaves (from 27.5 to 7040 Hz) of an
cg equally-tempered musical scale.
cg The frequency of the musical note Concert A (A4),
cg is 440 Hz. Middle C (C4) is 523.2511306012 Hz.
cg scaling See "scale".
cg scatter angle In a random walk, the scatter angle is the angle
cg between the initial and final direction vectors of
cg the point being walked. In the diffusion limit of
cg a walk of many mean free paths, the direction vectors
cg are not correlated.
cg See command "walk".
cc SCENH In command "plot", option "center", the horizontal
cc coordinate of the plot center.
cc SCENV In command "plot", option "center", the vertical
cc coordinate of the plot center.
cg screen To save all screen contents, see "record of session".
cg search See "search geom_base", command "search".
cc search Command to search for specified names or values among
cc objects, marker texts, symbol replacements, and alias
cc replacements, including full names, stems and
cc subscripts, and for specified values of variables,
cc big integers, the coordinates of points and vector
cc bound points, and the components of vectors.
cc See "named objects".
cc If any of the specified names contain blanks, use
cc a non-blank delimiter. See command "delimit".
cc
cc See "TOL" for the method of comparing floating point
cc values.
cc
cc See "character set" for the ASCII characters which
cc may be used in command "search".
cc
cc Command "search" 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 search
ccin search [help]
cc
cc Display the command options.
cc
ccin search OBJNAME1 OBJNAME2 OBJNAME3 ...
cc
cc Display all objects, marker texts, symbol
cc replacements and alias replacements with names,
cc name stems or subscripts the same as OBJNAME1,
cc OBJNAME2, OBJNAME3, ..., or if any of the latter
cc are a single character, beginning with that
cc character.
cc
cc Display all variables with numerical values
cc equal to OBJNAME1, OBJNAME2, OBJNAME3, ....
cc
cc Display all points and vector bound points with
cc coordinates equal to OBJNAME1, OBJNAME2, OBJNAME3,
cc ..., in the current coordinate system.
cc
cc Display all vectors with components equal to
cc OBJNAME1, OBJNAME2, OBJNAME3, ..., in the current
cc coordinate system.
cc
cc Display all big integers with integer or floating
cc point values, or base 10 logarithms equal to
cc OBJNAME1, OBJNAME2, OBJNAME3, ....
cc
cc search all
cc
cc Display all objects with object names.
cc See "named objects".
cc
cc Synonyms: [help, h], [search, find].
cg search geom_base To find an entry, e.g. ENTRY, in this file using a
cg general search tool, specify the entry as
cg "cc ENTRY" for commands, or "cg ENTRY" for
cg other definitions or descriptions.
cg
cg To find the line preceding the first entry beginning
cg with a specified letter, "L" for example, saarch for
cg "L-----" (upper case).
cg
cg To set the environment for UNIX text editor VI,
cg to search for entries in this file:
cg
cg vi geom_base<return>
cg
cg Start VI, open this file.
cg
cg :set ic<return>
cg
cg Ignore case (optional).
cg
cg /^......ENTRY/<return>
cg
cg Search for entry "ENTRY" in this file, starting in
cg column 7 of each line.
cg
cg See "extract entries", "grep".
cc SECONDS In command "angles", the number of seconds of an angle
cc specified in degree, minutes and second units.
cc May be integer, floating point or a variable.
cc See "DEGREES", "MINUTES".
cg seconds To find the machine time charged to the problem, use
cg command "time".
cg seer The equivalent of a magic 8-ball, which may be asked
cg questions with GEOM using input file seer.mac.
cg selected Randomly selected.
cg semiaxis See "ellipse", "ellipsoid", "transverse", "conjugate".
cg semicolon The semicolon character, ";".
cg semitone See "equally-tempered".
cg separation For circles, spheres or circular cylinders, the surface
cg separation is the distance the first object would
cg have to be moved toward (positive) or away from
cg (negative) the second object, along the axis of their
cg proximal points, to become tangent. There are four
cg possibilities:
cg
cg d12 - r1 - r2 tangent externally
cg d12 + r1 + r2 tangent externally, opposite side
cg d12 - r1 + r2 tangent internally
cg d12 + r1 - r2 tangent internally, opposite side
cg
cg where d12 is the distance between centers, and r1 and
cg r2 are the radii of the two objects.
cg
cg For a plane and a sphere, the surface separation is
cg the distance along the axis through the center of the
cg sphere, in the direction of the normal vector of the
cg plane, from the plane to the nearest and furthest
cg points on the sphere.
cg
cg For a plane and a circular cylinder whose axis is
cg parallel to the plane, the surface separation is the
cg distance along the axis normal to the plane and to
cg the axis of the cylinder, from the plane to the
cg nearest and furthest points on the cylinder.
cg
cg See command "kiss RAD1 RAD2 RAD3 [RAD4]".
cg separator See ';", "delimiter".
cg separator line A separator line is displayed in the output file
cg following the display from each command, in columns
cg 1-80. It's format is 1 "#", 71 "=", and 8 "*"
cg characters.
cg sequence See "name sequence", "family", "series".
cg The ASCII order of keyboard characters is:
cg blank !"#$%&'()*+,-./ 0-9 :;<=>?@ A-Z [\]^_` a-z
cg {|}^_`{|}~
cg For the order of entries in this file, case is
cg ignored. See "character set".
cc serial An option in command "operator", to create a tensor
cc operator that performs one, two or three sequential
cc rotations around different major axes by specified
cc angles (counterclockwise, with the axis pointed at
cc the observer), in any order.
cc See "axial", "planar", "triple", "vector".
cc series An option in command "variable", to create a series of
cc variables, by recursively generating a linear
cc function of the value of the previous variable in the
cc series, starting from a specified variable. The
cc series may be arithmetic, geometric, or more general.
cc The values of each variable, and the sum of the
cc series, including the value of the base variable,
cc will be displayed.
cc
cc The values of the variables in the series are
cc generated from the single-term recursion rule
cc (n = 1, NUMVAR):
cc
cc t(n) = FADD + FMULT * t(n-1), equivalent to:
cc t(n) = FADD * Z(n) + FMULT^n * t(0).
cc
cc where t(0) is the value of variable VAR(1), and
cc Z(n) = (1 - FMULT^n) / (1 - FMULT).
cc
cc The sum of the terms, including t(0), are:
cc s(1) = t(0) + t(1), ...,
cc s(n) = FADD * ((n + 1) - Z(n+1)) / (1 - FMULT) +
cc Z(n+1) * t(0).
cc
cc An arithmetic series results if FMULT = 1.
cc
cc A geometric series results if FADD = 0.
cc If FADD = 0 and VAR(1) has a value of 1, a family of
cc integral powers of FMULT is created.
cc
cc To find FMULT, when t(1) and s(NUMVAR+1) are known,
cc and FADD = 0, use command "ratio".
cc
cc These commands can be replaced, using do loops.
cc See "do loop use", commands "do", "enddo".
cg series See "arithmetic series", "family of objects",
cg "series sum", "reciprocal series".
cg series sum Calculations of the sum of series may be made by using
cg command "input" to repeat blocks of input lines
cg previously entered from your terminal, or in
cg a specified file, which calculate the terms of the
cg series and increment the sum. Conditions for ending
cg the summation may be specified with command "if",
cg followed by a "input tty" command to change the input
cg medium to your terminal, and command "endif".
cg session See "record of session".
cc set A synonym for variable.
cg set See "character set".
cc setup A synonym for environment.
cg shape To change the shape of a quadric surface, use command
cg "scale".
cg Scaling parallel or perpendicular to an axis may
cg change the shape of nonplanar quadric surfaces.
cg shape See "array shape", macros "shape1-3.mac" and
cg "shape3-1.mac".
cg shape 1-D to 3-D To shape that part of a 1-D array VIN(N), N = 1, NMAX,
cg with indices N = N1 to N2, to a 3-D array
cg VOUT(I,J,K), with indices I from 1 to IMAX, J from
cg 1 to JMAX, and K from 1 to KMAX, use macro
cg shape1-3.mac, or use the following set of commands:
cg
cg do n = N1 N2
cg
cg s nm = n - 1
cg s jkmax = jmax * kmax
cg s nmokmax = nm / kmax
cg s nmokmax = 1 int nmokmax
cg
cg s i = nm / jkmax
cg s i = 1 int i
cg s j = nmokmax mod jmax
cg s k = nm mod kmax
cg
cg s i = i + 1
cg s j = j + 1
cg s k = k + 1
cg
cg copy s VIN(n) VOUT(i,j,k) ! Use one of these.
cg rename s VIN(n) VOUT(i,j,k) ! Use one of these.
cg
cg enddo
cg
cg To get a 2-D array, set KMAX = 1, omit index k.
cg shape 3-D to 1-D To shape that part of a 3-D array VIN(I,J,K),
cg I = 1, IMAX, J = 1, JMAX, K = 1, KMAX,
cg with indices I = I1 to I2, J = J1 to J2, and
cg K = K1 to K2, to a 1-D array VOUT(N), with index
cg N = 1, NMAX, where NMAX = IMAX * JMAX * KMAX,
cg use macro shape3-1.mac, or use the following set of
cg commands:
cg
cg s jkmax = JMAX * KMAX
cg do i = I1 I2
cg s im = i - 1
cg s imjkmax = im * jkmax
cg do j = J1 J2
cg s jm = j - 1
cg s jmkmax = jm * KMAX
cg do k = K1 K2
cg s n + k jmkmax imjkmax
cg rename s VIN(i,j,k) VOUT(n) ! Use one of these.
cg copy s VIN(i,j,k) VOUt(n) ! Use one of these.
cg enddo
cg enddo
cg enddo
cg
cg To use a 2-D array, set KMAX = 1, omit index k.
cn shape1-3.mac A macro file for shaping a 1-D array into a 3-D array.
cn shape3-1.mac A macro file for shaping a 3-D array into a 1-D array.
cg sheet A sheet is a single continuous surface, either bounded,
cg as in a sphere or an ellipsoid, or as in a brick with
cg zero thickness in one coordinate direction, or
cg unbounded, as in other quadric surfaces.
cg See "hyperboloid".
cg shell To create zones partially or completely bounded by
cg concentric cylindrical or spherical surfaces, use
cg command "cylinder" or "sphere", options
cg "concentric" or "scale".
cg shortcuts To reduce the amount of typing, use one or more of the
cg following shortcuts:
cg
cg Use the character ";" to put multiple commands on the
cg same input line.
cg
cg Use short synonyms of command words. See
cg "synonyms". See commands "help",
cg "help COMMAND", "COMMAND help", and "synonym".
cg Create your own synonyms with command "symbol".
cg
cg Use short aliases for partial or complete commands
cg that must be repeated verbatim. See "alias".
cg
cg Put blocks of commands that must be repeated into
cg external files. Use them with command
cg "input IN_FILE [options]". See "macro".
cg
cg Reuse previously used commands. The last ncmdm
cg (currently 2000) input lines are saved in memory, and
cg may be displayed and repeated with command
cg "input command [options]" or "i c [options]".
cg To repeat commands with arguments replaced or
cg incremented, use command "redo" or "indo",
cg respectively.
cg All commands needed to reproduce the current GEOM run
cg are saved in file geom_cmd, and may be displayed and
cg repeated with command "input geom_cmd [options]"
cg
cg Use the special character "+" or "-"in place of a
cg new object name, to use the name formed by
cg incrementing or decrementing the base name for that
cg object type. See commands "last", "increment names".
cg
cg Create a family of objects with one command.
cg See "family of objects".
cg
cg Use the mouse to cut and paste data from the window
cg display into new commands.
cg SI See "SI units".
cg SI units See "conv.mac", "conversion factors".
cg side A quadric surface has sides. The positive side is in
cg the direction of the normal vector at any point in
cg the surface.
cg side See "side measure", command "alias", "right side",
cg "left side".
cc side A synonym for edge in command "trig".
cc side Command to find, for a point and a quadric surface,
cc the side of the surface the point is on, the nearest
cc (proximal) point on the surface, the normal vector
cc of the family of surfaces at the point and at the
cc proximal point, the vector from the point to the
cc proximal point, and the vector and distance from the
cc point to the proximal point. Also, to find the
cc member of the same family of quadric surfaces, that
cc passes through the point.
cc
cc Command "side" relates to objects: axisym, cone,
cc cylinder, ellipsoid, hyperb, plane, point, quadric,
cc sphere, symbol.
cc
cc Forms include:
cc
ccin help side
ccin side [help]
cc
cc Display the command options.
cc
ccin side PNAME QNAME
cc
cc For point PNAME and quadric surface QNAME, find the
cc side of the surface the point is on, the nearest
cc (proximal) point on the surface, the normal vector
cc of the family of surfaces at the point and at the
cc proximal point, the vector from the point to the
cc proximal point, and the vector and distance from
cc the point to the proximal point.
cc
cc The family of quadric surfaces which includes QNAME
cc have the implicit equation F(x,y,z) = constant
cc (zero for QNAME), and the normal vector
cc N(x,y,z) = (NX, NY, NZ). See "normal".
cc Find and display F(PNAME) as "Side measure", zero
cc if point PNAME is on quadric surface QNAME.
cc Flag as "***ON SURFACE***" if on the surface.
cc Find and display N(PNAME) as "Normal pnx,pny,pnz".
cc Find and display the point on QNAME nearest PNAME
cc as "Proximal x,y,z" (there may be more than one
cc proximal point). Find and display N at each
cc proximal point as "Normal qnx,qny,qnz".
cc Find the vector from point PNAME to the proximal
cc point, and display as "Proximal dx,dy,dz". This
cc is parallel (side is negative) or antiparallel
cc (side is positive) to vector N at the proximal
cc point.
cc Display the distance from point PNAME to quadric
cc surface QNAME as "Exact distance".
cc
cc To check the result, try command "proximal".
cc
cc To get the equation for the quadric surface in
cc family QNAME that passes through point PNAME,
cc subtract the displayed "Side measure" from the
cc constant term QC of the implicit equation of
cc quadric surface QNAME.
cc
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
ccin side PNAME [all]
cc
cc Do as above for point PNAME and all quadric
cc surfaces.
cc
ccin side all QNAME
cc
cc Do as above for all points and quadric surface
cc QNAME.
cc
ccin side all [all]
cc
cc Do as above for all points and all quadric
cc surfaces.
cc
cc Notes:
cc
cc The minimum distance may be stored in variable
cc VARNAME with command:
cc variable VARNAME distance
cc
cc The proximal point found on the last quadric surface
cc may 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 Synonyms: [help, h], [proximal, prox].
cg side measure For a quadric surface F(x,y,z) = 0, the measure of
cg which side of the surface a point PNAME = (x1,y1,z1)
cg is on is S = F(x1,y1,z1). The side is positive if
cg the point PNAME is on the same side of the surface
cg as the direction of the normal vector N = grad F.
cg An estimate of the distance of point PNAME from the
cg surface is S / |N|.
cg The minimum distance from point PNAME to the quadric
cg surface is calculated when command "side" or
cg "distance" is used.
cc SIGMA The decay constant of an exponentially varying
cc relative differential probability in a probability
cc bin specified with command "bin". May be zero or
cc negative. In the latter case, the relative
cc differential probability increases with an increasing
cc value of the random variable.
cc
cc For relative differential probabilities PL at VRANL
cc and PR at VRANR, SIGMA = log (PL / PR) / DVAL,
cc where DVAL = VRANR - VRANL.
cc sign A variable name used in a command in place of an
cc integer or floating point value may be prefixed
cc (without a delimiter) with a minus sign or a plus
cc sign, to indicate use of the negative of the actual
cc variable value or use of the actual variable value,
cc respectively. I.e., the plus sign has no effect.
cc sign An option in commands "icalc" and "variable", argument
cc FUNCTION, to indicate the sign function.
cc In command "icalc", M sign N means isign (M, N)
cc (M with the sign of N).
cc In command "variable", A sign B means sign (A, B)
cc (A with the sign of B).
cf sign The sign function. Example: z = sign (x, y) means
cf that z has the magnitude of x, and the same sign
cf as y.
cf See commands "icalc", "variable", argument FUNCTION.
cg significant See "significant figures".
cg significant figures
cg Calculations in GEOM may be done with integers
cg (stored in a 64-bit format on the DEC machines),
cg big integers (stored with one decimal digit per
cg machine word), and with floating point (stored in
cg a 64-bit format on the DEC machines).
cg
cg 64-bit integers are limited to a positive decimal
cg value of 1152921504606846975 (FFFFFFFFFFFFFFF hex).
cg GEOM only recognizes 64-bit decimal integer input
cg with up to 18 decimal digits.
cg
cg Big integers are limited to a positive decimal
cg value of 10^1000.
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 or results
cg with digits to the right of the decimal point by
cg multiplying one or more of the numbers by a
cg sufficiently large power of ten, then moving the
cg decimal point in the final result of the calculation.
cg For example, an accurate inverse of integer N may be
cg obtained by dividing 10^1000 by N, then moving the
cg decimal point 1000 places to the left in the result.
cg
cg Floating point numbers are limited to an accuracy
cg of about 16 digits in the mantissa, with an exponent
cg within the range from -308 to 308.
cg GEOM stores floating point variables as integers if
cg they are in the range from 0 to 10^16, and differ
cg from an integer by less than TOL times their value.
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 To use format 1pe23.15 for the display of variables,
cg execute command "tol 0" first.
cg
cg To use format 1pe23.15 for the display of any
cg internal variable, use command "debug".
cg
cg To use format 1pe25.17 for the display of any
cg variable, use command "base", with NBASE = 10.
cg
cg The UNIX text editor VI may be used to reduce the
cg number of significant figures in the output.
cg For example, to remove the final seven figures after
cg the decimal point of floating point output:
cg
cg :%s/.......E/E/g
cg silhouette See "outline".
cg simple See "simple plane".
cg simple plane A simple plane is a flat surface or a quadric surface
cg for which the standard equation is x = 0.
cg The general quadric surface equation is:
cg QC + QX * x + QY * y + QZ * z = 0.
cg This is a limiting case of either a parabolic
cg cylinder, a hyperbolic paraboloid or an elliptic
cg paraboloid.
cg simultaneous To solve three simultaneous equations in three
cg unknowns, when all of the equations are linear,
cg use command "solve".
cg To solve three simultaneous equations in three
cg unknowns, when one or more of the equations is
cg quadratic, use command "quadric" to define each
cg equation as a quadric surface in the coordinate space
cg of the three unknowns, then use command "triple" to
cg search for solutions.
cc sin An option in command "variable", argument FUNCTION,
cc to indicate the trigonometric sine function.
cc A sin B means A * sin (B).
cf sin The trigonometric sine function. Example: y = sin (x)
cf means y is the sine of x. x must be in radians.
cg single In output from commands "root" and "roots", indicates
cg a root of order 1 of a polynomial equation.
cc sinh An option in command "variable", argument FUNCTION,
cc to indicate the hyperbolic sine function.
cc A sinh B means A * sinh (B).
cf sinh The hyperbolic sine function. Example: y = sinh (x)
cf means y = (1/2) * (e^x - e^(-x)).
cc size An option in command "mesh", to display or specify the
cc size of the logical mesh.
cc size An option in command "plot", to display or specify the
cc size of the plot array.
cg size To change the size of an object, use command "scale".
cg For the maximum numbers of various objects, see
cg "array sizes", "maximum number" or use command
cg "tables".
cc sizes A synonym for tables.
cg skew A line or vector not coincident with or parallel to
cg another line or vector. See commands
cg "cross", "distance", option "line", "dot",
cg "vector", option "angle".
cg slab To create zones partially bounded by two parallel
cg planes, use command "plane", options "parallel" and
cg "scale", and command "zone".
cg See "shell".
cg
cg See commands "brick" and "point".
cg slash The division or slash character, "/".
cg slice A slice is a cut through a 3-D geometric object with
cg a plane parallel to a major plane (x, y or z),
cg defining all intersections of the object with the
cg cutting plane.
cc slice Command to find the intersection of a major plane with
cc a quadric surface or the quadric surfaces bounding a
cc zone or to find points in a quadric surface.
cc See command "quadric QNAME slice ...".
cc
cc Command "slice" relates to objects: axisym, cone,
cc cylinder, ellipsoid, hyperb, plane, quadric, sphere,
cc symbol, variable, zone.
cc
cc Forms include:
cc
ccin help slice
ccin slice [help]
cc
cc Display the command options.
cc
ccins slice quadric QNAME [x,y,z|x] [VAL|0]
ccins [[y,z,x] VMIN DV VMAX] (all on one line)
cc
cc Slice quadric surface QNAME with the major plane
cc [x,y,z|x] = [VAL|0], and display any resulting
cc quadric curve, and any minimum, maximum or
cc intersection points it may have. Optionally, find
cc all points on any such quadric curve with the
cc coordinate axis [y, z, x] values VMIN (DV) VMAX,
cc where that coordinate axis is different from that
cc of the slice plane, and display them in a form
cc suitable for creating new points by cutting and
cc pasting, or be editing the output file into a new
cc input file.
cc
ccin slice zone ZNAME [[x,y,z|x] [[VAL|0]]]
cc
cc Slice the quadric surfaces bounding zone ZNAME with
cc the major plane [x,y,z|x] = [VAL|0], and display
cc any resulting quadric curves, and any minimum,
cc maximum and intersection points they may have.
cc
cc Synonyms: [help, h], [quadric, q, quad],
cc [slice, cut], [x, X], [y, Y], [z, Z], [zone, z, zn].
cc slice An option in command "quadric", to find a quadric curve
cc that is the intersection of a quadric surface with a
cc plane parallel to a major plane, and which defines an
cc infinite cylinder with an axis perpendicular to that
cc plane.
cc SMAX In command "plot limits", the maximum value of the
cc horizontal or vertical axis coordinate.
cc SMIN In command "plot limits", the minimum value of the
cc horizontal or vertical axis coordinate.
cg solution See "triangle solution", command "trig".
cc solve An option in command "sudoku", to try to solve a
cc 9 x 9 Sudoku puzzle, give the current set of digits
cc assigned to the cells.
cg solve See "triangle solution", command "trig".
cc solve Command to solve three simultaneous linear equations
cc in three unknowns or two simultaneous linear
cc equations in two unknowns.
cc
cc Command "solve" relates to objects: symbol, vector.
cc
cc Forms include:
cc
ccin help solve
ccin solve [help]
cc
cc Display the command options.
cc
ccin solve VA VB VC VD
cc
cc Solve for the three unknowns x, y and z in the
cc three simultaneous linear equations:
cc A1 * x + A2 * y + A3 * z = D1,
cc B1 * x + B2 * y + B2 * z = D2,
cc C1 * x + C2 * y + C3 * z = D3,
cc where VA, VB, VC and VD are the vectors
cc VA = (A1, A2, A3), VB = (B1, B2, B3),
cc VC = (C1, C2, C3), VD = (D1, D2, D3).
cc
cc To solve for the two unknowns x and y in two
cc simultaneous linear equations, set
cc C1 = C2 = A3 = B3 = D3 = 0, C3 = 1.
cc
cc Synonyms: [help, h].
cg solve triangle See "triangle solution", "trig".
cg sort To sort objects into increasing, decreasing or random
cg order of their ASCII or integer names (left-adjusted
cg and right-filled with blanks), and any ASCII or
cg integer subscripts, use command "sort". ASCII names
cg or subscripts are sorted in ASCII order. Integer
cg names or subscripts are sorted in numerical order.
cg
cg To get numerical sorting without a completely
cg integer name or without parenthesized explicit
cg integer subscripts, make all names the same length,
cg ending with digits, e.g., p001, p002, ..., p099,
cg p100, p101, ...
cg
cg The order of available keyboard characters is:
cg blank !"#$%&'()*+,-./ 0-9 :;<=>?@ A-Z [\]^_` a-z
cg {|}^_`{|}~
cg
cg See "character set".
cc sort Command to sort aliases, markers, variables, vectors,
cc tensor operators, points, clusters, meshes, lines,
cc triangles, regular polygons, annular disks, planes,
cc quadric surfaces, regular polyhedrons, tetrahedrons,
cc bricks, zones, probability bins, probability
cc distribution functions (pdfs) and symbols,
cc by name.
cc
cc Except for object names that are integers or contain
cc parenthesized explicit integer subscripts, sorting is
cc optionally into increasing, decreasing or random
cc ASCII order of the object names (left-adjusted,
cc right-filled with blanks).
cc
cc Integer object names are sorted according to their
cc integer values, with options as above.
cc
cc In the case of object names containing parenthesized
cc explicit integer subscripts, each subscript is tested
cc separately, and if an explicit integer, optionally
cc put into increasing, decreasing or random numerical
cc order, for each unique set of preceding characters.
cc
cc The order of available keyboard characters is:
cc blank !"#$%&'()*+,-./ 0-9 :;<=>?@ A-Z [\]^_` a-z
cc {|}^_`{|}~
cc See "character set".
cc
cc Command "sort" relates to objects: alias, axisym,
cc bin, brick, cluster, cone, cylinder, disk, ellipsoid,
cc hyperb, line, marker, mesh, operator, pdf, plane,
cc point, polygon, polyhedron, quadric, sphere, symbol,
cc tetrahedron, triangle, variable, vector, zone.
cc
cc Forms include:
cc
ccin help sort
ccin sort [help]
cc
cc Display the command options.
cc
ccin sort all [+,-,?|+]
cc
cc Sort all objects and mesh points in ascending,
cc descending, or random (default ascending) order,
cc and display names.
cc
ccin sort OBJTYPE [+,-,?|+]
cc
cc Sort objects of the type indicated by OBJTYPE and
cc display names. OBJTYPE may be alias, marker,
cc variable, vector, operator, point, cluster,
cc line, triangle, polygon, disk, quadric, polyhedron,
cc tetrahedron, brick, zone, bin, pdf or symbol.
cc Sort in ascending, descending or random (default
cc ascending) order.
cc
ccin sort quadric [+,-,?|+]
cc
cc Sort all quadric surfaces and display names
cc (includes planes, spheres, circular cylinders,
cc circular cones, hyperbolic paraboloids, ellipsoids,
cc axially symmetric quadric surfaces, and general
cc quadric surfaces).
cc Sort in ascending, descending or random (default
cc ascending) order.
cc
cc sort QTYPE [+,-,?|+]
cc
cc Sort all quadric surfaces and display the names of
cc quadric type QTYPE (plane, sphere, cylinder, cone,
cc hyperb, ellipsoid, axisym or quadric).
cc Sort in ascending, descending or random (default
cc ascending) order.
cc
ccin sort mesh [+,-,?|+]
cc
cc Sort all mesh points and display names.
cc Sort in ascending, descending or random (default
cc ascending) order.
cc
cc Synonyms: [?, random], [alias, al, a], [brick, br],
cc [cluster, cl], [cylinder, cyl], [disk, disc, dk],
cc [ellipsoid, el, ell, ellipse], [help, h],
cc [hyperb, saddle], [line, l, ln], [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 [sphere, sph], [symbol, sym, symb],
cc [tetrahedron, tet, tetra], [triangle, tri],
cc [variable, s, set, var], [vector, v, vect],
cc [zone, z, zn].
cg space See "coordinate system".
cg spatial See "spatial derivatives", "special characters".
cg spatial derivatives
cg The implicit equation for a quadric surface is:
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), whose
cg components are the spatial partial derivatives of F:
cg NX = dF/dx =
cg QX + 2 * QXX * x + QXY * y + QZX * z
cg NY = dF/dy =
cg QY + QXY * x + 2 * QYY * y + QYZ * z
cg NZ = dF/dz =
cg QZ + QZX * x + QYZ * y + 2 * QZZ * z
cg
cg The first and second partial derivatives of each
cg spatial coordinate with respect to the other spatial
cg coordinates (in the surface) are:
cg dx/dy = -NY / NX, dx/dz = -NZ / NX
cg dy/dz = -NZ / NY, dy/dx = -NX / NY
cg dz/dx = -NX / NZ, dz/dy = -NY / NZ
cg
cg d(dx/dy)/dy = (QXY * NY - 2 * QYY * NX) / NX^2
cg d(dx/dy)/dz = (QZX * NY - 2 * QYZ * NX) / NX^2
cg d(dx/dz)/dz = (QZX * NZ - 2 * QZZ * NX) / NX^2
cg
cg d(dy/dz)/dz = (QYZ * NZ - 2 * QZZ * NY) / NY^2
cg d(dy/dz)/dx = (QXY * NZ - 2 * QZX * NY) / NY^2
cg d(dy/dx)/dx = (QXY * NX - 2 * QXX * NY) / NY^2
cg
cg d(dz/dx)/dx = (QZX * NX - 2 * QXX * NZ) / NZ^2
cg d(dz/dx)/dy = (QYZ * NX - 2 * QXY * NZ) / NZ^2
cg d(dz/dy)/dy = (QYZ * NY - 2 * QYY * NZ) / NZ^2
cg
cg The change in one coordinate due to changes in the
cg other two coordinates (in the surface) may be found
cg by solving the following quadratic equation for
cg dx, dy or dz:
cg (NX * dx + NY * dy + QXX * dx^2 + QYY * dy^2) +
cg (NZ + QZX * dx + QYX * dy) * dz + QZZ * dz^2 = 0
cg (convenient forms of the equation for dx and dy may
cg be obtained by permuting x, y and z).
cg Note that there may be 0, 1 or 2 solutions.
cg special See "special characters".
cg special characters
cg Some characters have special meaning in commands.
cg See the entries for the following characters:
cg
cg (blank) ! " # $ % & ' * ^ + - . / ; < = > ? @ C ^ c
cg
cg See "character set".
cg spectrum See "Planck spectrum", "Wien spectrum".
cg specular See "reflection".
cg specular reflection
cg See "reflection".
cg speed In a display of data resulting from command
cg "accelerate", the magnitude of the velocity vector.
cc sph A synonym for sphere.
cc sph A synonym for spherical in command "coordinate".
cc SPH(1) In command "sphere", the name of a base sphere, used to
cc create a family of spheres.
cg sphere A sphere is a quadric surface in 3-D space,
cg described by an implicit quadric equation.
cg A sphere is symmetric around a central point, and
cg is a limit case of an ellipsoid.
cg
cg The standard forms of the implicit equations for
cg spheres (with all coefficients positive) are:
cg
cg Real sphere:
cg - 1 + QXX * (x^2 + y^2 + z^2) = 0
cg or - rsph^2 + x^2 + y^2 + z^2 = 0
cg Imaginary sphere:
cg 1 + QXX * (x^2 + y^2 + z^2) = 0
cg or rsph^2 + x^2 + y^2 + z^2 = 0
cg
cg To find the distance from a point to a sphere,
cg use command "distance", "proximal" or "side".
cg
cg To find the distance from a sphere to a point, use
cg command "side" or "distance". To find the distance
cg and proximal point, and/or the intersections
cg of a sphere and a line, a triangle, plane, another
cg sphere or a circular cylinder, use command
cg "distance".
cg
cg To find the distance between a sphere and any other
cg quadric surface, first find the distance from the
cg point at the center of the sphere to the quadric
cg surface, using command "distance", "proximal" or
cg "side", then subtract the radius of the sphere.
cg A negative distance indicates overlap.
cg
cg To find two spheres, each tangent to four mutually
cg tangent spheres with specified radii, use command
cg "kiss". See "tangent spheres".
cg
cg The following commands relate to spheres:
cg accelerate, arc, axisym, copy, debug, delete,
cg distance, extrema, help, invert, last, list, move,
cg operator, point, project, proximal, quadric, reflect,
cg rename, repack, rotate, scale, search, side, slice,
cg sort, sphere, synonym, symbol, tables, track, triple,
cg vector, zone.
cg
cg Also see "rsph sphere".
cc sphere Command to display or create one or more spheres
cc (except that linear or radial scaling may create
cc ellipsoids). Creating a sphere or ellipsoid replaces
cc any existing quadric surface having the same name.
cc The display for a sphere includes the display for a
cc general quadric, plus the radius, area and volume.
cc The normal vectors will be pointed radially outward.
cc See "quadric" for other options.
cc See "do loop use".
cc
cc Command "sphere" relates to objects: axisym,
cc operator, point, quadric, sphere, symbol, variable,
cc vector.
cc
cc Forms include:
cc
ccin help sphere
ccin sphere [help]
cc
cc Display the command options.
cc
ccin sphere [all,list]
cc
cc Display all spheres (short display).
cc
ccin sphere list SPHNAME1 SPHNAME2 SPHNAME3 ...
cc
cc Display spheres SPHNAME1, SPHNAME2, SPHNAME3,
cc ..., with or without subscripts (short display).
cc
ccin sphere SPHNAME
cc
cc Display sphere SPHNAME (long display).
cc
cc In the following commands, use "+" or "-" instead of
cc SPHNAME or SPH(2) to form the name by incrementing or
cc decrementing the base name for spheres.
cc See command "last", "increment names".
cc
ccin sphere SPHNAME = PCEN RADIUS
cc
cc Create sphere SPHNAME, centered at point PCEN, with
cc positive radius RADIUS.
cc
ccin sphere SPHNAME fit PAXIS VAXIS PNAME1 PNAME2
cc
cc Create sphere SPHNAME, centered on the axis through
cc point PAXIS in the direction of vector VAXIS, and
cc with its surface through the two points PNAME1 and
cc PNAME2.
cc
ccin sphere SPHNAME point PNAME1 PNAME2 PNAME3 PNAME4
cc
cc Create sphere SPHNAME through the four points
cc PNAME1-4, which must not be coincident, collinear
cc or coplanar.
cc
ccin sphere SPHNAME concentric NUMSPH INC PCEN RAD1 RAD2
cc
cc Create a family of NUMSPH concentric spheres
cc SPHNAME, ..., with names incremented by INC
cc characters, centered at point PCEN, with radii
cc equally spaced from radius RAD1 to radius RAD2.
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 sphere SPH(2) move NUMSPH INC SPH(1) VMOVE
cc
cc Create a family of NUMSPH spheres SPH(2), ..., with
cc names incremented by INC characters, and spaced at
cc intervals of vector VMOVE, starting from sphere
cc SPH(1). See "increment names".
cc See command "quadric" for an equivalent set of
cc commands using subscripted names and a do loop.
cc
ccin sphere SPH(2) rotate NUMSPH INC SPH(1) OPNAME PINV
cc
cc Create a family of NUMSPH spheres SPH(2), ..., with
cc names incremented by INC characters, by rotating
cc the preceding sphere with operator OPNAME and
cc invariant point PINV, starting from sphere SPH(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 sphere SPH(2) scale NUMSPH INC SPH(1) OPNAME PINV
cc
cc Create a family of NUMSPH quadric surfaces SPH(2),
cc ..., with names incremented by INC characters, by
cc scaling the preceding quadric surfaces with
cc operator OPNAME and invariant point PINV, starting
cc from sphere SPH(1). WARNING: a linear or radial
cc scaling operator will produce 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: [concentric, conc], [help, h],
cc [move, mv, trans, translate], [rotate, rot],
cc [sphere, sph].
cc sphere An option in commands "copy", "delete", "last", "list",
cc "rename", "repack" and "sort", to perform the
cc specified operation on one or more spheres.
cc Synonyms: [sphere, sph].
cc sphere An option in command "project", to project onto the
cc surface of a sphere.
cg sphere See "sphere, circumscribed", "sphere, inscribed".
cg sphere arrays Spheres may be created as arrays with subscripted
cg names. See "subscript", "subscripted names",
cg commands "do" and "enddo".
cg sphere, circumscribed
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 "sphere, inscribed".
cg sphere, inscribed
cg 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 "sphere, circumscribed".
cg A sphere may also be externally tangent to four
cg other mutually tangent spheres. See command "kiss".
cg spheres A family of spheres may be created with command
cg "sphere", options "concentric", "move", "rotate" or
cg "scale" (with restrictions) or as follows:
cg
cg p pinv (options)
cg
cg Create an invariant point, if needed.
cg
cg sph SPH(1) (options)
cg
cg Create a base sphere.
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 SPH(1) SPH(2)
cg
cg Copy sphere SPH(1) to the first sphere.
cg
cg Repeat the following block of commands as many times
cg as needed to create the rest of the family of
cg spheres. See "shortcuts".
cg
cg mv sph SPH(1) vmove
cg
cg Move the base sphere by amount vmove, if needed.
cg
cg [invert,reflect,rotate,scale] sph SPH(1) opr [pinv]
cg
cg Invert, reflect, rotate or scale the base sphere
cg with tensor operator opr, if needed.
cg
cg cp sph SPH(1) +
cg
cg Copy the base sphere to the next sphere.
cc spherical An option in command "coordinate", to use a 3-D
cc coordinate system with the spherical radial,
cc azimuthal angle and polar angle coordinates
cc (rsph, theta, phi).
cc Specified with command "coordinate".
cc Synonyms: [spherical, sph].
cg spherical To create volume elements bounded by surfaces of a
cg spherical coordinate system, use command "brick".
cg To create zones partially or completely bounded by
cg spherical surfaces, use command "sphere", options
cg "concentric", "scale", and command "zone".
cg
cg To create arrays of points on surfaces of a
cg spherical coordinate system, see command
cg "point PMESH brick BRNAME mesh block [volume]".
cg spheroid A spheroid is a circular ellipsoid, for which two of
cg the three semiaxes are equal. If the remaining
cg semiaxis is larger, the spheroid is prolate.
cg Otherwise, the spheroid is oblate.
cg See "circular ellipsoid", "oblate spheroid",
cg "prolate spheroid".
cc SPHNAME The name of a sphere. May have up to 24 characters,
cc including any subscripts, and may be ASCII, integer,
cc or floating point.
cc No sphere name may be "+", "-", "all", "h", "help",
cc "list" or "thru", begin with "!" or contain ";".
cc Specified with command "sphere", "axisym" or
cc "quadric".
cc Also referred to as SPHNAME1, SPHNAME2, SPHNAME3,
cc ..., SPH(1), SPH(2), ...
cc
cc Spheres 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, rotate,
cc scale, search, side, slice, sort, sphere, track,
cc triple, vector.
cg spin See "spin the bottle", command "spin".
cc spin Command to spin the random number sequence.
cc
cc Command "spin" relates to objects: symbol, variable.
cc
cc Forms include:
cc
ccin help spin
ccin spin [help]
cc
cc Display the command options.
cc
ccin spin N
cc
ccin spin random
cc
cc Set the random number seed to a new value based on
cc the date and clock. This will make any results
cc based on random numbers not reproducible.
cc
cc Go past the next N random numbers in the random
cc number sequence.
cc
cc Synonyms: [help, h], [random, ?].
cg spin the bottle To randomly choose between K 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 icalc 1 ran K
cg
cg Randomly sample one of the integers from 1 to K.
cg spiral See "archimedean spiral".
cc split A synonym for end.
cc sqrt An option in command "variable", argument FUNCTION,
cc to indicate the real square root function.
cc A sqrt B means A * sqrt (B).
cf sqrt The real square root function. Arguments must not be
cf negative. Example: y = sqrt (x) means
cf y = x^(1/2) (real root only).
cf
cf To find the real or complex square roots of any
cf numerical value B, use command:
cf
cf roots B 0 -1
cg square bracket The left or right square bracket, "[" or "]".
cn square.mac A macro file for finding the smallest square that two
cn rectangles will fit in, with a 1 x A rectangle
cn horizontal in the lower right corner, and a 1 x B
cn rectangle diagonally from the lower left to upper
cn right, with its lowest corner to the left of the
cn 1 x A rectangle. Type:
cn
cn s %a = A
cn s %b = B
cn in square.mac #1 #2
cn
cn After displays from square.mac, choose a value
cn of cosine (theta), and type:
cn
cn s %costh = COSINE
cn in square.mac #3 #4
cg standard See "standard deviation", "standard input",
cg "standard output", "expected value".
cg standard deviation
cg See "mean", "expected value". See command "mean".
cg standard input The standard input medium is the user's terminal.
cg It may be redirected to file IN_FILE with the GEOM
cg execution line "geom IN_FILE" or
cg "geom IN_FILE OUT_FILE" or with command "input".
cg Do NOT use command "geom < IN_FILE", or prompts will
cg not be seen by the user, and the run will hang up.
cg See "GEOM execution", "input", "redirect input".
cg standard output The standard output medium is the user's terminal.
cg All output is also initially sent to file geom_hsp,
cg unless redirected to file OUT_FILE with the execution
cg line "geom IN_FILE OUT_FILE".
cg Do not use the execution line "geom > OUT_FILE" or
cg prompts for input data will not be displayed.
cg
cg When input is from file IN_FILE (not the user's
cg terminal), you may redirect the standard output from
cg the user's terminal to a file STDOUT by using the
cg execution line
cg "geom IN_FILE OUT_FILE > STDOUT", or
cg "geom IN_FILE OUT_FILE >! STDOUT".
cg Use the latter form to write over an existing file
cg STDOUT.
cg
cg See "GEOM execution", "input", "output",
cg "redirect output".
cg star The asterisk or star character, "*".
cg starting See "starting up".
cg starting up To begin using GEOM, look up entries "object types",
cg "commands" and "key words" in this file. Also see
cg "help" and command "define".
cg Also see "arithmetic", "distance", "intersection",
cg "Monte Carlo", "project", "symmetry ops", "track".
cg
cg Or just execute "geom", then type the problem title,
cg then type "help", and then use commands "help"
cg and "define" to find more information about GEOM.
cc stat A synonym for status.
cc statement label Commands saved in memory or in the command summary file
cc geom_cmd may be referred to either by their line
cc number or by their initial character string.
cc Comment lines may be used to provide unique initial
cc character strings for use with commands "input" or
cc "goto".
cc status Command to display certain environmental data.
cc
cc Forms include:
cc
ccin status [help]
cc
cc Display the problem title, the name and position of
cc the input file, the name of the output file,
cc the name of the command file, the word delimiter,
cc the coordinate system, the angle unit,
cc the tolerance limit, the integer mode for
cc command "icalc", the binary display option for
cc command "icalc", and the time charged to the run.
cc Also see command "plot status".
cc
cc Synonyms: [help, h], [status, stat].
cc status An option in command "plot", to display the current
cc plot parameters.
cc stddev An option in command "variable", to create a variable
cc with the standard deviation of the mean value of a
cc group of variables, as previously found with command
cc "mean".
cc stein A synonym for steiner.
cg Steiner See "Steiner vertex", "Steiner wall".
cc steiner Command to display the Steiner vertex of a triangle.
cc
cc Command "steiner" relates to objects: symbol,
cc triangle.
cc
cc Forms include:
cc
ccin help steiner
ccin steiner [help]
cc
cc Display the command options.
cc
ccin steiner TRNAME
cc
cc Display the attributes of triangle TRNAME,
cc including the Steiner vertex, its distances from
cc the vertices of the triangle, the sum of those
cc distances, and the relative vertex weights.
cc
cc Synonyms: [help, h], [steiner, stein]
cg Steiner vertex For a given triangle, with vertices A = (AX, AY, AZ),
cg B = (BX, BY, BZ), and C = (CX, CY, CZ), the Steiner
cg vertex is the point S = (SX, SY, SZ) within the
cg triangle for which the sum of the three lengths
cg SA + SB + SC is minimized.
cg
cg The Steiner vertex S = (SX, SY, SZ) satisfies:
cg
cg SA = sqrt ((SX-AX)^2 + (SY-AY)^2 + (SZ-AZ)^2))
cg SB = sqrt ((SX-BX)^2 + (SY-BY)^2 + (SZ-BZ)^2))
cg SC = sqrt ((SX-CX)^2 + (SY-CY)^2 + (SZ-CZ)^2))
cg
cg WA = SB * SC / (SA * SB + SB * SC + SC * SA)
cg WB = SC * SA / (SA * SB + SB * SC + SC * SA)
cg WC = SA * SB / (SA * SB + SB * SC + SC * SA).
cg
cg SX = WA * AX + WB * BX + WC * CX
cg SY = WA * AY + WB * BY + WC * CY
cg SZ = WA * AZ + WB * BZ + WC * CZ.
cg
cg If the angle at any vertex of the triangle is 120
cg degrees or more, the Steiner vertex is at that
cg vertex. Otherwise, the three lines from the
cg triangle vertices to the Steiner vertex meet at
cg angles of 120 degrees.
cg
cg If vertex A is at the origin, and vertex B is on the
cg x axis, and vertex C is in the x-y plane, then the
cg Steiner vertex is on a circle centered at x = BX / 2,
cg y = -BX * sqrt(3)/6, with a radius of sqrt(3)/3.
cg This circle passes thru vertices A and B.
cg
cg For the triangle above, with vertex angles a, b and
cg c, and edge lengths AB, BC and CA:
cg SA = (2 / sqrt(3)) * CA * AB * sin (120 - a) /
cg sqrt (CA^2 + AB^2 + 2*CA*AB*cos (120 - a)),
cg SB = (2 / sqrt(3)) * AB * BC * sin (120 - b) /
cg sqrt (AB^2 + BC^2 + 2*AB*BC*cos (120 - b)),
cg SC = (2 / sqrt(3)) * BC * CA * sin (120 - c) /
cg sqrt (BC^2 + CA^2 + 2*BC*CA*cos (120 - c)).
cg Steiner wall Three lines meeting at a vertex, with 120 degree angles
cg between the lines. See command "steiner".
cg stem In an object name, the part of the name preceding any
cg integer subscripts. Such a stem may itself be
cg an integer, except for variable names.
cg For example, if OBJNAME = STEM(NSUB1,NSUB2,...), and
cg NSUBi is either an explicit integer or a variable
cg with an integer value, the stem is STEM, and the
cg subscripts are NSUB1, NSUB2, ...,, which may also be
cg subscripted, but only with an explicit integer
cg subscript.
cg See "subscripted names".
cg stem name See "stem".
cg step One step in a random walk process is a displacement
cg by a distance randomly sampled from an exponential
cg distribution with an average value equal to the
cg mean free path, and in a direction randomly sampled
cg from an isotropic distribution.
cn store.h File containing specifications of all GEOM internal
cn parameters and variables used in more than one
cn subroutine, and their definitions. Their names may
cn be used as input arguments in command "debug".
cn See file geom_internal.
cg straight See "straight line".
cg straight line A straight line is the shortest distance between two
cg points. See "line".
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 A quadric surface (a hyperbolic paraboloid) may be
cg created that contains the four vertices, the four
cg edges and the center of a nonplanar quadrangle.
cg See command "hyperb".
cc STRING In command "alias", any character string up to 72
cc characters long or whatever will fit in the
cc 80-character input line of command "alias".
cc Must be bracketed by single (') or double (") quotes.
cc If STRING is bracketed by single quotes, double any
cc internal single quotes.
cc If STRING is bracketed by double quotes, double any
cc internal double quotes.
cc STRING will be substituted for the alias name,
cc whenever the alias name is the first word of an input
cc line, as long as the combined total number of
cc characters does not exceed 80.
cc
cc Do not begin STRING with "cc", "!", "*" or "/", or
cc with a delimited "c", "C", "cxxxx", or "Cxxxx",
cc where x is ".", "+", "-", or "_",
cc unless you intend for the alias to be a comment.
cc STRING In command "increment", any delimited string with no
cc more than 24 characters, that does not begin with "!"
cc or contain ";".
cc STRING In command "prompt", any delimited ASCII string with no
cc more than 8 characters, that does not begin with "!"
cc or contain ";", and is not an integer or floating
cc point value.
cc STRING In commands "redo" and "indo", a delimited string
cc (not an integer or "$", and may not begin with "!" or
cc contain ";").
cc The command to be redone, after making any specified
cc replacements or increments of arguments, is the last
cc preceding command that begins with the string.
cc STRING In command "symbol", a delimited ASCII string which
cc does not begin with "!" or contain ";", and which
cc is to be incremented (or decremented) by a specified
cc number of characters. See "increment'.
cg STRING In UNIX text editor VI command "/^......STRING",
cg a character string beginning in column 7 of this
cg file, to be searched for.
cc STRING1 In command "input", a delimited string (not an integer
cc or "?", and may not begin with "!" or contain ";").
cc The specified input file or input lines saved in
cc memory will be set at the first occurrence of STRING1
cc at the beginning of a line. This will fail if any
cc variable has the name STRING1.
cc Lines that begin with one of the characters "#", "*"
cc and "/" are comment lines, so lines that begin with
cc unique strings may be placed anywhere in an input
cc file.
cc For example:
cc
cc input IN_FILE #macro.123 #macro.end
cc
cc Open file IN_FILE, and position it at the first
cc line that begins with the string "#macro.123".
cc File IN_FILE will be read until an end-of-file is
cc reached, another command "input" is executed or a
cc line that begins with the delimited string
cc "#macro.end" is read.
cc
cc This allows a large number of macros (a set of GEOM
cc command lines) to be kept in a single file.
cc Each such macro should end with command "return",
cc to get the next command from the previous input
cc medium.
cc STRING1 In command 'hex', a hexadecimal integer or floating
cc point value (option "in"), or a decimal floating
cc point or integer value or variable name (option
cc "out"). Likewise for STRING2, STRING3, ...
cc STRING2 In command "input", a delimited string (not an integer
cc or "?", and may not begin with "!" or contain ";").
cc If not specified, defaults to STRING1. The specified
cc input file or list of input lines saved in memory
cc will be read until a line is read that begins with
cc STRING2, or an end-of-file is reached, or another
cc command "input" is executed. This will fail if any
cc variable has the name STRING2.
cc If STRING2 is the same as STRING1, a single line will
cc be read.
cg subroutines See "geom_files". GEOM subroutines, not including APT
cg subroutines, are as follows:
cg accel, alias, aliasget, aloha, alout,
cg angles, angout, aptsubr, arc, area,
cg axisym, bin, bindep, binfind, binout,
cg binstore, bisectx, blankout, brick, brout,
cg brstore, brtest, circle, clfind, closey,
cg clout, cluster, compsubs, cone, contfr,
cg coord, copy, cross, crt, cute,
cg cuts, cylinder, debugx define, delete,
cg delim, disk, distance, dkout, doenddo,
cg dot,
cg ellipsoid, extrema, factor, ffeed, geomend,
cg getio, getlun, helpall, helpalph, helpenv,
cg helperr, helpgen, helpmath, helpmisc, helpobj,
cg helpran, hex, hyperb, icalc, ifendif,
cg incr, incrsubs, indo, input, intcirc,
cg intcyl, intplcy, intplqu, intpls, intquad,
cg intspcy, intsph, iris, kiss, last,
cg line, linefind, list, lout, lstore,
cg main, markdep, marker, markget, markout,
cg matrix, mcvol, mesh, meshblk, meshchk,
cg meshchk1, meshdel, meshexch, meshfind, meshins,
cg meshinv, meshline, meshmove, meshout, meshrel,
cg meshrlxb, meshrlxm, meshtest, meshvout, movex,
cg openx, operator, opout, output, parse,
cg pdf, pdfout, pgout, phout, plane,
cg plotmap, plotpt, plotvar, plotx, plout,
cg point, polygon, polyhed, pout, project,
cg prompt, prox, ptbrick, ptcoord, ptdep,
cg ptdist, ptfind, ptoper, ptstore, pttet,
cg ptzone, quadfind, quadric, qumove, quoper,
cg quout, qustore, qutype, ranf, ratio,
cg redo, renamea, renamex, repack, root,
cg rootf, roots, sample, savecmd, search,
cg side, slice, sliceq, solve, sort,
cg sphere, spin, status, subsrepl, sudo,
cg symbdisp, symbfind, symbol,
cg symbout, symbrepl, symstore, synonym, synout,
cg tables, tbank, tetout, tetra, tetstore,
cg tettest, timex, title, tolset, track,
cg triangle, trig, triple, tristore, trout,
cg twist, undo, varfind, variable, varmode,
cg varout, varstore, vectfind, vector, volume,
cg voper, vout, vstore, walk, when,
cg where, znmake, znstore, zone, zout.
cg
cg These are all FORTRAN source files, except ranf,
cg which is a C source file.
cg
cg In addition, store.h is a macro or cliche to define
cg all GEOM parameters and common blocks.
cc subscript An option in command "debug", to display GEOM internal
cc variables for the stem and subscripts of input word
cc LASTWORD: astem, fstem, istem, mstem, lstem, nsub,
cc asub, fsub, isub, msub, lsub.
cc Must be used in the form:
cc
cc debug subscript LASTWORD
cg subscript An object name may have up to 16 subscripts, the first
cg preceded by the stem of the name, a left parenthesis,
cg and the last followed by a right parenthesis (the
cg last character of the name), and each separated by
cg commas from other subscripts, all following the
cg unsubscripted stem of the name.
cg A subscript may be any positive, zero or negative
cg integer or the name of a previously defined integer
cg variable. The later may itself be subscripted, but
cg only with a single explicit integer subscript.
cg This does not apply to markers.
cg See "subscripted names".
cg
cg Commands "do" and "enddo" may be used to loop over
cg ranges of subscripts for any subscripted object name.
cg
cg subscripted See "subscripted names".
cg subscripted names
cg An object name may be subscripted, with the form
cg STEM(NSUB1,NSUB2,..., NSUBi,...), where STEM is the
cg stem name, and the subscripts NSUBi, are either
cg explicit integers or variables with an integer value.
cg In the latter case, the variable name NSUBi may
cg itself be subscripted, but only with an explicit
cg integer subscript. The object name, including all
cg subscripts, must not contain delimiter characters.
cg
cg Use of subscripts allows arrays of objects to be
cg treated with a minimum of commands, using do loops.
cg
cg For example, the commands:
cg
cg set n = 53
cg set v(n) = 123
cg
cg are equivalent to the command:
cg
cg set v(53) = 123
cg
cg and the commands:
cg
cg set nn = 17
cg set v(nn) = 12
cg set y = 1 + v(nn)
cg set q(v(17)) = 1.234
cg set z(nn) = q(v(17))
cg
cg are equivalent to the commands:
cg
cg set v(17) = 12
cg set y = 1 + 12
cg set q(12) = 1.234
cg set z(17) = 1.234
cg
cg Any object defined with a name containing an
cg explicit integer subscript will have any leading
cg zeros (following any initial sign) removed from the
cg subscript.
cg
cg For example, the command:
cg
cg set x(-005) = 3
cg
cg will result in the creation of variable x(-5), with
cg the value 3.
cg substitute See "alias", "replace arguments", "replace name",
cg "replace object", "symbol", "synonym", "variable".
cc su A synonym for sudoku.
cc sudoku Command to specify and try to solve a Sudoku puzzle.
cc At least 17 digits, from 1 to 9, must be initially
cc assigned to some of the 81 cells of the 9 x 9 Sudoku
cc grid, with no digits repeated in any row, column or
cc box.
cc
cc Forms include:
cc
cc help sudoku
cc sudoku [help]
cc
cc sudoku new
cc
cc Start a new 9 x 9 Sudoku puzzle.
cc
cc sudoku [row,col,box] N [=] I1 I2 I3 I4 I5 I6 I7 I8 I9
cc
cc Assign up to 9 initial digits I1, I2, I3, ... to
cc [row, column, box] N.
cc Use zero for blank or unassigned digits.
cc
cc sudoku cell NROW NCOL [=] I1
cc
cc Assign the digit I1 to row NROW, column NCOL.
cc Use zero for blank or unassigned digits.
cc Show all digits permitted in each cell.
cc
cc sudoku perm
cc
cc Display all digits permitted in each Sudoku cell.
cc
cc sudoku save
cc
cc Save and display the current assignments. They can
cc be restored later with option "reset".
cc
cc sudoku reset
cc
cc Reset the assignments to those of the last save.
cc
cc sudoku solve
cc
cc Save the current assignments and try to solve this
cc Sudoku puzzle.
cc
cc Synonyms: [help, h], [sudoku, su].
cg Sudoku puzzle A numerical puzzle requiring a 9 x 9 grid of cells,
cg in nine rows and nine columns, subdivided into nine
cg 3 x 3 boxes, to be filled with the nine digits 1-9,
cg such that each row, each column, and each box
cg contains all nine digits, without repeats.
cg See command "sudoku".
cc SUM In command "ratio", the sum of the first NUMT terms of
cc a geometric series, for which the first term is
cc TERM1.
cc sum An option in command "quadric", to find the weighted
cc sum or difference of two quadric surfaces.
cc Synonyms: [sum, add].
cc sum An option in command "vector", to find the weighted
cc sum or difference of two vectors.
cc Synonyms: [sum, add].
cg sum To find the weighted sum or difference of two vectors,
cg use command "vector".
cg To find the sum of an arithmetic or geometric series,
cg or a more general series, use command "variable",
cg option "series".
cg sum See "partial sum", "series sum".
cg summary See "geom_cmd", "saved input lines".
cg superscript The power, superscript or up arrow symbol, "^".
cg surface A plane or quadric surface. All are stored in a single
cg list, with a single set of unique names, and
cg classified as quadric surfaces.
cg See: plane, sphere, cylinder, cone, ellipsoid,
cg axisym, quadric. Surfaces may be used to specify the
cg boundaries of zones.
cg Also see "surface of coord", "ruled surface".
cg surface See "surface area", "surface element",
cg "surface of coord", "surface reflection".
cg surface area See "area".
cg surface element A surface element in the mesh is a set of four mesh
cg points with the same value of one of the mesh
cg indices, and two adjacent values of each of the other
cg two indices, forming a quadrangle bounded by adjacent
cg layers in each of the other two mesh directions.
cg See "point element", "line element",
cg "volume element".
cg surface of coord To create a surface at a constant value of one of the
cg three coordinates of an orthogonal coordinate system,
cg see "x plane", "y plane", "z plane", "rcyl cylinder",
cg "theta plane", "rsph sphere", "phi cone".
cg
cg To make a family of surfaces at different constant
cg values of one of the three coordinates of an
cg orthogonal coordinate system, use one of the
cg following commands:
cg
cg command options(s) constant coordinate
cg
cg plane scale, parallel (x, y and z)
cg cylinder scale, concentric (rcyl)
cg plane rotate (theta)
cg sphere scale, concentric (rsph)
cg cone nest (phi)
cg
cg To create a 2-D array of points on a surface of a
cg coordinate system, see command
cg "point PMESH brick BRNAME mesh block [volume]".
cg surface of revolution
cg A quadric surface with only x and z terms, with
cg coordinate X replaced by coordinate RCYL, where
cg RCYL^2 = X^2 + Y^2, is equivalent to a surface of
cg revolution around the z axis. See "toroid".
cc surface reflection
cc See "normal", "track reflection".
cc sw A synonym for rename.
cc switch A synonym for rename.
cc sym A synonym for symbol.
cc symb A synonym for symbol.
cc SYMBNAME The name of a symbol with a replacement value. May
cc have up to 24 characters, including any subscripts.
cc No symbol may be "+", "-", "all", "h", "help",
cc "list" or "thru", begin with "!" or contain ";".
cc Specified with command "symbol".
cg symbol A symbol may be used to replace any long or frequently
cg uses word in input lines with a shorter or simpler
cg substitute, for simpler input.
cg
cg A symbol is a user-defined synonym, to be used like
cg the preset synonyms, and in addition, applies to the
cg stem and subscripts (but not subscripted subscripts)
cg of subscripted names. See "synonyms".
cg A symbol may also be used to correct errors in
cg previously executed commands. See command "input".
cg
cg NOTE! A warning message will be displayed if a
cg symbol has the same name as a preset synonym or an
cg alias or its replacement.
cg
cg A symbol may be created with command "symbol".
cg The name of the symbol may be any single delimited
cg character string up to 24 characters long, including
cg any subscripts, and must remain delimited when used.
cg The replacement value of the symbol must also be a
cg single delimited character string, and may be up to
cg 72 characters long or whatever will fit on the
cg command "symbol" input line, and may be an ASCII
cg character string, an integer or floating point value,
cg a key word used as a command or command option, the
cg name of another symbol, the name of a variable or any
cg other object, including the stem and subscripts (but
cg not subscripted subscripts) of a subscripted name.
cg
cg The name of a symbol is a synonym for its ultimate
cg replacement value, which is the result of making up
cg to 10 successive replacements of replacement names
cg which are themselves the names of other symbols.
cg
cg Before a command is executed, and after any alias is
cg replaced, all symbols in the command, including the
cg stems and subscripts (but not subscripted subscripts)
cg of subscripted words, are replaced by their ultimate
cg replacement values, with the following exceptions:
cg comment lines, appended comments, commands skipped
cg because of an "if" command result, commands "alias",
cg "copy alias", "copy symbol", "debug", "define",
cg "delete alias", "delete symbol", "rename alias",
cg "rename symbol", "symbol".
cg
cg The following commands relate to symbols:
cg copy, debug, delete, help, last, list, rename,
cg repack, search, sort, symbol, symbol, synonym,
cg tables.
cg
cg The following symbols are created at the beginning of
cg each run, and may be replaced by using their name in
cg command "symbol":
cg
cg pi 3.14159265358979324 math constant pi
cg deg/rad 57.29577951308232 degrees per radian
cg rad/deg 0.01745329251994330 radians per degree
cg ebase 2.71828182845904523 base of natural logs
cg rgold 1.61803398874989485 golden ratio
cg euler 0.577215664901533861 Euler's constant
cc symbol An option in command "debug", to display GEOM internal
cc parameters and variables for symbols and their
cc replacements: nsymbm, nsymbs, asymbl, asymbr,
cc fsymb, isymb, lsymb, msymb, asymbls, lasymbls.
cc Synonyms: [symbol, sym, symb]
cc symbol Command to create, modify or display a symbol
cc and its replacement value.
cc See "do loop use".
cc
cc Before a command is executed, and after any alias is
cc replaced, all symbols in the command, including the
cc stems and subscripts (but not subscripted subscripts)
cc of subscripted words, are replaced by their ultimate
cc replacement values, with the following exceptions:
cc comment lines, appended comments, commands skipped
cc because of an "if" command result, commands "alias",
cc "copy alias", "copy symbol", "debug", "define",
cc "delete alias", "delete symbol", "rename alias",
cc "rename symbol", "symbol".
cc
cc NOTE! A warning message will be displayed if a
cc symbol or its replacement has the same name as a
cc command or option or any of their preset synonyms.
cc
cc Command "symbol" relates to objects: alias, axisym,
cc big, bin, brick, cluster, cone, cylinder, disk,
cc ellipsoid, help, hyperb, line, marker, mesh,
cc operator, pdf, plane, plot, point, polygon,
cc polyhedron, quadric, sphere, symbol, tetrahedron,
ccc triangle, variable, vector, zone.
cc
cc Forms include:
cc
ccin help symbol
ccin symbol [help]
cc
cc Display the command options.
cc
ccin symbol [all,list]
cc
cc Display all symbols and their ultimate
cc replacement values.
cc All symbols may also be displayed with command
cc "synonym".
cc
ccin symbol SYMBNAME
cc
cc Display symbol SYMBNAME and its ultimate
cc replacement value.
cc
ccin symbol list SYMBNAME1 SYMBNAME2 SYMBNAME3 ...
cc
cc Display symbols SYMBNAME1, SYMBNAME2, SYMBNAME3,
cc ..., with or without subscripts, and their
cc ultimate replacement values.
cc
ccin symbol list SYMBNAME1 thru SYMBNAME2
cc
cc Display all symbols with names in the range
cc from SYMBNAME1 to SYMBNAME2, with or without
cc subscripts, and their ultimate replacement values.
cc
ccin symbol SYMBNAME [=] REPLNAME
cc
cc Create symbol SYMBNAME with replacement value
cc REPLNAME. If REPLNAME is itself a symbol name,
cc SYMBNAME will have a different ultimate replacement
cc value, when used in a command. The "=" is optional
cc if REPLNAME is explicitly integer or floating
cc point.
cc
ccin symbol SYMBNAME [=] STRING increment [INCR|1]
cc
cc Create symbol SYMBNAME with replacement value
cc STRING incremented by INCR characters.
cc INCR must be an integer or integer variable.
cc The "=" is optional if STRING is explicitly integer
cc or floating point. If STRING is subscripted, its
cc final length may not exceed 24 characters.
cc See "increment names".
cc
ccin symbol SYMBNAME increment [INCR|1]
cc
cc Increment the initial replacement value of symbol
cc SYMBNAME by INCR characters.
cc INCR must be an integer or integer variable.
cc If SYMBNAME is subscripted, its final length may
cc not exceed 24 characters.
cc See "increment names".
cc
ccin symbol SYMBNAME = deg/rad
cc
cc Create symbol SYMBNAME with the value of the
cc number of degrees per radian: 57.29577951308232
cc
ccin symbol SYMBNAME = ebase
cc
cc Create symbol SYMBNAME with the value of e, the
cc base of natural logarithms: 2.718281828459045
cc
ccin symbol SYMBNAME = euler
cc
cc Create symbol SYMBNAME with the value of Euler's
cc constant: 0.577215664901533861
cc
ccin symbol SYMBNAME = rgold
cc
cc Create symbol SYMBNAME with the value of the
cc golden ratio: 1.618033988749894
cc
ccin symbol SYMBNAME = pi
cc
cc Create symbol SYMBNAME with the value of the
cc mathematical constant pi: 3.141592653589793
cc
ccin symbol SYMBNAME = rad/deg
cc
cc Create symbol SYMBNAME with the value of the
cc number of radians per degree: 0.01745329251994330
cc
cc Synonyms: [euler, Euler], [help, h],
cc [symbol, sym, symb].
cc symbol An option in commands "copy", "delete", "last",
cc "list", "rename", "repack" and "sort", to perform the
cc specified operation on one or more symbols.
cc Synonyms: [symbol, sym, symb]
cg symbol arrays Symbols may be created as arrays with subscripted
cg names. See "subscript", "subscripted names",
cg commands "do" and "enddo".
cg symbolic A symbolic command, option or argument is a word which,
cg after replacement of all symbols with their ultimate
cg values, satisfies the requirements of a command,
cg option or argument.
cg See "symbol", "argument", "alias", "marker".
cg symbols See commands "symbol", "alias", "marker".
cg symmetric Two geometric objects are symmetric in a general sense
cg if one may be transformed to the other one by
cg translation, inversion, reflection, rotation, linear
cg scaling, radial scaling or uniform scaling or any
cg combination of these.
cc symmetry See "operator", "symmetric", "symmetry ops".
cg symmetry See "symmetry axis", "symmetry ops", "symmetry point".
cg symmetry axis See "axis of symmetry".
cg symmetry ops Symmetry operations include translation, inversion,
cg reflection, rotation, linear scaling, radial scaling,
cg and uniform scaling.
cg See "move", "operator", "invert", "reflect",
cg "rotate", "scale".
cg Symmetry operations may be performed on points,
cg clusters, vectors and quadric surfaces (including
cg planes, spheres, circular cylinders, circular cones,
cg hyperbolic paraboloids, ellipsoids, axially symmetric
cg quadric surface, and general quadric surfaces).
cg
cg For quadric surfaces, see "center", "align".
cg symmetry point See "quartic". A quartic equation P(z) = 0, has
cg a symmetry point wherever the third derivative,
cg P'''(z) = 0.
cc syn A synonym for synonym.
cc SYNAME1 In command "synonym", a command or key word or the
cc first letter of a command or key word, for which
cc any synonyms and restrictions on usage are to be
cc displayed. Likewise for SYNAME2, SYNAME3, ...
cg synonym A synonym is a word used in place of another word.
cg GEOM uses a list of preset synonyms, which may be
cg displayed with command "synonym", and which have
cg exactly the same meaning as the original word.
cg You may also create your own synonyms with commands
cg "symbol" and "alias", which may or may not have the
cg same meaning as the original word. See "synonyms".
cc synonym An option in command "debug", to display GEOM internal
cc parameters and variables for synonyms:
cc nsynm, nsyns, asyn1, asyn2, asyncmd.
cc Synonyms: [synonym, syn].
cc synonym Command to display the preset synonyms for commands and
cc key words, and any symbols and aliases defined
cc by the user. To create your own synonyms, use
cc commands "symbol" and "alias". See "synonyms".
cc Use command "alph" to display an alphabetic list of
cc all commands and their preset synonyms.
cc
cc Command "synonym" 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 synonym
ccin synonym [help]
cc
cc Display the command options.
cc
ccin synonym all
cc
cc Display all synonyms, symbols and aliases.
cc
cc synonym none
cc
cc Display all command words and key words for which
cc there is no preset synonym.
cc
ccin synonym SYNAME1 SYNAME2 SYNAME3 ...
cc
cc Display the synonyms, symbols and aliases for
cc SYNAME1, SYNAME2, SYNAME3, ..., or if any of these
cc have a single character, display all synonyms,
cc symbols and aliases beginning with that character.
cc Also, display any restrictions on usage of
cc synonyms. SYNAME1 may not be "all", "help" or "h".
cc Subscripts are not ignored, so must match exactly.
cc
cc Synonyms: [help, h], [synonym, syn].
cg synonyms The preset synonyms (in brackets) listed below may be
cg used anywhere for comments, command words and key
cg words (except when restricted to a particular use or
cg command, specified in parentheses after the synonyms)
cg and have exactly the same meaning as the initial word
cg in each bracketed group.
cg
cg Command "symbol" may be used to create new synonyms,
cg or to change the meaning of any preset synonym.
cg
cg Command "alias" may be used to create a new synonym
cg for any symbol or word at the beginning of any
cg comment or command.
cg
cg [", ' (in pairs)], [#, *, / (comment)],
cg [/=, .ne. (if)], [/=, =/ (if)],
cg [0, . (indo)], [0, zero (big)],
cg [<, .lt. (if)], [<=, .le. (if)], [<=, =< (if)],
cg [=, .eq. (if)], [>, .gt. (if)], [>=, .ge. (if)],
cg [>=, => (if)], [?, random], [^, **],
cg [.and., .int.], [.or., .un.],[.xnor., .eqv.],
cg [accelerate, accel], [alias, a, al], [angle, ang],
cg [angles, ang, angle (angles)],
cg [binary, bin (icalc, big)], [block, bl], [brick, br],
cg [c, C (comment)], [c++++, C++++ (comment)],
cg [c----, C---- (comment)], [c...., C.... (comment)],
cg [cartesian, cart, Cartesian, rect, rectangular, xyz],
cg [cc, CC (comment)], [circle, circ], [cluster, cl],
cg [command, c, cmd], [commands, alph, comm],
cg [concentric, conc], [contfr, cf],
cg [coordinate, coord, cs, sys, system],
cg [copy, cp, dup, duplicate], [cosine, cos],
cg [cross, outer], [cylinder, cyl],
cg [cylindrical, cyl (coordinate)],
cg [cylindrical, cyl (project)],
cg [c____, C____ (comment)],
cg [define, def], [degrees, deg, degree],
cg [delete, del, remove, rm],
cg [delimiter, delim, delimit], [disk, disc, dk],
cg [distance, dist (variable)],
cg [distance, dist, int, inter, intersect (distance)],
cg [dot, inner], [duplicate, copy, cp, dup],
cg [edge, side (trig)], [ellipsoid, el, ell, ellipse],
cg [end, exit, quit, split], [environment, env, setup],
cg [error, err], [euler, Euler], [exchange, exch],
cg [extrema, extr], [family, fam], [focus, foc, focal],
cg [grads, grad], [help, h], [hyperb, saddle],
cg [increment, incr], [input, call, i, in, r, rd, read],
cg [insert, ins], [intersect, int, inter (quadric)],
cg [invert, inv], [limits, lim], [line, l, ln],
cg [list, display, print (list)], [lock, zzz],
cg [marker, m, mark], [maxwell, Maxwell],
cg [move, mv, trans, translate], [normal, norm],
cg [object, obj], [octal, oct (icalc, big)],
cg [operator, op, oper], [outer, cross],
cg [output, o, out, w, wr, write], [parallel, par],
cg [parameter, param], [perspective, pers],
cg [planck, Planck], [plane, pl], [point, p, pnt, pt],
cg [polygon, pg, polyg], [polyhedron, ph, polyh],
cg [power, pow], [project, proj], [proximal, prox],
cg [quadric, q, quad], [radians, rad, radian],
cg [random, ?], [ratio, rat], [reflect, refl],
cg [rel, * (vector)],
cg [rename, sw, switch], [return, rtn], [rotate, rot],
cg [scale, sc], [scale, uniform (oper)], [search, find],
cg [slice, cut], [sphere, sph],
cg [spherical, sph (coord)], [status, stat],
cg [steiner, stein], [sudoku, su], [sum, add],
cg [symbol, sym, symb], [synonym, syn], [tables, sizes],
cg [tangent, tan (plane)], [tetrahedron, tet, tetra],
cg [time, t], [title, id], [track, trk],
cg [triangle, tri], [twist, tw], [undo, u],
cg [variable, s, set, var], [vector, v, vect],
cg [viewfactor, view], [volume, vol], [wien, Wien],
cg [x, X], [y, Y], [yes, y (output)],
cg [z, Z], [zone, z, zn].
cg
cg In addition, you may create your own synonym for any
cg delimited character string, by creating a symbol
cg whose replacement value is the character string.
cg With certain exceptions (see "symbol"), subsequent
cg use of the delimited name of the symbol will be the
cg same as using the character string, unless the
cg character string is itself a symbol.
cg
cg For example:
cg
cg symb x = delete
cg
cg Create symbol x with replacement value
cg "delete".
cg
cg x p PNAME
cg
cg Same as command "delete p PNAME".
cg
cg See "alias", "marker", "symbol", "commands",
cg "key words".
cg SYS A user of machine time. See "CPU", "I/O".
cc sys A synonym for coordinate.
cc system A synonym for coordinate.
T-------------------------------------------------------------------------------
cc t A synonym for time.
cc T1 See "T1, T2, T3".
cc T1, T2, T3 In command "accelerate", option "fit", the times on the
cc parabolic trajectory of a uniformly accelerated
cc particle, at points P1, P2, P3.
cc If only one point P1 is specified, T1 need not be.
cc T2 See "T1, T2, T3".
cc T3 See "T1, T2, T3".
cg table See "table sizes".
cg table of data See command "variable", option "data".
cg table sizes See "maximum number", command "tables".
cc tables Command to display the current numbers and maximum
cc allowed numbers of various objects, and the commands
cc that may be used to display or create them.
cc
cc Command "tables" relates to objects: alias, axisym
cc bin, brick, cluster, cone, cylinder, disk, ellipsoid,
cc hyperb, line, marker, mesh, operator, pdf, plane,
cc plot, point, polygon, polyhedron, quadric, sphere,
cc symbol, tetrahedron, triangle, variable, vector,
cc zone.
cc
cc Forms include:
cc
ccin help tables
ccin tables [help]
cc
cc Display the current numbers and maximum allowed
cc numbers of various objects, and the commands that
cc may be used to create and display them.
cc
cc Synonyms: [help, h], [tables, sizes].
cc
cc See "maximum number".
cg tag See commands "alias", "big", "marker", "variable".
cc tag In command "big", an option to specify a descriptive
cc tag for a big integer, with up to 32 characters
cc specified between single or double quotes.
cc tag In command "variable", an option to specify a
cc descriptive tag for a variable, with up to 48
cc characters specified between single or double quotes.
cc tan A synonym for tangent.
cf tan The trigonometric tangent function.
cf Example: y = tan (x) means y is the tangent of x.
cf x must be in radians.
cc tangent An option in command "variable", argument FUNCTION,
cc to indicate the trigonometric tangent function.
cc A tangent B means A * tan (B).
cc Synonyms [tangent, tan].
cc tangent An option in command "plane", to create a plane tangent
cc at a specified point to one of a family of quadric
cc surfaces represented by a specified quadric surface.
cc Synonyms: [tangent, tan].
cg tangent Two curves are tangent at a point if they both pass
cg through that point, and their direction vectors are
cg parallel or antiparallel at that point, but not
cg elsewhere in an arbitrarily small neighborhood of
cg that point.
cg
cg Two surfaces are tangent at a point if they both pass
cg through that point, and their normal vectors are
cg parallel or antiparallel at that point, but not
cg elsewhere in an arbitrarily small neighborhood of
cg that point.
cg See commands "intcirc", "intersect", "distance",
cg "triangle".
cg
cg Two quadric surfaces may be made tangent if they both
cg have extrema in a specified direction, and one or
cg both are moved to make the corresponding extreme
cg points coincident. See commands "extrema", "move".
cg tangent See "tangent circles", "tangent plane",
cg "tangent spheres".
cg tangent circles Given the centers of three tangent circles, their
cg radii and the coordinates of the tangent points may
cg be found with command "triangle", specifying the
cg three centers as the vertices of a triangle.
cg Given the radii of three tangent circles, their
cg center coordinates and the radii and center
cg coordinates of two additional circles tangent to the
cg first three may be found with command "kiss".
cg
cg Given two tangent circles in the x-y plane, with
cg coordinates and radii (x1,y1,r1) and (x2,y2,r2), the
cg tangent point (xt,yt) satisfies the two equations:
cg (xt - x1)^2 + (yt - y1)^2 = r1^2
cg (xt - x2)^2 + (yt - y2)^2 = r2^2
cg The coordinates of the centers of the circles satisfy
cg the equation:
cg (x2 - x1)^2 + (y2 - y1)^2 = (r1 + r2)^2.
cg
cg Additional tangent circles satisfy similar equations.
cg Problems requiring tangent circles to be found may
cg sometimes be solved by treating systems of such
cg equations as implicit quadric surface equations in
cg (x.y,r) space, and using commands "intersection",
cg "distance" and "triple" to find their intersections.
cg
cg The problem of four tangent circles, each tangent to
cg the other three, has an elegant solution, due to
cg Descartes, and later Beecroft, for the radius of the
cg fourth circle, given the radii of the other three:
cg b1^2 + b2^2 + b3^2 + b4^2 =
cg (b1 + b2 + b3 + b4)^2 / 2,
cg where b is the curvature (the reciprocal of the
cg radius), negative if the circle is "kissed" on the
cg inside, zero if the circle is actually a straight
cg line.
cg If b1, b2 and b3 are known, two solutions for b4 are:
cg b4 = b1 + b2 + b3 + 2 * sqrt (b1*b2 + b2*b3 + b3*b1),
cg b4 = b1 + b2 + b3 - 2 * sqrt (b1*b2 + b2*b3 + b3*b1).
cg For example, if b1 = b2 = b3, then
cg b4 = (2 * sqrt (3) + 3) * b1 or
cg b4 = -(2 * sqrt (3) - 3) * b1, and
cg r4 = r1 / (2 * sqrt (3) + 3) or
cg r4 = -(2 * sqrt (3) + 3) * r1 / 3.
cg
cg To find two circles, each tangent to three tangent
cg circles with radii RAD1, RAD2 and RAD3, use command
cg "kiss RAD1 RAD2 RAD3", with a negative value for the
cg radius of a circle with internal tangencies, and a
cg very large value for a straight line.
cg tangent plane To create a plane tangent to a quadric surface at a
cg specified point on the surface, find the normal to
cg the quadric surface at that point, using command
cg "side", then create the plane at that point with that
cg normal vector, using command "plane".
cg
cg To create a plane tangent to a quadric surface, with
cg a specified normal vector, use command "extrema" to
cg find the point on the quadric surface with that
cg normal vector, then create the plane at that point,
cg with command "plane".
cg tangent spheres Two spheres are tangent if a line joining their centers
cg passes through a point at which their surfaces
cg coincide. The condition of tangency, for two spheres
cg of radii r1 and r2, with centers at points
cg p1 = (x1,y1,z1), p2 = (x2,y2,z2) is as follows:
cg (x2-x1)^2 + (y2-y2)^2 + (z2-z1)^2 = (r1+r2)^2.
cg The tangent point pt = (xt,yt,zt) is given by:
cg xt = (r2 * x1 + r1 * x2) / (r1 + r2)
cg yt = (r2 * y1 + r1 * y2) / (r1 + r2)
cg zt = (r2 * z1 + r1 * z2) / (r1 + r2)
cg
cg The problem of five tangent spheres, each tangent to
cg the other four, has an elegant solution for the
cg radius of the fifth circle, given the radii of the
cg other four:
cg b1^2 + b2^2 + b3^2 + b4^2 + b5^2 =
cg (b1 + b2 + b3 + b4 + b5)^2 / 3,
cg where b is the curvature (the reciprocal of the
cg radius), negative if the sphere is "kissed" on the
cg inside, zero if the sphere is actually a plane.
cg Given b1, b2, b3 and b4, the two solutions for b5
cg are:
cg b5 = 0.5 * (b1 + b2 + b3 + b4) + 0.5 * sqrt (fact)
cg b5 = 0.5 * (b1 + b2 + b3 + b4) - 0.5 * sqrt (fact)
cg where fact =
cg 6 * (b1*b2 + b2*b3 + b3*b4 + b4*b1 + b1*b3 + b2*b4)
cg - 3 * (b1^2 + b2^2 + b3^2 + b4^2)
cg For example, if b1 = b2 = b3 = b4, then
cg b5 = (sqrt (6) + 2) * b1 or -(sqrt (6) - 2) * b1, and
cg r5 = r1 / (sqrt (6) + 2) or -(sqrt (6) + 2) * r1 / 4.
cg
cg The center coordinates of the spheres can be rotated
cg and translated from the following:
cg x1 = 0.0
cg y1 = 0.0
cg z1 = 0.0
cg x2 = r1 + r2
cg y2 = 0.0
cg z2 = 0.0
cg x3 = r1 + r3 * (r1 - r2) / (r1 + r2)
cg = r1 if r1 = r2
cg = r1 if r1 + r2 = 0.
cg y3 = 2.0 * sqrt (r1*r2*r3*(r1+r2+r3)) / (r1 + r2)
cg = sqrt (r3 * (2.0 * r1 + r3)) if r1 = r2
cg = 0 if r1 + r2 = 0.
cg z3 = 0.0
cg x4 = r1 + r4 * (r1 - r2) / (r1 + r2)
cg = r1 if r1 = (+/-)r2,
cg arg2 = b1 + b2
cg arg3 = b1 * b2 + b3 * arg2
cg y4 = (b4*arg2 + (b3*arg2 - (b1^2 + b2^2)))/
cg (b4*arg2*sqrt(arg3)),
cg with the sign of y4 the same as
cg y4 = (r1*(r1 + r3) + r4*(r1 - r3) - x3*x4) / y3
cg y4 = 0.0 if y3 = 0 or r1 = -r2,
cg arg5 = b1 + b2 + b3
cg z4 = sqrt ((4.0*arg3 - (b4-arg5)^2)/(b4^2*arg3))
cg x5 = r1 + r5 * (r1 - r2) / (r1 + r2)
cg = r1 if r1 = (+/-)r2,
cg arg2 = b1 + b2
cg arg3 = b1 * b2 + b3 * arg2
cg y5 = (b5*arg2 + (b3*arg2 - (b1^2 + b2^2)))/
cg (b5*arg2*sqrt(arg3)),
cg with the sign of y5 the same as
cg y5 = (r1*(r1 + r4) + r5*(r1 - r4) - x4*x5) / y4
cg y5 = 0.0 if y4 = 0 or r1 = -r2,
cg arg5 = b1 + b2 + b3
cg z5 = sqrt ((4.0*arg3 - (b4-arg5)^2)/(b4^2*arg3))
cg with the sign of z5 the same as
cg z5 = (r1*(r1+r4)+r5*(r1-r4)-x4*x5 - y4*y5) / z4,
cg
cg See command "kiss".
cc tanh An option in command "variable", argument FUNCTION,
cc to indicate the hyperbolic tangent function.
cc A tanh B means A * tanh (B).
cf tanh The hyperbolic tangent function.
cf Example: y = tanh (x)
cf means y = (e^x - e^(-x)) / (e^x + e^(-x)).
cc temp An option in command "debug", to display the temporary
cc arrays, currently used only for option "prime" in
cc command "variable".
cc temp An option in command "debug", to display GEOM internal
cc parameters and variables for temporary arrays:
cc ntempm, ntemps, atempa, atempb, ftempa, ftempb,
cc itempa, itempb.
cc These are currently used only for option "prime" in
cc command "variable".
cg tempered See "tempered scale".
cg tempered scale See "equally-tempered", "frequency ratios",
cg "scale.mac", "cents".
cg tensor A object with components having row and column indices,
cg which may operate on the coordinates of a point, the
cg components of a vector, or the coefficients of an
cg implicit quadric equation, to rotate, scale, invert
cg or reflect the operand. In GEOM, the only tensor
cg operator used is a 3 by 3 array. See command
cg "operator".
cc TERM1 In command "ratio", the value of the first term of a
cc geometric series with NUMT terms and sum SUM.
cd test Directory for test problems.
cd In Open YANA Cluster: ~edwards/work/geom/test
cd Archives: /users/u47/edwards/geom/test
cd See "doc", "make".
cn test As a suffix, indicates a standard test problem input
cn file.
cn In Open YANA Cluster: ~edwards/work/geom/test
cn FILENAME.test is the input file for producing output
cn file FILENAME.hsp, and may be identical to the
cn command summary file geom_cmd produced by running
cn GEOM with the input file FILENAME.test, except for
cn certain commands not needed for producing the desired
cn output (preceded by the character "<" in
cn FILENAME.hsp).
cc test An option in command "mesh", to see if all indices in
cc the mesh block have been assigned to mesh points, and
cc to see if any line, surface or volume elements in the
cc mesh block violate certain geometric restrictions.
cc See "mesh tests".
cg test See "test problems".
cg test problems All standard test problem input and output files are in
cg ~edwards/work/geom/test on the open YANA cluster.
cg In that directory there are also command files
cg "geom.mv.hsp.all" to rename the output files,
cg "geom.exe.all" to rerun all the test problems, and
cg "geom.diff.all" to compare the old and new output
cg files.
cg
cg There are 286 test problems as of June 16, 2006.
cg Input files havve the added suffix ".test", and
cg output files have the added suffix ".hsp".
cg The beginning of each file name generally refers to
cg the command being tested, and any suffix generally
cg refers to an option within that command.
cg
cg The list:
cg accel, accel.cl, accel.fit, accel.ln,
cg accel.pl, accel.pt, accel.quad, alias,
cg alias.new, ang, arc, area, axisym, axisym.foc,
cg axisym.mv, axisym.op, basb, base, big,
cg big.add, big.div, big.exp, big.fac, big.mod,
cg big.mul, big.perm, big.pi, big.pow, big.root,
cg big.sub, bin, bisect, bp, br, br.incr, circle,
cg cl.bool, cl.cl, cluster, cone, cone.circ,
cg cone.ell, cone.line, cone.move, cone.nest,
cg cone.op, contfr, conv, coord, copy, cross,
cg crt, crtinv, curve, cute, cutline, cuts, cyl,
cg cyl.conc, cyl.ell, cyl.hyp, cyl.move, cyl.op,
cg data, deal, debug, debug.sub, define, delete,
cg delim, disk, dist.brick, dist.cl, dist.cyl,
cg dist.plcy, dist.plqu, dist.plsp, dist.q,
cg dist.spcy, dist.sph, dist.tet, dist.zone,
cg distal, distance, do, do.error, dot, ell,
cg ell.gen, ell.hard, ell.move, ell.obl, ell.op,
cg ell.prol, extrema, factf.mac, factor, fib.mac,
cg general, goto, helpall, hex, hyp.1, hyp.fit,
cg hyp1.6th, hyp1.circ, hyp1.ell, hyp2.6th,
cg hyp2.circ, hyp2.ell, hyperb, icalc, if,
cg if.loop, if.mac, incr, indo, input, intcirc,
cg invert, iris, jcyl, jell, jnew, junk.test,
cg kiss.circ, kiss.sph, line, link, list,
cg list.obj, marker, mcvol, mesh, mesh.bl,
cg mesh.del, mesh.del.bl, mesh.exch, mesh.ins,
cg mesh.ins.bl, mesh.inv, mesh.junk, mesh.linear,
cg mesh.move, mesh.new, mesh.proj, mesh.pt.br,
cg mesh.relax, mesh.relaxm, mesh.subs, mesh.tst,
cg mesh.void, meshpoint, move, new, nonsimple,
cg op.quad, ortho, par.ell, par.ell.5th, par.hyp,
cg par.hyp.5th, pdf, perm.comb, pl, pl.fit,
cg pl.move, pl.par, pl.proj, pl.rot, pl.scale,
cg pl.tangent, pl.tri, plot, plot.array,
cg plot.center, plot.del, plot.limits, plot.mark,
cg plot.point, plot.zoom, polyg, polyh, proj.cyl,
cg proj.pers, proj.pl, proj.sph, proj.view, prox,
cg pt, pt.br.frac, pt.br.mesh, pt.br.ran,
cg pt.move, pt.op, pt.z.ran, q, q.fit, q.move,
cg q.mult, q.nonstd, q.op, q.order, q.proj,
cg q.simple, ratio, redo, refl, rename,
cg rename.array, root, root4, rootf, roots,
cg rot.axial, rot.planar, rot.serial, rot.triple,
cg rot.vector, sample, scale, scale.mac, search,
cg seg, shape1, shape3, side, side.all,
cg side.cone, side.ell, side.parab, slice,
cg slice.ell, sliceq, solve, sort, sph, sph.conc,
cg sph.fit, sph.move, sph.op, sph.shell,
cg square.mac, stein, su, subs, sudoku, sym, syn,
cg tet, tet.incr, track, tri, tri.new, trig,
cg triple, triple.pl, triple.q, tw.axial,
cg tw.radial, undo, v.ang, v.ran.ang, v.ran.cos,
cg var, var.ASCII, var.series, var.sumprod,
cg var.tag, varmode, vect, vect.cos, vect.norm,
cg vect.op, vect.proj, vect.quad, vect.sum,
cg vect.triple, volume, walk, where, whis,
cg zeta.mac, zone, zone.family, zone.incr.
cc tet A synonym for tetrahedron.
cc TETNAME The name of a tetrahedron. May have up to 24
cc characters, and may be ASCII, integer or floating
cc point. No tetrahedron name may be "+", "-", "all",
cc "h", "help", "list" or "thru", begin with "!" or
cc contain ";".
cc Also referred to as TETNAME1, TETNAME2, TETNAME3, ...
cc TET(1), TET(2), ...
cc Specified with command "tetrahedron".
cc
cc Tetrahedrons may appear in commands:
cc copy, delete, distance, point, rename, tetrahedron,
cc where.
cc Also see commands: last, list, repack, search, sort.
cc tetra A synonym for tetrahedron.
cg tetrahedron A tetrahedron is a geometric object in 3-D space,
cg specified by four points. The tetrahedron will
cg change if any of the points used to specify it are
cg moved. No point may be deleted if it is being used
cg to specify an existing tetrahedron. However, a
cg change in the name of such a point will not change
cg the tetrahedron.
cg The maximum number of tetrahedrons is now 1000.
cg See commands: tetrahedron, copy, delete, rename,
cg list, search, sort, trig, point, vector, where.
cg
cg A regular tetrahedron has 4 vertices, 6 edges, and
cg 4 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 edge = 2.0 * sqrt (6.0) / 3.0 = 1.632993161855
cg area = 2.0 / sqrt (3.0) = 1.154700538379
cg volume = 8.0 * sqrt (3.0) / 27.0 = 0.5132002392797
cg rinsc = 1.0 / 3.0 = 0.3333333333333
cg angdih = acos (1.0 / 3.0) = 70.52877936551
cg angcent = acos (-1.0 / 3.0) = 109.4712206345
cg See "polyhedron".
cg
cg To find if a point is in a tetrahedron, use command
cg "distance" or "where".
cg
cg To find the distances from a point to the four
cg surfaces bounding a tetrahedron, use command
cg "distance".
cg
cg The following commands relate to
cg tetrahedrons:
cg cluster, copy, debug, delete, distance, help, last,
cg list, point, rename, repack, search, sort, symbol,
cg synonym, tables, tetrahedron, where.
cg
cg Also see "angle", "brick", "vertex", "zone".
cg
cg Given a tetrahedron with the three vertex points P1,
cg P2 and P3, to find the fourth vertex point P4 such
cg that the edge from P1 to P4 makes angle ANG12 with
cg the edge from P1 to P2, makes angle ANG13 with the
cg edge from P1 to P3, and has length EDGE14, use the
cg following commands:
cg
cg vector V12 point P1 P2
cg vector V13 point P1 P3
cg vector V14 angle ANG12 V12 ANG13 V13
cg copy point P1 P4
cg move point P4 V14 EDGE14
cc tetrahedron An option in command "debug", to display GEOM internal
cc parameters and variables for tetrahedrons:
cc ntetm, ntets, atetra, atetpt, latetpt, voltet,
cc atetras, latetras.
cc Synonyms: [tetrahedron, tet, tetra].
cc tetrahedron Command to display or create tetrahedrons. Creating a
cc tetrahedron replaces any existing tetrahedron having
cc the same name. The display for a tetrahedron
cc includes (short) the volume and the names and (long)
cc the coordinates of the four vertex points.
cc See "do loop use".
cc
cc Other commands useful in creating tetrahedrons
cc include "trig" and "vector", option "angle".
cc
cc Command "tetrahedron" relates to objects: point,
cc symbol, tetrahedron, variable.
cc
cc Forms include:
cc
ccin help tetrahedron
ccin tetrahedron [help]
cc
cc Display the command options.
cc
ccin tetrahedron [all,list]
cc
cc Display (short) all tetrahedrons.
cc
ccin tetrahedron list TETNAME1 TETNAME2 TETNAME3 ...
cc
cc Display tetrahedrons TETNAME1, TETNAME2, TETNAME3,
cc ..., with or without subscripts (short display).
cc
ccin tetrahedron TETNAME
cc
cc Display (long) tetrahedron TETNAME.
cc
cc In the following commands, use "+" or "-" instead of
cc TETNAME or TET(2) to form the name by incrementing or
cc decrementing the base name for tetrahedrons.
cc See command "last", "increment names".
cc
ccin tetrahedron TETNAME = PNAME1 PNAME2 PNAME3 PNAME4
cc
cc Create tetrahedron TETNAME, with vertex points
cc PNAME1, PNAME2, PNAME3 and PNAME4. Any later
cc change in the coordinates or names of points
cc PNAME1, PNAME2, PNAME3 or PNAME4 will change the
cc definition of tetrahedron TETNAME.
cc
ccin tetra TET(2) incr NUMTET INC TET(1) IP1 IP2 IP3 IP4
cc
cc Create a family of NUMTET tetrahedrons TET(2), ...,
cc with names incremented by INC characters, by
cc incrementing the names of the four vertex points of
cc the preceding tetrahedron by the number of
cc characters indicated by the integers IP1, IP2, IP3,
cc IP4, starting from tetrahedron TET(1). Final
cc arguments of "0" may be omitted.
cc See "increment names".
cc Note: tetra is a synonym for tetrahedron.
cc Note: incr is a synonym for increment.
cc
cc This command can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc
cc Synonyms: [help, h], [increment, incr],
cc [tetrahedron, tet, tetra].
cc
cc See command "zone".
cc tetrahedron An option in commands "copy", "cluster", "delete",
cc "last", "list", "rename", "repack" and "sort", to
cc perform the specified operation on one or more
cc tetrahedrons.
cc Synonyms: [tetrahedron, tet, tetra].
cg tetrahedron arrays
cg Tetrahedrons may be created as arrays with
cg subscripted names. See "subscript",
cg "subscripted names", commands "do" and "enddo".
cg tetrahedrons A family of tetrahedrons may be created with
cg command "tetrahedron", option "increment".
cc TEXT In command "marker", any character string up to 72
cc characters long or whatever will fit in the
cc 80-character input line of command "marker".
cc Must be bracketed by single (') or double (") quotes.
cc If TEXT is bracketed by single quotes, double any
cc internal single quotes.
cc If TEXT is bracketed by double quotes, double any
cc internal double quotes.
cc The TEXT assigned to marker AMNAME will be displayed
cc whenever command "marker AMNAME" or
cc "marker all" is typed.
cc then An option in a command "if", which may be left blank,
cc to indicate that the following commands are to be
cc executed if the condition of this command "if" is
cc satisfied, up to the next command "endif", unless
cc this command "if" is not executed because of failure
cc of a condition of a preceding command "if".
cg theorum See "Chinese Remainder".
cc THETA The angular coordinate theta of a point, in a
cc cylindrical coordinate system. THETA is the azimuth
cc angle of the point, measured counterclockwise around
cc the z axis from the y plane, and ranging from 0 to
cc 360 degrees (0 to 2 * pi radians) or from -180 to
cc 180 degrees (-pi to pi radians). THETA is
cc indeterminate for a point on the z axis.
cc May be integer, floating point or a variable.
cc Used in commands "point", "vector", "plane".
cc THETA = arctan (Y / X)
cc theta An option in command "brick", to make surfaces of
cc constant theta the shared coordinate surfaces between
cc adjacent bricks.
cg theta The azimuth angle, measured counterclockwise around the
cg z axis from the y plane, and ranging from 0 to 360
cg degrees (0 to 2 * pi radians) or from -180 to 180
cg degrees (-pi to pi radians). You may put any angle
cg into that range with the function:
cg amod (theta + 360.0, 360.0) (degrees)
cg amod (theta + 2.0 * pi, 2.0 * pi) (radians)
cg repeated if necessary.
cg Used in cylindrical and spherical coordinate systems.
cg See "phi", "polar".
cg theta See "theta plane".
cg theta plane To create a plane at a constant value THETA of theta in
cg an (rcyl, theta, z) orthogonal coordinate system:
cg
cg coord cyl
cg plane plth = 1 THETA 0 0 1 0
cg
cg To create a family of planes at equally spaced angles
cg around an axis, use command "plane" with option
cg "rotate".
cg three See "three edges", "three equations".
cg three edges To solve a triangle, given the lengths of the three
cg edges, see "triangle solution".
cg three equations To solve three simultaneous linear equations for three
cg unknowns, use command "solve".
cc thru An option in commands "cluster", "delete", "pdf",
cc "variable" and "symbol", to include all existing
cc objects with names in the increasing ASCII sequence
cc from the name of the preceding object to the name of
cc the following object.
cc The order of available keyboard characters is:
cc blank !"#$%&'()*+,-./ 0-9 :;<=>?@ A-Z [\]^_` a-z
cc {|}^_`{|}~
cc See "character set".
cc The names need not be the names of existing objects,
cc but may be any convenient limiting names to include
cc the desired family of objects.
cc Note: for the stems of names, and for non-integer
cc subscripts, the sequence is left-adjusted ASCII, and
cc not numerical, so "x1 thru x10" will not include x2,
cc but "x2 thru x3" will include x200. For integer
cc stems and subscripts, the sequence is in numerical
cc order. To include unsubscripted object names in a
cc numerical sequence, make all names the same length,
cc ending in digits. E.g., x001, x002, ..., x100, x101,
cc ..., x200, ... will be included in the family of
cc objects "x000 thru x999", in that order.
cc
cc Do not name any object "thru", or if you do, change
cc its name with command "rename" before using a command
cc with option "thru".
cc thru An option in command "factor", to include all integers
cc with values from the preceding integer to the
cc following integer.
cg tic-tac-toe A game (O's and X's) which may be played with GEOM,
cg using input file tictactoe.mac to set up the pdf.
cg tile A tile is a surface bounded by two different constant
cg values of each of two coordinates of an orthogonal
cg coordinate system, at a constant value of the third
cg coordinate. To create a tile, create a brick with
cg the minimum and maximum values of one of the three
cg coordinates the same.
cc time Command to display the milliseconds of machine time
cc currently charged to the problem. To time a
cc procedure, do before and after, find the difference.
cc NOT CURRENTLY AVAILABLE ON THE OPEN YANA CLUSTER.
cc
cc Forms include:
cc
ccin help time
ccin time help
cc
cc Display the command options.
cc
ccin time
cc
cc Display the milliseconds of machine time charged to
cc the problem.
cc
cc Synonyms: [help, h], [time, t].
cc time An option in command "accelerate", to find the particle
cc position, velocity and path length at one or more
cc specified times.
cg time In a display of data resulting from command
cg "accelerate", the time measured from the initial time
cg at which the particle position and velocity are
cg specified. For option "time", ranges from TMIN to
cg TMAX, with time increment DT.
cc TITLE In command "title", a problem title and/or user ID,
cc with up to 72 characters.
cc If the input file is specified on the execution line,
cc e.g. "geom IN_FILE", the first line of the input
cc file should be command "title".
cc title An option in command "plot", to display or specify the
cc title of a plot.
cc title Command to display or specify the problem title.
cc If the input file is specified on the execution line,
cc the first input line should be command "title".
cc If input is from the user's terminal, the user is
cc prompted for command "title". The title may be
cc respecified at any time during a run, and is
cc displayed at the end of the run.
cc
cc Forms include:
cc
ccin help title
ccin title help
cc
cc Display the command options.
cc
ccin title
cc
cc Display the problem title.
cc
ccin title TITLE
cc
cc Specify the new problem title and/or user ID,
cc TITLE (up to 72 characters).
cc
cc Synonyms: [help, h], [title, id].
cg title A title for a table, plot or plot axis may be created
cg with command "marker".
cc TMAX In command "accelerate", the final time for which
cc path length, velocity and position are to be
cc displayed. If not specified, defaults to TMIN.
cc If specified, must not be less than TMIN.
cc See "DT", "TMIN".
cc TMIN In command "accelerate", the initial time for which
cc path length, velocity and position are to be
cc displayed. Must not be more than TMAX.
cc See "DT", "TMAX".
cc TOL The numerical tolerance limit, for controlling
cc truncation and round-off errors, and determining the
cc convergence of various iterative procedures.
cc May be specified with command "tol".
cc If not specified, TOL defaults to 10^(-11).
cc If the specified value of TOL is less than 0 or
cc larger than 0.001, TOL defaults to 10^(-11).
cc If TOL exceeds 10^(-14), floating point variables
cc will be displayed with 13 digits, else 16 digits.
cc Note: TOL is not a named variable.
cc
cc Used to provide a practical numerical way of dealing
cc with such geometrical concepts as objects that
cc coincide or intersect, are coaxial, congruent,
cc parallel, perpendicular, collinear, coplanar, etc.,
cc while avoiding meaningless residual values near zero.
cc
cc For example, if C = A - B or if C = A + B,
cc C is replaced by zero if less than the error estimate
cc TOL * (abs (A) + abs (B)).
cc
cc To test for numerical truncation effects, repeat the
cc calculations with various values of TOL, including
cc zero (but see below), and compare the results.
cc
cc Certain procedures require a nonzero value of TOL,
cc and will temporarily use the default value 10^(-11)
cc if the current value of TOL is zero. Such procedures
cc include evaluating quadric surfaces, finding the
cc real and complex roots of cubic and quartic
cc polynomials, and finding real roots, extrema and
cc inflection points of a higher order polynomial
cc equation by Newtonian iteration.
cc
cc The value of TOL should also be consistent with the
cc precision of data used as input in commands.
cc For example, if a value is cut and pasted from a
cc display with 13 significant figures, TOL should be
cc not smaller than 10^(-13) (input as "1.e-13").
cc
cc See "error estimate".
cc tol Command to display or specify the current value of the
cc tolerance limit TOL. See "error estimate".
cc
cc Command "tol" relates to objects: big, point,
cc symbol, variable, vector.
cc
cc Forms include:
cc
ccin help tol
ccin tol help
cc
cc Display the command options.
cc
ccin tol
cc
cc Display the current value of TOL.
cc
ccin tol [=] TOL
cc
cc Change the numerical tolerance limit to TOL, which
cc must be between 0.0 and 0.001.
cc The default value is 10^(-11).
cc
cc Synonyms: [help, h].
cg tolerance The amount of numerical truncation error allowed.
cg See "TOL", "tol", "error estimate".
cg toroid A toroid is a surface generated by rotating a closed
cg plane curve around an axis in the plane of the curve,
cg but not containing or passing through the curve.
cg A torus is a toroid for which the closed plane curve
cg is a circle.
cg
cg If the generating curve is an ellipse centered at
cg point (R0, Z0), with major and minor axes SR and SZ,
cg respectively, and the axis of rotation is the z axis,
cg then the equation of the toroid is
cg ((RCYL - R0)/SR)^2 + ((Z - Z0)/SZ)^2 = 1.
cg
cg See "surface of revolution".
cg
cg To find the distance from a point (X,Y,Z) to such a
cg toroid, create an elliptic cylinder whose equation is
cg ((X - R0) / SR)^2 + ((Z - Z0) / SZ)^2 = 1:
cg p PCEN = R0 0 Z0
cg ellipsoid QNAME PCEN SR 1E99 SZ
cg then create a new point ((X^2 + Y^2)^0.5, 0, Z):
cg s X2 = X ^ 2
cg s Y2 = Y ^ 2
cg s RCYL = X2 + Y2
cg s PNAME = RCYL 0 Z
cg then find the distance from the new point to the
cg quadric surface:
cg dist p PNAME quadric QNAME
cg torus See "toroid".
cg total See "total path".
cg total path The total path of a random walk process is the sum of
cg the displacements of each step.
cg Totient See "Totient function".
cc totient A synonym for factor. See "totient function".
cg totient See "totient function".
cg Totient function 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 are relatively prime to
cg N (have no common factors other than 1).
cg totient function Euler's Totient Function J(N) for modulo N arithmetic
cg is the number of integers in the set (1, ..., N - 1)
cg that are relatively prime to N (have no common factor
cg other than 1). This is the number of integers that
cg have multiplicative inverses for multiplication in
cg modulo N, and thus allow a solution when N is in the
cg denominator of a division problem in modular
cg arithmetic.
cg Zero can never have a multiplicative inverse in
cg modular multiplication.
cg Euler's totient function is also used when doing
cg exponentiation in modulo N arithmetic:
cg (K^M) mod N = (K mod N)^(M mod J(N)).
cg
cg If N has the prime factors P1, P2, ..., PN, i.e.,
cg N = P1^a1 * P2^a2 * ... * PN^aN, where a1, a2, ...,
cg aN are positive integer exponents, then J(N) can be
cg calculated as follows:
cg J(N) = N * (1-1/P1) * (1-1/P2) * ... * (1-1/PN).
cg This is because for each prime factor P, a fraction
cg 1/P of the numbers less than N are divisable by P,
cg and thus are not relatively prime to N.
cg
cg See command "factor".
cc trace Command to display the return path from the current
cc input file. Input files that have already executed
cc command "return" will not be displayed.
cc To see the entire input file history, use a system
cc command to display every line in the output file(s)
cc beginning with the pattern "input (new)", and the
cc following line.
cc
cc Forms include:
cc
ccin help trace
ccin trace [help]
cc
cc Display the command options for commands "trace"
cc and "input".
cc
ccin trace
cc
cc Display the return path from the current input
cc file.
cc
cc Synonyms: [help, h].
cg trace An invariant of a quadric surface, for translations and
cg rotations. See "invariants of quadric".
cg track A track is a straight line through 3-D space, specified
cg by an initial point and by a vector specifying the
cg direction of the track from that point. Command
cg "track" may be used to find the distance to, and
cg coordinates of, the intersection or point of closest
cg proximity of the track to a point, line, triangle,
cg plane, sphere, circular cylinder, circular cone,
cg ellipsoid, axially symmetric quadric surface, general
cg quadric surface or to all bounding surfaces of a
cg zone.
cg For planes and quadric surfaces, the angle between
cg the track, the normal vector at the intersection
cg point, and the direction of a reflected track are
cg also displayed.
cg
cg For the intersection of the parabolic trajectory of a
cg uniformly accelerated particle with a point, a plane
cg or any general quadric surface, see command
cg "accelerate".
cg
cg See commands "distance", "proximal", "side".
cc track Command to find the distance to and coordinates of
cc the point of intersection or closest proximity of a
cc linear track (specified by an initial point and a
cc direction vector) to a target, which may be a point,
cc line, triangle, plane, sphere, circular cylinder,
cc circular cone, hyperbolic paraboloid, ellipsoid,
cc axially symmetric quadric surface, general quadric
cc surface or all of the quadric surfaces bounding a
cc zone.
cc
cc For a linear track along a line ALNAME, first create
cc a vector VNAME parallel to the line with command:
cc vector VNAME line ALNAME
cc and use an endpoint of ALNAME for PNAME.
cc
cc The minimum distance found between the track and the
cc last target may be stored in variable VARNAME with
cc command:
cc variable VARNAME distance
cc
cc The point found on the track nearest to the last
cc target may be stored in point PNAME with the
cc command "point PNAME proximal".
cc
cc The vector between the proximal points on the track
cc and on the target may be stored in vector VNAME with
cc command "vector VNAME proximal".
cc
cc Command "track" relates to objects: axisym, cone,
cc cylinder, ellipsoid, hyperb, line, plane, point,
cc quadric, sphere, symbol, triangle, vector, zone.
cc
cc Forms include:
cc
ccin help track
ccin track [help]
cc
cc Display the command options.
cc
ccin track PNAME VNAME point PNAME2
cc
cc Find the distance to and coordinates of the point
cc nearest the point PNAME2 on the track from point
cc PNAME in direction VNAME, and the proximal
cc distance.
cc
ccin track PNAME VNAME line ALNAME
cc
cc Find the distance to and coordinates of the point
cc nearest the line ALNAME on the track from point
cc PNAME in direction VNAME, and the proximal
cc distance. Find the angle between the track and the
cc line at an intersection.
cc
ccin track PNAME VNAME triangle TRNAME
cc
cc Find the distance to and coordinates of the point
cc of intersection with the triangle TRNAME, on the
cc track from point PNAME in direction VNAME, and if
cc the track is parallel to the triangle, the proximal
cc distance. Find the angle between the track and the
cc normal vector at an intersection, and the reflected
cc track direction.
cc
ccin track PNAME VNAME plane PLNAME
cc
cc Find the distance to and coordinates of the point
cc of intersection with the plane PLNAME, on the track
cc from point PNAME in direction VNAME, and if the
cc track is parallel to the plane, the proximal
cc distance. Find the angle between the track and the
cc normal vector at an intersection, and the reflected
cc track direction.
cc
ccin track PNAME VNAME zone ZNAME
cc
cc Find the distances to and coordinates of all points
cc of intersection with the surfaces bounding zone
cc ZNAME, on the track from point PNAME in direction
cc VNAME. If the track is parallel to any planar
cc boundaries, the proximal distance will be found and
cc displayed. For all intersections, find the angle
cc between the track and the normal vector and the
cc reflected track direction.
cc
cc
ccin track PNAME VNAME QTYPE QNAME
cc
cc Find the distance to and coordinates of any
cc point(s) of intersection with the quadric surface
cc of type QTYPE (axisym, cone, cylinder, ellipsoid,
cc hyperb, plane, quadric or sphere), with name QNAME,
cc on the track from point PNAME in direction VNAME.
cc There may be zero, one or two intersections.
cc If no intersection is found, find the point of
cc minimum F(x,y,z) along the track, where F = 0 is
cc the equation of the quadric surface. This may not
cc be the proximal point, except for a sphere or
cc cylinder. For each intersection, find the normal
cc vector of the quadric surface, the angle between
cc the track and the normal vector, and the reflected
cc track direction. If the angle is zero or 180
cc degrees, the intersection is a proximal point for
cc nearby points on the track.
cc
cc Synonyms: [cylinder, cyl],
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], [triangle, tri],
cc [zone, z, zn].
cc
cc See commands "accelerate", "distance", "proximal",
cc "side", "variable".
cg track See "track reflection".
cg track reflection The reflection direction VREFL of a track with initial
cg direction vector VINIT, reflected in a surface with
cg normal vector VNORM, is found as follows:
cg UNORM = VNORM / |VNORM|,
cg VREFL = VINIT - 2 * (VINIT dot UNORM) * UNORM
cg tracking See "track", "accelerate".
cg trajectory See "accelerate", "acceleration", "path", "track".
cc trans A synonym for move.
cg transform See "transform the mesh".
cg transform the mesh
cg To rotate, invert, reflect or move the logical mesh,
cg see command "mesh", options "exchange", "invert" and
cg "move".
cg
cg Rotations, inversions and reflections of the mesh
cg block all have the net effect of replacing the mesh
cg index directions (k, l, m) with a different
cg permutation of the index directions, with an
cg arbitrary sign for each index direction. To do such
cg a transformation of the mesh block, first invert
cg each index that changes sign, using command "move",
cg option "invert". Then change the order of the
cg indices, using command "mesh", option "exchange".
cg No more than two exchanges are needed.
cg For example, to transform the indices from (k, l, m)
cg to (l -m, -k):
cg
cg mesh invert block k ! (-k, l, m)
cg mesh invert block m ! (-k, l, -m)
cg mesh exchange block k m ! (-m, l, -k)
cg mesh exchange block k l ! ( l, -m, -k)
cg Note that the last command exchanges the current k
cg and l indices, not the original k and l indices.
cg
cg To move the mesh block, use command "mesh", option
cg "move".
cg transformation To transform from one coordinate system to another,
cg see "coordinate system", "coordinate conversion",
cg "vector conversion" or use command "coordinate".
cc translate A synonym for move.
cg translation Vectors may be used as translation operators to move
cg points, clusters, vectors, planes or quadric
cg surfaces, with command "move", or to create a family
cg of translated points with command "point", option
cg "move", or to create a family of translated quadric
cg surfaces with command "axisym", "cone", "cylinder",
cg "ellipsoid", "plane", "quadric" or "sphere",
cg option "move".
cg
cg See "transform the mesh".
cg transverse On a hyperbola or hyperbolic cylinder, the transverse
cg semiaxis is the line from the origin to a vertex.
cg Its length is half the distance between the vertices.
cg See "conjugate".
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 transverse semiaxis is:
cg sqrt (-QC / QYY), if QC > 0, and
cg sqrt (-QC / QXX), 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 transverse semiaxis is:
cg sqrt (-QC / QYY), in the plane x = 0, and
cg sqrt (-QC / QXX), in the 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 length of the transverse semiaxis is:
cg sqrt (-QC / QZZ).
cc tri A synonym for triangle.
cg triangle A geometric object in 3-D space, specified by three
cg points. The triangle 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 triangle. However, a change in the name of such a
cg point will not change the triangle.
cg A triangle may be used to create a plane with command
cg "plane" or a circular cylinder with command
cg "cylinder".
cg The maximum number of triangles is now 1000.
cg
cg Triangles may be sorted by name with command "sort".
cg
cg The following commands relate to triangles:
cg copy, debug, delete, distance, help, last, list,
cg plane, point, rename, repack, search, sort, steiner,
cg symbol, synonym, tables, track, triangle.
cg
cg The following are included in the data display for
cg a triangle: its vertices, its normal vector, its
cg area, its edge lengths, its vertex angles,
cg its centroid, at the intersection of the medians
cg (from the vertices to the midpoints of the opposite
cg edges); its orthocenter, at the intersection of the
cg altitudes (from the vertices perpendicularly to the
cg opposite edge); the center of its inscribed circle,
cg at the intersection of the bisectors of the vertex
cg angles; and the center of its circumscribed circle,
cg at the intersection of the perpendicular bisectors
cg of the edges.
cg To solve a triangle with missing parts, see "trig".
cc triangle An option in command "debug", to display GEOM internal
cc parameters and variables for triangles:
cc ntrigm, ntrigs, atrig, atript1, latript1, atript2,
cc latript2, atript3, latript3, atrigs, latrigs.
cc Synonyms: [triangle, tri].
cc triangle Command to display or create triangles. Creating a
cc triangle replaces any existing triangle having the
cc same name. The display for a triangle includes
cc the names and coordinates of the vertex points,
cc the normal vector, the area, the edge lengths,
cc the angles, the Steiner vertex, and other data,
cc including tangent circles at the vertices, each
cc straight line cut that partitions the triangle into
cc sections with equal perimeters and areas, and the
cc equilateral triangle that projects onto the triangle.
cc See commands "kiss", "trig", "cuts", "cute".
cc See "do loop use".
cc
cc To find the missing parts of a triangle for which
cc only three of the six parts (three angles and three
cc edges) are known, use command "trig".
cc
cc Command "triangle" relates to objects: point,
cc symbol, triangle, variable.
cc
cc Forms include:
cc
ccin help triangle
ccin triangle [help]
cc
cc Display the command options.
cc
ccin triangle [all,list]
cc
cc Display all triangles (short display).
cc
ccin triangle list TRNAME1 TRNAME2 TRNAME3 ...
cc
cc Display triangles TRNAME1, TRNAME2, TRNAME3, ...,
cc with or without subscripts (short display).
cc
ccin triangle TRNAME
cc
cc Display triangle TRNAME (long display).
cc
cc In the following command, use "+" or "-" instead of
cc TRNAME to form the name by incrementing or
cc decrementing the base name for triangles.
cc See command "last", "increment names".
cc
ccin triangle TRNAME = PNAME1 PNAME2 PNAME3
cc
cc Create triangle TRNAME, through points PNAME1,
cc PNAME2 and PNAME3. Any later change in the
cc coordinates or names of points PNAME1, PNAME2 or
cc PNAME3 will change the definition of triangle
cc TRNAME.
cc
ccin triangle TR(2) incr NUMTR INC TR(1) IP1 [IP2 [IP3]]
cc
cc Create a family of NUMTR triangles TR(2), ...,
cc with names incremented by INC characters, by
cc incrementing the names of the three vertex points
cc of the preceding triangle by the number of
cc characters indicated by the integers IP1, IP2, IP3,
cc starting from triangle TR(1). Final arguments of
cc "0" may be omitted. See "increment names".
cc Note: incr is a synonym for increment.
cc
cc This command can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc
cc Synonyms: [help, h], [increment, incr],
cc [triangle, tri].
cc triangle An option in commands "copy", "delete", "last", "list",
cc "rename", "repack" and "sort", to perform the
cc specified operation on one or more triangles.
cc Synonyms: [triangle, tri].
cg triangle See "triangle cutting", "triangle solution".
cg triangle arrays Triangles may be created as arrays with subscripted
cg names. See "subscript", "subscripted names",
cg commands "do" and "enddo".
cg triangle cutting Any triangle may be cut by a single line into two
cg figures (a triangle and a triangle or a quadrangle)
cg of equal areas and perimeters. Some triangles may
cg be so cut in three different ways. There is always
cg one cut from the longest edge to the shortest edge.
cg There may be no cuts, one cut or two cuts of equal
cg lengths from the longest edge to the middle edge.
cg An isosceles triangle always has one cut from the
cg symmetric vertex to the center of the opposite edge.
cg An equilateral triangle has three such cuts.
cg
cg All equally partitioning cuts pass through the center
cg of the inscribed circle, which is at the intersection
cg of the lines bisecting the vertex angles. For a
cg triangle with vertex coordinates A = (Ax, Ay, Az),
cg B = (Bx, By, Bz) and C = (Cx, Cy, Cz), with opposite
cg edges a, b and c, and perimeter p = a + b + c, the
cg center of the inscribed circle is at point D:
cg D = (a * A + b * B + c * C) / p.
cg
cg For a triangle with vertices A, B an C, and opposite
cg edges a, b, and c, respectively, the fractional
cg distances alpha and beta of the ends of any cutting
cg line along edges a and b from vertex C (two possible
cg solutions) are as follows:
cg alpha(1) = 0.25 * (p + sqrt (p^2 - 8*a*b)) / a
cg beta(1) = 0.5 / alpha(1)
cg alpha(2) = 2 * b / (p + sqrt (p^2 - 8*a*b))
cg beta(2) = 0.5 / alpha(2)
cg where p = a + b + c, the perimeter of the triangle.
cg To be valid, alpha must be real (p^2 cannot be less
cg than 8*a*b), and alpha must be between 0.5 and 1.
cg These results follow from the requirements of equal
cg area, alpha * beta = 0.5, and equal perimeters,
cg alpha * a + beta * b = 0.5 * p.
cg If either alpha or beta is 1, the cut is through
cg vertex B or A, respectively.
cg
cg The small triangular section that includes vertex C
cg has sides a', b' and c', where a' = alpha * a,
cg b' = beta * b, and the length of the cutting line
cg is c' = sqrt (p * (c - 0.25 * p)).
cg
cg The fractional distance, F, of the center of the
cg inscribed circle along the cutting line from edge
cg a to edge b is given by:
cg F = 2 * alpha * a / p = 1 - 2 * beta * b / p
cg
cg Equations equivalent to those above for vertex C
cg apply to vertices A and B, also, by suitable
cg permutation of edges a, b, and c. The total number
cg of equally partitioning cuts may be one, two or
cg three.
cg
cg For any triangle created or used in GEOM, the
cg equally partitioning cutting lines are displayed.
cg
cg See "triangle cutting", commands "cuts", "cute",
cg "triangle", and "trig".
cg triangle solution
cg The missing parts of a triangle may be found, given
cg three of its parts (three angles and three edges),
cg with command "trig".
cg
cg A triangle may be defined by specifying:
cg
cg the lengths of three edges, each shorter than the sum
cg of the other two;
cg
cg the lengths of two edges and any positive angle less
cg than 180 degrees or pi radians (zero, one or two
cg solutions are possible, if the specified angle is not
cg between the two specified edges);
cg
cg the length of one edge and two positive angles, each
cg less than 180 degrees or pi radians, and not totaling
cg more than 180 degrees or pi radians.
cg
cg The temporary vertex points displayed may be used to
cg create points, and then used to create a triangle.
cg
cg The triangle may be scaled, translated, inverted or
cg rotated as desired after it is specified.
cg
cg See "Law of Cosines", "Law of Sines", command "trig".
cg triangles A family of triangle may be created with command
cg "triangle", option "increment".
cg trig See "trig functions", command "trig".
cc trig Command to find the missing parts of a triangle, given
cc three of its 6 parts (three edges and three angles).
cc Vertex points in the plane z = 0 will be displayed
cc also, with one at the origin, and one on the x axis.
cc The vertex points and triangle are given temporary
cc names, but are not saved in memory for later use.
cc The display for the triangle also includes the normal
cc vector, the area, the edge lengths, the angles, and
cc other data, including tangent circles at the
cc vertices, and each straight line cut that partitions
cc the triangle into sections with equal perimeters and
cc areas. See commands "kiss", "triangle", "cuts",
cc "cute".
cc
cc Command "trig" relates to objects: symbol, variable.
cc
cc Forms include:
cc
ccin help trig
ccin trig [help]
cc
cc Display the command options.
cc
ccin trig [angle,edge] VA [angle,edge] VB [angle,edge] VC
cc
cc Find the missing triangle parts, given an angle
cc (in the current angle units) or edge length VA, the
cc adjacent angle or edge length VB, and the adjacent
cc angle or edge length VC, in counterclockwise order
cc around the triangle. Edge lengths must be
cc positive. If three edges are specified, the sum of
cc the lengths of any two edges must exceed the length
cc of the other edge. Vertex angles must be positive.
cc Any single angle or sum of two angles must be less
cc than 180 degrees or pi radians. If three angles
cc are specified, their sum must be 180 degrees or pi
cc radians. If the last two arguments are missing,
cc they will default to "edge 1".
cc Arguments VA, VB and VC may be integer or floatint
cc point values or variables.
cc
cc Synonyms: [angle, ang], [help, h], [edge, side].
cg trig functions To do math using the FORTRAN direct and inverse trig
cg functions sin, cos, tan, asin, acos, atan, atan2,
cg use command "variable", option "function".
cg trigonometry See "triangle", "trig", "trig functions".
cg trip count The number of times a do loop is executed, including
cg the first time, is the trip count. For the arguments
cg in command "do", the trip count is the integer value
cg not exceeding 1 + (N2 - N1) / INC.
cg triple See "triple point", "triple product", command "triple".
cg triple In output from command "roots", indicates a root of
cg order 3 of a polynomial equation.
cg In output from command "root", indicates a root of
cg order 3 or more of a polynomial equation.
cc triple An option in command "variable", to find the scalar
cc triple product of three vectors.
cc triple An option in command "vector", to find the vector
cc triple product of three vectors.
cc triple An option in command "operator", to create a tensor
cc operator that performs a rotation of vector A to be
cc parallel to vector C, and the plane containing
cc vectors A and B to be parallel to the plane
cc containing vectors C and D.
cc If vectors A and B are perpendicular, and vectors
cc C and D are perpendicular, then this rotates the
cc positive triple (A, B, A cross B) to be aligned with
cc the positive triple (C, D, C cross D), or
cc equivalently, respecifies the coordinate axes to be
cc (A, B, A cross B) instead of (C, D, C cross D).
cc See "axial", "planar", "serial", "vector".
cc triple A command to search for triple points of intersection
cc between three quadric surfaces. This is equivalent
cc to solving three simultaneous second order equations
cc in three unknowns. With 0, 1, 2 or 3 planes, no more
cc than 8, 4, 2 or 1 triple points are possible. The
cc method always works if two or three of the surfaces
cc are planes.
cc
cc If two or more of the quadric surfaces are nonplanar,
cc an iterative method is used, with 64 randomly sampled
cc initial guesses, and all apparently distinct results
cc are displayed. To search for more triple points,
cc repeat the command.
cc
cc TOL is used both as an absolute and relative
cc convergence criterion. Any triple point coordinates
cc less than 10 * TOL are treated as insignificant.
cc
cc This command may not find some triple points, where
cc two or three of the surfaces are tangent or nearly
cc so. Use command "distance", with pairs of surfaces,
cc to search for intersections or tangencies.
cc
cc Command "triple" relates to objects: axisym, cone,
cc cylinder, ellipsoid, hyperb, plane, point, quadric,
cc symbol, sphere.
cc
cc Forms include:
cc
ccin help triple
ccin triple [help]
cc
cc Display the command options.
cc
ccin triple QNAME1 QNAME2 QNAME3 [point PNAME]
cc
cc Search for triple points of intersection of the
cc three quadric surfaces QNAME1, QNAME2 and QNAME3,
cc with initial guess point PNAME.
cc Always works when two or three of the surfaces are
cc planes. May not find all triple points of three
cc nonplanar surfaces.
cc Try to find linear combinations of the quadric
cc surface equations that eliminate the quadric terms.
cc
cc The relationship between any pair of quadric surfaces
cc QNAMEi and QNAMEj may be found with command
cc "distance quadric QNAMEi quadric QNAMEj". Repeat
cc with option "plane" instead of "quadric" for planes,
cc for more information.
cc
cc The last triple point found by command "triple" may
cc be stored in point PNAME with command
cc "point PNAME proximal". The vector from the origin
cc to the last triple point may be stored in vector
cc VNAME with command "vector VNAME proximal".
cc
cc The normal vectors and extreme curvatures of each of
cc the surfaces at a triple point may be displayed with
cc command "arc". The angles of incidence of each
cc pair of surfaces at a triple point may be displayed
cc with command "variable", option "angle" with the
cc two normal vectors.
cc
cc Synonyms: [help, h], [point, p, pnt, pt].
cg triple point A triple point is the point of intersection of three
cg surfaces. If two or more of those surfaces are
cg planes, any triple points can be found with command
cg "triple".
cg
cg If all three surfaces are nonplanar, use command
cg "triple" to search for triple points, or use
cg command "distance" with each of the three pairs of
cg surfaces, to find if they intersect or are tangent.
cg triple product The scalar triple product of the three vectors VNAME1,
cg VNAME2 and VNAME3 is the scalar value
cg (VNAME1 cross VNAME2) dot VNAME3 =
cg VNAME1 dot (VNAME2 cross VNAME3),
cg which may be found with the command:
cg
cg variable VARNAME triple VNAME1 VNAME2 VNAME3
cg
cg Create variable VARNAME with the value of the
cg scalar triple product:
cg (VNAME1 cross VNAME2) dot VNAME3.
cg
cg The scalar triple product is the volume of a
cg parallelepiped which has the three edge vectors
cg VNAME1, VNAME2 and VNAME3 at one vertex, in the
cg sequence given by the right hand rule (thumb out,
cg forefinger out, middle finger up).
cg
cg The scalar triple product is also the value of the
cg 3 by 3 determinant for which the three vectors are
cg either the row or column vectors.
cg
cg The vector triple product of the three vectors
cg VNAME1, VNAME2 and VNAME3 may be either the vector
cg VNAME1 cross (VNAME2 cross VNAME3) or the vector
cg (VNAME1 cross VNAME2) cross VNAME3
cg These may be found with the commands:
cg
cg vector VNAME triple VNAME1 VNAME2 VNAME3
cg
cg Create vector VNAME with the value of the
cg vector triple product:
cg VNAME1 cross (VNAME2 cross VNAME3)
cg
cg vector VNAME triple VNAME3 VNAME2 VNAME1
cg
cg Create vector VNAME with the value of the
cg vector triple product:
cg (VNAME1 cross VNAME2) cross VNAME3
cc trk A synonym for track.
cc TRNAME The name of a triangle. May have up to 24 characters,
cc including any subscripts, and may be ASCII, integer,
cc or floating point.
cc No triangle name may be "+", "-", "all", "h", "help",
cc "list" or "thru", begin with "!" or contain ";".
cc Specified with command "triangle".
cc Also referred to as TRNAME1, TRNAME2, TRNAME3, ...
cc
cc Triangles may appear in commands:
cc copy, delete, distance, intersect, plane, point,
cc rename, track, triangle.
cc Also see commands: last, list, repack, search, sort.
cg truncation See "truncation error".
cg truncation error Numerical truncation errors occur in almost all
cg numerical calculations. Numerical tolerance limit
cg TOL is used to control the effects of such numerical
cg truncation errors, and to allow decisions to be made
cg without having exact data. To test for such effects,
cg repeat the calculations with various values of TOL,
cg including zero. See "error estimate".
cg tuning See "piano tuning".
cg turn See "turn off".
cg turn off Some environmental parameters may be turned on and off.
cg See "create environment", "off".
cc tw A synonym for twist.
cg twist A twist is an operation that rotates a point around
cg an axis in proportion to the axial distance of the
cg point from a specified point on the axis (option
cg "axial"), or in proportion to the radial distance
cg from a specified radius from the axis (option
cg "radial"). The pitch is the axial or radial distance
cg for one complete revolution around the axis.
cc twist Command to twist points around an axis, depending on
cc the axial or radial distance of each point.
cc
cc Command "twist" relates to objects: cluster, point,
cc symbol, variable, vector.
cc
cc Forms include:
cc
ccin help twist
ccin twist [help]
cc
cc Display the command options.
cc
ccin twist point PNAME PAXIS VAXIS axial PITCH
cc
cc Twist point PNAME around the axis through point
cc PAXIS in the direction of vector VAXIS, in
cc proportion to its axial distance from point PAXIS,
cc with one clockwise rotation per axial distance
cc increment PITCH. Make PITCH negative for a
cc counterclockwise rotation. A row of points in a
cc straight line parallel to the axis would be twisted
cc to lie on a curve like a coil spring.
cc
ccin twist point all PAXIS VAXIS axial PITCH
cc
cc As above, for all points.
cc
ccin twist cluster CLNAME PAXIS VAXIS axial PITCH
cc As above, for all points in cluster CLNAME.
cc
ccin twist point PNAME PAXIS VAXIS radial PITCH RINV
cc
cc Twist point PNAME around the axis through point
cc PAXIS in the direction of vector VAXIS, in
cc proportion to its radial distance from radius RINV,
cc with one clockwise rotation per radial distance
cc increment PITCH outward from radius RINV. Make
cc PITCH negative for a counterclockwise rotation.
cc A row of points in a straight line perpendicular to
cc the axis would be twisted to lie on an Archimedean
cc spiral.
cc
ccin twist point all PAXIS VAXIS radial PITCH RINV
cc
cc As above, for all points.
cc
ccin twist cluster CLNAME PAXIS VAXIS radial PITCH RINV
cc
cc As above, for all points in cluster CLNAME.
cc
cc Synonyms: [help, h], [cluster, cl],
cc [point, p, pnt, pt].
cg type See "quadric types", "object type".
cg type of bin See "bin types".
cg type of quadric See "quadric types".
cc types An option in command "quadric", to display a list of
cc quadric surface types.
cg types See "commands".
cc types of commands
cc See "commands".
cc typing See "shortcuts".
U-------------------------------------------------------------------------------
cc U In commands "point", "vector" or "plane", cc a coordinate in the x, rcyl or rsph direction, cc for a rectangular, cylindrical or spherical cc coordinate system, respectively. cc May be integer, floating point or a variable. cc cc In command "operator", the name of an axis (x, y cc or z). cc u A synonym for undo. cg u Indicates a major axis of the current coordinate cg system. The first coordinate displayed by cg command "coordinate". cc U1 See "U1, U2". cc U1, U2 The u coordinates (x, y or z) of the centers of two cc circles, in command "intcirc". cc U2 See "U1, U2". cc UANGLE In command "operator", option "serial", the angle cc (counterclockwise, with the axis pointed at the cc observer) of rotation around the u axis (x, y or z). cc May be integer, floating point or a variable. cg ultimate See "value", "symbol". cg ultimate value See "value", "symbol". cc UMAX In command "brick", the maximum value of the first cc coordinate, which may be x, rcyl or rsph, cc for a rectangular, cylindrical or spherical cc coordinate system, respectively. cc Must not be less than UMIN. cc UMIN In command "brick", the minimum value of the first cc coordinate, which may be x, rcyl or rsph, cc for a rectangular, cylindrical or spherical cc coordinate system, respectively. cc Must not be more than UMAX. cg UNASSIGNED In the display for a point, indicates the point has cg not been assigned mesh indices. cg undefined A geometric object is undefined if it has not yet been cg created, or if it has been created and deleted, so cg no geometric object of the required type currently cg exists with the name specified in a command. cg underline The underline character, "_". cc undo Command to undo one or more preceding commands. cc This is most useful when you have unintentionally cc deleted or changed an object, but can double the cc amount of machine time used up to the time of the cc error, because is undoes the last N commands by cc repeating all but the last N commands used in the cc current session. See commands "redo", "indo". 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 Command "undo" relates to objects: symbol, variable. cc cc Forms include: cc ccin help undo ccin undo help cc cc Display the command options. cc ccin undo [$] cc cc Delete the preceding command, by repeating all cc preceding commands except the last command. cc ccin undo [N|1] cc cc Undo the preceding N commands, by repeating all cc preceding commands except the last N cc commands. cc cc Synonyms: [help, h], [undo, u]. cc cc If the error may be undone by correcting and cc repeating one or a few of the preceding commands, cc use command "input command -N $ ?" to display the cc last N + 1 commands, then use command cc "input command LINE1 LINE2" and/or command cc "redo LINE [R1 R2 R3 ...]" and/or command cc "indo LINE [I1 I2 I3 ...]'" to correct the error. cg undo See "undo errors". cg undo errors See commands "undo", "redo", "indo". If you have ended cg the run, see "input file" for how to make a new input cg file to repeat the run without the error(s). cc uniform A synonym for scale in command "operator". cg uniform A uniform probability distribution is a type of cg probability bin or probability distribution function cg for which the relative differential probability is cg constant for a range of values of a random variable. cg A type of probability distribution function (pdf) cg consisting of a set of such probability bins is cg called a histogram. cg See "discrete event", "discrete value", "linear", cg "power-law", "exponential". cg See commands "bin", "pdf", "sample". cc uniform An option in command "bin", to create a probability cc bin with a uniform differential probability cc (a histogram). cg uniform For uniform acceleration, see command "accelerate". cg union See "Boolean". cc unit An option in command "vector", to make the vector cc length 1. cc Also see "unit conversion", "conv.mac". cg unit See "unit conversion". cg unit conversion Selected metric conversion factors are in macro file cg conv.mac in directory ~edwards/work/geom/test . cg For angle unit conversion, see "rad/deg", "deg/rad". cg units See commands "angles", "coordinates". cg See "conv.mac", "conversion factors". cg UNIX A computer operating system. cg unrecognized A command or command argument is unrecognized if it cg is not found in a list of possible commands or cg command arguments. cg up arrow The power, superscript or up arrow symbol, "^". cg user The person or program that causes the GEOM code to be cg executed, and controls it while it is executing. cg user's terminal The computer station where the user executes the cg GEOM code from a keyboard and monitor. cg A copy of most output is displayed here, unless cg redirected to a file (e.g. STDOUT) by ending cg the execution line with " > STDOUT". Use ">!" to cg write over an existing file STDOUT.
V-------------------------------------------------------------------------------
cc V In command "variable", the same as VARNAME. cc V In commands "point", "vector" or "plane", cc a coordinate in the y, theta or theta direction, cc for a rectangular, cylindrical or spherical cc coordinate system, respectively. cc May be integer, floating point or a variable. cc cc In command "operator", the name of an axis (x, y cc or z). cc v A synonym for vector. cc v An option im command "plot axis', to specify the cc coordinate and label of the vertical plot axis. cc An option in command "plot limits", to specify the cc minimum and maximum values SMIN and SMAX of the cc vertical axis coordinate. cc An option in command "plot zoom", to specify a cc multiplier ZOOMULT for the zoom factor for the cc vertical axis. cg v Indicates a major axis of the current coordinate cg system. The second coordinate displayed by cg command "coordinate". cc V1 See "V1, V2". cc V1, V2 The v coordinates (x, y or z) of the centers of two cc circles, in command "intcirc". cc V2 See "V1, V2". cc VA, VB, VC In command "variable", one of two arguments of a cc mathematical function. May be integer, floating cc point, or the name of an integer or floating point cc variable. cc cc In command "if", one of two arguments which are to cc be tested to see if they satisfy a specified cc relationship. May be integer, floating point, or the cc name of an integer or floating point variable. cc cc In command "trig", the angles and/or edges specified. cc May be integer, floating point, or the name of an cc integer or floating point variable. cc Must be positive. If an angle, in the current angle cc units, and less that 180 degrees or pi radians. cc cc In commands "cute" and "cuts", the triangle edges cc specified. cc May be integer, floating point, or the name of an cc integer or floating point variable. cc Must be positive, but less than the sum of the cc other two edge lengths. cc cc In command "solve", the vector of the coefficients cc of the unknown variable x in the three equations. cc VACC In command "accelerate", the name of the vector cc specifying the constant acceleration applied to a cc particle to create a parabolic trajectory. cg VACUUM In the display for a point, indicates a vacuum mesh cg point, with coordinates cg (x, y, z) = (-10^99, -10^99, -10^99) cg vacuum A mesh point represents a vacuum, if it has coordinates cg (x, y, z) = (-10^99, -10^99, -10^99). Such mesh cg points may be created with commands cg "mesh point PMESH", "mesh point PMESH block". cc vacuum An option in command "cluster", to make a cluster of cc all mesh points with vacuum coordinates. cc VAL The coordinate of a plane parallel to a major plane in cc command "quadric". Defaults to 0. cc VALUE A command argument. May be a symbol or an cc integer or floating point variable or constant. cc VALUE In an optional command argument, indicated by cc [ARGUMENT|VALUE], the default value of ARGUMENT. cc value An option in command "bin", to create a bin for a cc discrete value of a random variable. cc value An option in command "roots", to evaluate a polynomial cc at a specified argument, and to try to find a root cc of the polynomial near that argument. cc See command "variable", option "polynomial". cg value The value of a variable is the integer or cg floating point value that will replace the name of cg the variable, when the variable is used in a cg command, unless the variable name is prefixed (with cg no delimiter) with a minus sign, in which case cg the negative of that value will be used. cg cg To display the replacement value of any variable cg VARNAME, type "var VARNAME". cg cg See "search", command "variable". cc VALUE1, VALUE2 In command "base", a decimal value to be converted to cc a specified number base. May be integer or floating cc point. Also referred to as VALUE3, VALUE4, ... cg vane To create a family of planes at equally spaced angles cg around a common axis, use command "plane" with cg option "rotate". cc VANGLE In command "operator", option "serial", the angle cc (counterclockwise, with the axis pointed at the cc observer) of rotation around the v axis (x, y or z). cc May be integer, floating point or a variable. cc var A synonym for variable. cc VAR(N), VAR(N-1) Names of variables. See "names". cg variable A variable may be of type integer or floating cg point, and is created with command "variable". cg The maximum number of variables is now 1000. cg For ASCII variables, see "symbol". cg cg In all commands (except for LINE in commands "indo" cg and "redo"), after replacement of aliases and cg symbols, all integer or floating point arguments are cg replaced with their values, and all floating point cg variable names used for floating point arguments are cg replaced with their values. See option "data". cg cg A variable must have a single-character name that is cg a lower case letter, or a multi-character name that cg begins with a lower or upper case letter or one of cg the characters "$", "%" or "@", is not "random" or cg "tol", and does not contain ";", and may be up to 24 cg characters long, including any subscripts. cg cg An integer variable has an integer value and an cg equivalent floating point value, and may be used in cg place of any command argument that allows either an cg integer or a floating point value. Either an integer cg or an integer variable must be used for any command cg argument that requires an integer value (as indicated cg by an upper case name that begins with I-N). cg See command "big" for operations with big integers, cg with up to 1001 digits. cg cg A floating point variable has a floating point value, cg and may be used in place of any command argument that cg allows a floating point value (as indicated by an cg upper case name that begins with A-H or O-Z, that cg is not an object name). 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 cg The following commands relate to variables: cg accelerate, angles, axisym, big, bin, brick, cone, cg contfr, copy, crt, cute, cuts, cylinder, debug, cg delete, disk, ellipsoid, factor, goto, help, hex, if, cg increment, indo, input, intcirc, iris, kiss, last, cg line, list, mcvol, mesh, move, operator, plane, plot, cg point, polygon, polyhedron, project, proximal, cg quadric, ratio, redo, rename, repack, root, rootf, cg roots, sample, search, slice, solve, sort, sphere, cg spin, symbol, synonym, tables, tetrahedron, tol, cg triangle, trig, twist, undo, variable, vector, walk, cg zone. cg cg It may be useful to create a series of floating point cg variables, with names differing by the same number cg of characters, to use as arguments in a command which cg will be repeated with modifications by using command cg "indo". This allows floating point arguments to be cg incremented by making integer increments in the cg floating point variable names. cc variable An option in command "debug", to display GEOM internal cc parameters and variables for variables: cc nvarm, nvars, avarbl, fvar, ivar, lvar, mvar, cc avartag, avarbls, lavarbls. cc Synonyms: [variable, s, set, var]. cc variable Command to display or create an integer or floating cc point variable. Creating a variable replaces any cc existing variable having the same name. cc See "do loop use". cc cc The name of an integer or floating point variable cc may be up to 24 characters long, including any cc subscripts, and must begin with an upper or lower cc case letter or one of the characters "$", "%" or "@", cc may not contain the character ";", and must not be cc "random" or "tol". cc cc Such a variable will be integer if its fractional cc difference from an integer is no more than TOL, cc otherwise it will be floating point. cc cc The display includes the variable name, value and cc type. Integer variables will be displayed in integer cc format, and if larger in absolute value than 1000000, cc in floating point format also. Floating point values cc will be displayed with 13 digits if TOL exceeds cc 10^(-15), otherwise 16 digits. Results may be cc rounded to zero or the nearest integer, depending on cc TOL. cc cc Use command "icalc" to enter or display integer data cc in decimal, octal or hexadecimal integer mode or to cc display integer data in binary mode. cc 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 other info, cc with up to 48 characters specified between single or cc double quotes. cc cc Command "variable" relates to objects: symbol, cc variable, vector. cc cc Forms include: cc ccin help variable ccin variable [help] cc cc Display the command options. cc ccin variable [all,list] cc cc Display all variables. cc ccin variable list VARNAME1 thru VARNAME2 cc cc Display all variables with names in the range cc from VARNAME1 thru VARNAME2, with or without cc subscripts. cc ccin variable list VARNAME1 VARNAME2 VARNAME3 ... cc cc Display variables VARNAME1, VARNAME2, cc VARNAME3, ..., with or without subscripts. cc ccin variable VARNAME cc cc Display variable VARNAME. cc cc In the following commands, use "+" or "-" instead of cc VARNAME or VAR(2) to form the name by incrementing or cc decrementing the base name for variables. cc See command "last", "increment names". cc ccin variable VARNAME [=] FVAR cc cc Create floating point variable VARNAME with cc floating point value FVAR, which may be explicitly cc floating point, a floating point variable or a cc symbolic word which after symbol replacement, is cc floating point. The "=" is optional if FVAR is cc explicitly floating point or a symbolic word which, cc after symbol replacement, is floating point. cc ccin variable VARNAME [=] IVAR cc cc Create integer variable VARNAME with integer value cc IVAR, which may be explicitly integer, an integer cc variable or a symbolic word which, after symbol cc replacement, is an integer. The "=" is optional if cc IVAR is explicitly integer or a symbolic word cc which, after symbol replacement, is an integer. cc cc variable VARNAME tag 'VARTAG' cc cc Assign the descriptive tag VARTAG to variable cc VARNAME. The tag VARTAG may have up to 48 cc characters bracketed between single or double cc quotes. cc ccin variable VARNAME big BIGNAME cc cc Create variable VARNAME with the value of big cc integer BIGNAME, if it will fit in a machine word. cc ccin variable VARNAME random cc cc Create variable VARNAME with a random value cc uniformly distributed between 0 and 1. cc ccin variable VARNAME random VMIN VMAX cc cc Create variable VARNAME with a random value cc uniformly distributed between VMIN and VMAX. cc ccin variable VARNAME random VMIN VMAX cc (followed by) ccin variable VARNAME = 1 nint VARNAME cc cc Create variable VARNAME with a random integer value cc uniformly distributed between VMIN and VMAX. cc ccin variable VARNAME vector VNAME cc cc Create variable VARNAME with the value of the cc length of vector VNAME. cc ccin variable VARNAME angle VNAME1 VNAME2 cc cc Create variable VARNAME with the value of the angle cc between the two vectors VNAME1 and VNAME2, in the cc current angle units. See "angles". cc ccin variable VARNAME dot VNAME1 VNAME2 cc cc Create variable VARNAME with the value of the dot cc (inner) product of the two vectors VNAME1 and cc VNAME2. cc ccin variable VARNAME triple VNAME1 VNAME2 VNAME3 cc cc Create variable VARNAME with the value of the cc scalar triple product of the three vectors VNAME1, cc VNAME2, VNAME3: (VNAME1 cross VNAME2) dot VNAME3. cc ccin variable VARNAME distance cc cc Create variable VARNAME with the value of the last cc distance stored as a result of using command cc "distance", "proximal", "side", "track" or "walk". cc See each command for details. cc ccin variable VARNAME [mean,stddev,min,max] cc cc Create variable VARNAME with the value last cc found with command "mean" of the [mean, standard dd deviation, minimum, maximum] of a specified group cc of variables. cc ccin variable VAR(N) series NUMVAR INC VAR(N-1) FADD FMULT cc cc Create a series of NUMVAR variables VAR(N), ..., cc with names incremented by INC characters, by adding cc FADD to FMULT times the value of the preceding cc variable, starting from the value of variable cc VAR(N-1). See "increment names". See "series", cc command "ratio". cc cc This command can be replaced, using a do loop. cc See "do loop use", commands "do", "enddo". cc ccin variable VAR(N) data F1 F2 F3 ... enddata cc cc Create a set of variables VAR(N), VAR(N+1), ... cc VAR(N+2), ..., with the values F1, F2, F3, ..., cc which may be variables or explicit integer or cc floating point values, but not symbols, except on cc the first line. All arguments after "data" may be cc on separate input lines, singly or in groups. cc Blank lines are ignored. The first variable name cc may have any subscript, or be any allowed variable cc names, and the remaining variable names will be cc incremented by one subscript or character. cc ccin variable VAR(1) prime NUMPRIME cc cc Create a series of NUMPRIME variables VAR(1), ..., cc with names incremented by one character, with the cc values of the first NUMPRIME prime numbers. cc See "prime integers", for the first 100 values. cc E.g., with VAR(1) = x(1), and NUMPRIME = 1200, cc some of the other values would be: cc x(100) = 523, x(200) = 1217, x(300) = 1979, cc x(400) = 2731, x(500) = 3559, x(600) = 4397, cc x(700) = 5273, x(800) = 6131, x(900) = 6991, cc x(1000) = 7907, x(1100) = 8821, x(1200) = 9721. cc ccin variable VARNAME [=] K mod M cc cc Create a variable VARNAME, with the value K mod M, cc in the range from 0 to M - 1, where K and M are cc each an integer variable or value or a symbolic cc word which, after symbol replacement, is an cc integer. cc If K and M have no common factors other than 1, cc display the order of K and the multiplicative cc inverse of K in modulo M. cc ccin variable VARNAME [=] K [+,-,*,/,^] L mod M cc cc Create variable VARNAME with integer value cc (K [+, -, *, /, ^] L) mod M, where K, L and M are cc integers. M must not be less than 2. cc K / L has no answer unless L and M are relatively cc prime, and L is not zero. In that case, the order cc and multiplicative inverse of L in modulo M is cc displayed. cc K ^ L has no answer if L is negative or if the cc result exceeds the maximum integer size on the cc machine. cc If the function of K and L has no common factor cc with M other than 1, display the order of that cc function and the multiplicative inverse of that cc function in modulo M. cc See "modular arithmetic". cc ccin variable VARNAME [=] VA FUNCTION VB cc cc Create a variable VARNAME, with the value a cc function of VA and VB, which may be integer or cc floating point variables or values or symbolic cc words which, after symbol replacement, are integer cc or floating point: cc cc FUNCTION value FUNCTION value cc cc + VA + VB * VA * VB cc - VA - VB / VA / VB cc ^ VA^VB exp VA*exp(VB) cc sqrt VA*sqrt(VB) curt VA*VB^(1/3) cc root VA^(1/VB) cc log VA*log(VB) log10 VA*log10(VB) cc sin VA*sin(VB) asin VA*asin(VB) cc cos VA*cos(VB) acos VA*acos(VB) cc tangent VA*tan(VB) atan VA*atan(VB) cc atan2 atan2(VA,VB) cc sinh VA*sinh(VB) asinh VA*asinh(VB) cc cosh VA*cosh(VB) acosh VA*acosh(VB) cc tanh VA*tanh(VB) atanh VA*atanh(VB) cc abs VA*abs(VB) sign sign(VA,VB) cc int VA*aint(VB) nint VA*nint(VB) cc max amax1(VA,VB) min amin1(VA,VB) cc fact VA * VB! avg (VA+VB)/2 cc ran A random number cc between cc VA and VB. cc gcd Greatest common cc divisor of cc VA and VB cc lcm Least common cc multiple of cc VA and VB cc cc The "=" is optional if argument VA is an integer cc or floating point value or a symbol whose cc replacement value is an integer or floating point cc value. cc cc Note: use command "icalc" or "big" to find the cc number of permutations or combinations of M things cc taken N at a time. cc cc Angle inputs and outputs are in the current angle cc units. See "angles". cc cc Either VA or VB or both may be VARNAME. cc ccin variable VARNAME [+,*,/] F1 F2 F3 ... FN cc cc Create variable VARNAME whose value is the cc [sum, product, sum of reciprocals] of the factors cc F1, F2, F3, ... FN, each of which may be an integer cc or floating point variable or value or a symbolic cc word which, after symbol replacement, is integer or cc floating point. cc Any of the factors may be VARNAME. cc See commands "variable", option "series". cc cc variable V + F(N1) thru F(N2) cc cc Use the following set of commands: cc s V = 0 cc s n = N1 cc a f = "s V = V + F(n) ; s n = n + 1" cc f (repeat until n > N2) cc cc variable V * F(N1) thru F(N2) cc cc Use the following set of commands: cc s V = 1 cc s n = N1 cc a f = "s V = V * F(n) ; s n = n + 1" cc f (repeat until n > N2) cc cc variable V / F(N1) thru F(N2) cc cc Use the following set of commands: cc s V = 0 cc s n = N1 cc a f = "s R = 1 / F(n) ; s V = V + R ; s n = n + 1" cc f (repeat until n > N2) cc cc variable VARNAME polynomial XARG A(0) A(1) ... A(N) cc cc Create variable VARNAME with the value of the cc polynomial function of XARG: cc A(0) + A(1) * XARG + ... + A(N) * XARG^N. cc cc Synonyms: [alias, al, a], [angle, ang], cc [distance, dist], [dot, inner], [help, h], cc [polynomial, poly], [tangent, tan], cc [variable, s, set, var]. cc variable An option in commands "copy", "delete", "last", "list", cc "rename", "repack" and "sort", to perform the cc specified operation on one or more variables. cg variable arrays Variables may be created as arrays with subscripted cg names. See "subscript", "subscripted names", cg commands "do" and "enddo". cg variables A family or series of variable values may be created. cg See command "variable", option "series". cg variables See "internal", for GEOM internal variables. cc VARINT An option in command "debug", to display the value of cc any GEOM internal parameter or variable that is cc specified and defined in file store.h. cc VARINT, VARINT1, VARINT2, ... may be any of: cc cc abig, abigtag, abin, abinm, abins, cc abinx, abrick, cc abricks, acendk, acenpg, acenph, achar, cc aclust, aclusts, acmd, acones, acoordu, cc acoordv, acoordw, acoppg, acopph, acyls, cc adate, adisk, adisks, aellips, afd, cc afilcmd, afilout, ahyperbs, ainfbeg, ainfend, cc ainfile, ainput, ainputex, alias1, alias1s, cc alias2, aline, alines, alipt1, alipt2, cc aloddate, alodmach, alodtime, amach, amark, cc amarks, amarkt, amode, angbr, angunit, cc aoper, aopers, aoptype, aoptyps, aoutputex, cc apaircl, apairpt, apdf, apdfs, apdfx, cc aplanes, aplbuf, aplmarkh, aplmarkt, aplmarkv, cc aplotm, aplstat, apoint, apoints, apolyg, cc apolygs, apolyh, apolyhs, aprompt, aprompts, cc aptmark, aptmarks, aqform, aqtype, aqtypes, cc aquad, aquads, aseven, aspheres, astat, cc asym, asymbl, asymbls, asymbr, asyn1, cc asyn2, asyncmd, asys, asysbr, atempa, cc atempb, atetpt, atetra, atetras, atime, cc atitle, atrig, atrigs, atript1, atript2, cc atript3, atypeb, atypeh, avarbl, avarbls, cc avartag, avector, avectors, averpg, averph, cc aversion, aword, axish, axisv, axisyms, cc azone, azones, azquad, azsign, bignum, cc conbin, cosbin, dubr, dvbr, dwbr, cc ebase, euler, fsymb, ftempa, ftempb, cc fvar, fword, iamax, icmd, idig, cc idmax, cc iemax, iin, incycle, infbeg, infend, cc inline, iout, istat, isymb, itempa, cc itempb, ivar, iword, kbl1, kbl2, cc kmax, kpoint, labins, labricks, laclust, cc laclusts, lacones, lacyls, ladisks, laellips, cc lahyperbs, lalias1s, lalias2, lalines, lalipt1, cc lalipt2, laopers, lapdfs, laplanes, lapoint, cc lapoints, lapolygs, lapolyhs, laquad, laquads, cc laspheres, lasymbls, latetpt, latetras, latrigs, cc latript1, latript2, latript3, lavarbls, lavector, cc lavectors, laxisyms, lazones, lazquad, lbl1, cc lbl2, lmax, lpoint, lprompts, lsymb, cc lvar, lword, lwords, mbl1, mbl2, cc mmax, mpoint, msymb, mtype, mvar, cc nalias, naliasm, nbigm, nbinm, nbinpdf, cc nbins, cc nbitm, nbrickm, nbricks, ncharm, nclustm, cc nclusts, ncmdm, ncmds, ndig, ndigm, cc ndiskm, cc ndisks, cc nedgph, nedgphs, nerr, nfilem, nfiles, cc nflast, nifnest, ninput, nlinem, nlines, cc nmarkm, nmarks, noperm, nopers, npairm, cc npairs, npdfm, npdfs, npdfxm, npdfxs, cc nplcen, nplcolm, nplcols, nplimh, nplimv, cc nplinem, nplines, nplptm, nplpts, npointm, cc npoints, npolygm, npolygs, npolyhm, npolyhs, cc nptmarkm, nptmarks, nquadm, nquads, nsampb, cc nstat, nstats, nsymbm, nsymbs, nsynm, cc nsyns, ntempm, ntemps, ntetm, ntets, cc ntrigm, ntrigs, nvarm, nvars, nvectm, cc nvectors, nver1, nver2, nverpg, nverpgm, cc nverpgs, nverph, nverphs, nword, nwordm, cc nzonem, nzones, nzquadm, nzquads, pdist, cc pi, prbin, prbint, probl, probr, cc proxptx, proxpty, proxptz, proxvx, proxvy, cc proxvz, prpdf, prpdft, qc, qx, cc qxx, qxy, qy, qyy, qyz, cc qz, qzx, qzz, radkin, radkout, cc rgold, rotq, small, splhcen, splhmax, cc splhmin, splvcen, splvmax, splvmin, spthmax, cc spthmin, sptvmax, sptvmin, tol, tolmin, cc tolz, toper, topin, ubr, vbr, cc vexbin, vexpdf, vlen, volbr, voltet, cc volz, volzerr, vranl, vranr, wbr, cc x, xbnd, xcen, xcom, xnormdk, cc xplot, xver, xverh, y, ybnd, cc ycen, ycom, ynormdk, yplot, yver, cc yverh, z, zbnd, zcen, zcom, cc znormdk, zoomplh, zoomplv, zplot, zver, cc zverh. cc cc Note: the first character of each variable name cc indicates its data type: "a" for character data, cc "i" through "n" for integer data, and "b" through "h" cc and "o" through "z" for floating point data. cc VARNAME In command "do", the name of the loop variable, cc which if not already defined, will be created. cc Variable VARNAME may be used inside of the loop, cc but its value should not be changed, except to a cc value outside its range, immediatly preceding cc command "enddo", so as to end the do loop. cc VARNAME The name of a variable. May have up to 24 characters, cc including any subscripts. cc Also referred to as VARNAME1, VARNAME2, VARNAME3, cc ..., VAR(N-1), VAR(N).,... cc Specified with command "variable". cc Units or other information regarding a variable may cc be specified with command "variable", option "tag", cc with up to 48 characters. cc cc The name of an integer or floating point variable cc must begin with an upper or lower case letter or one cc of the characters "$", "%" or "@", may not be cc "random" or "tol", and may not contain the character cc ";". In command "point", the name may not be any of cc the key words used in other "point" commands. cc cc An integer or floating point variable may be used in cc any command in place of a numerical argument cc requiring a floating point value. cc An integer variable may be used in any command in cc place of a numerical argument requiring an integer cc value. cc Prefixing the variable name with "-", with no cc delimiter, results in use of the negative of the cc variable value. cc Prefixing the variable name with "+", with no cc delimiter, has no effect. cc See "arguments", "symbol". cc cc Integer and floating point variables may be used in cc commands: cc axisym, bin, brick, cone, contfr, copy, cylinder, cc delete, disk, do, ellipsoid, icalc, increment, indo, cc input, intcirc, mcvol, move, operator, plane, point, cc polygon, polyhedron, project, quadric, ratio, redo, cc rename, root, rootf, roots, sample, slice, sphere, cc tetrahedron, tol, undo, variable, vector, walk, zone. cc cc Also see commands: last, list, repack, search, sort. cc VARTAG In command "variable", option "tag", a tag or label cc for a variable, bracketed between single or double cc quotes. cc VAXIS The name of a vector used to specify an axis. cc Used in commands "axisym", "cone", "cylinder", cc "ellipsoid", "operator", "plane", "project", cc "twist", "vector", "volume". cc Note: vector VAXIS may be sampled randomly. cc vect A synonym for vector. cg vector A vector is a geometric object having a name VNAME, cg orthogonal components in three-dimensional space, cg a magnitude, and optionally, a bound point in 3-D cg space, which is needed to fully specify a vector in cg a non-Cartesian coordinate system. cg See "coordinate conversion", "vector conversion". cg cg A vector may also be a numerical object, consisting cg of three numerical components. See command "solve". cg cg When a vector is created, a warning will be displayed cg if the vector has no components larger than TOL. cg The length will not include any components less than cg TOL or less than TOL times the length, so may cg truncate to zero. cg NOTE! A vector is NOT dependent on the subsequent cg movement of any point(s) used to create it. cg See "line". cg The maximum number of vectors is now 1000. cg cg Vectors may be sorted by name with command "sort". cg cg Vectors may be used as translation operators, to move cg points, clusters, vectors, planes and quadric cg surfaces. cg cg The following commands relate to vectors: cg accelerate, area, axisym, cone, copy, cross, cg cylinder, debug, delete, disk, dot, ellipsoid, cg extrema, help, invert, last, list, move, operator, cg plane, point, project, quadric, reflect, rename, cg repack, rotate, scale, search, sort, sphere, symbol, cg synonym, tables, sphere, track, twist, variable, cg vector, volume, walk. cc vector An option in command "debug", to display GEOM internal cc parameters and variables for vectors: cc nvectm, nvectors, avector, lavector, xcom, ycom, cc zcom, xbnd, ybnd, zbnd, vlen, avectors, lavectors. cc Synonyms: [vector, v, vect]. cc vector Command to display, modify or create one or more cc vectors. Creating a vector replaces any existing cc vector with the same name. The display for a vector cc includes the components and the length. cc See "do loop use". cc cc Command "vector" relates to objects: axisym, cone, cc cylinder, ellipsoid, hyperb, line, operator, plane, cc point, quadric, sphere, symbol, variable, vector. cc cc Forms include: cc ccin help vector ccin vector [help] cc cc Display the command options. cc ccin vector [all,list] cc cc Display all vectors. cc ccin vector list VNAME1 VNAME2 VNAME3 ... cc cc Display vectors VNAME1, VNAME2, VNAME3, ..., with cc or without subscripts. cc ccin vector VNAME cc cc Display vector VNAME. cc ccin vector VNAME [unit,abs] cc cc Change the length of vector VNAME to 1. cc The vector must not be null. cc ccin vector VNAME abs ABSLEN cc cc Change the length of vector VNAME to ABSLEN. cc The vector must not be null. cc ccin vector VNAME rel RATIO cc cc Multiply the length and components of vector VNAME cc by RATIO. The vector must not be null. cc You may use "*" instead of "rel". cc ccin vector VNAME bound PNAME cc cc Change the bound point of vector VNAME to the cc current coordinates of point PNAME. cc cc In the following commands, use "+" or "-" instead of cc VNAME or VNAME(2) to form the name by incrementing or cc decrementing the base name for vectors. cc See command "last", "increment names". cc ccin vector VNAME [=] DU DV DW [[U|0] [[V|0] [[W|0]]]] cc cc Create vector VNAME, with components (DU, DV, DW) cc [at bound point (U, V, W)], in the current cc coordinate system. Any unspecified words at the cc end of the command, after DU, default to 0. cc The "=" is optional if argument DU is explicitly cc integer or floating point or a symbolic word cc which, after symbol replacement, integer or cc floating point. cc See "coordinate conversion", "vector conversion". cc This command could be inside a triple do loop cc over the U, V and W indices, using subscripted cc names, to create a family of vectors. cc ccin vector VNAME point PNAME1 PNAME2 cc cc Create vector VNAME, directed from point PNAME1 cc toward point PNAME2, with length equal to the cc distance from PNAME1 to PNAME2, and bound at cc PNAME1. cc This command could be inside a do loop, using cc subscripted names, to create a family of vectors. cc ccin vector VNAME point PNAME1 PNAME2 [unit,abs] cc cc Create unit vector VNAME, directed from point cc PNAME1 toward point PNAME2, with unit length, and cc bound at PNAME1. PNAME1 and PNAME2 must not cc coincide. cc ccin vector VNAME point PNAME1 PNAME2 abs [ABSLEN|1] cc cc Create vector VNAME, directed from point PNAME1 cc toward point PNAME2, with length ABSLEN, and bound cc at PNAME1. If not specified, ABSLEN defaults to 1, cc so that VNAME is a unit vector. PNAME1 and PNAME2 cc must not coincide. cc ccin vector VNAME point PNAME1 PNAME2 rel [RELLEN|1] cc cc Create vector VNAME, directed from point PNAME1 cc toward point PNAME2, with length equal to RELLEN cc times the distance from point PNAME1 to point cc PNAME2, and bound at point PNAME1. If not cc specified, RELLEN defaults to 1. cc ccin vector VNAME line ALNAME cc cc Create vector VNAME, with the direction and length cc of line ALNAME, and bound at the initial point of cc line ALNAME. cc ccin vector VNAME random cc cc Create unit vector VNAME, randomly sampled from cc an isotropic distribution. cc ccin vector VNAME random VAXIS cosine [POWER|1] cc cc Create unit vector VNAME, randomly sampled from a cc cosine-power distribution with power POWER, around cc the axis vector VAXIS. cc ccin vector VNAME random VAXIS angle ANGLE cc cc Create unit vector VNAME, randomly sampled from a cc uniform azimuthal distribution at angle ANGLE from cc the axis vector VAXIS. cc ccin vector VNAME proximal cc cc Create vector VNAME, between the proximal points cc found by the last command "accelerate", "distance", cc "proximal", "side" or "track", or from the origin cc to a triple point found by the last command cc "triple", or between the initial and final cc coordinates of a point moved with command "walk". cc ccin vector VNAME cross VNAME1 VNAME2 cc cc Create vector VNAME, the cross (outer) product of cc vectors VNAME1 and VNAME2. cc ccin vector VNAME triple VNAME1 VNAME2 VNAME3 cc cc Create vector VNAME, the vector triple product of cc vectors VNAME1, VNAME2 and VNAME3: cc VNAME1 cross (VNAME2 cross VNAME3) = cc (VNAME3 cross VNAME2) cross VNAME1. cc ccin vector VNAME angle ANG1 VNAME1 ANG2 VNAME2 cc cc Create unit vector VNAME with angle ANG1 from cc vector VNAME1 and angle ANG2 from vector VNAME2, cc in the current angle units. cc ccin vector VNAME project VNAME1 PLNAME cc cc Create vector VNAME, the projection of vector cc VNAME1 onto the plane PLNAME. cc ccin v VNAME sum VNAME1 VNAME2 [[VMULT1|1] [[VMULT2|1]]] cc cc Create vector VNAME, the weighted sum of vectors cc VNAME1 and VNAME2: cc VNAME = VMULT1 * VNAME1 + VMULT2 * VNAME2, cc where VMULT1 and VMULT2 default to 1. cc ccin vector VNAME QTYPE QNAME cc cc Create vector VNAME, from the central point of the cc quadric surface of type QTYPE (plane, sphere, cc cylinder, cone, hyperb, ellipsoid, axisym or cc quadric) with name QNAME (see "names"), to the cc origin. The command "move quadric QNAME VNAME" cc will then move the center of the quadric surface cc to the origin. cc See "operator". cc ccin vector VNAME normal PNAME QNAME cc cc Create vector VNAME, the normal vector at point cc PNAME of the family of quadric surfaces represented cc by quadric surface QNAME. cc ccin vector VNAME(2) rotate NUMV INC VNAME(1) OPNAME PINV cc cc Create a family of NUMV vectors VNAME(2), ..., with cc names incremented by INC characters, by rotating cc the preceding vector with operator OPNAME and cc invariant point PINV, starting from vector cc VNAME(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 vector VNAME(2) scale NUMV INC VNAME(1) OPNAME PINV cc cc Create a family of NUMV vectors VNAME(2), ..., with cc names incremented by INC characters, by scaling the cc preceding vector with operator OPNAME and invariant cc point PINV, starting from vector VNAME(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 cc Synonyms: [angle, ang], [cosine, cos], cc [cylinder, cyl], [ellipsoid, el, ell, ellipse], cc [help, h], [hyperb, saddle], [normal, norm], cc [plane, pl], [proximal, prox], [quadric, q, quad], cc [random, ?], [rel, *], [rotate, rot], [sphere, sph], cc [sum, add], [vector, v, vect]. cc vector An option in commands "copy", "delete", "last", "list", cc "rename", "repack" and "sort", to perform the cc specified operation on one or more vectors. cc Synonyms: [vector, v, vect]. cc vector An option in command "operator", to create a tensor cc operator that performs a rotation of one vector to be cc parallel to another vector, around an axis cc perpendicular to both, and through the origin. cc This is only one of an infinite number of ways to cc align the two vectors, since any arbitrary rotation cc around either vector may be done before or after the cc rotation. cc See "axial", "planar", "serial", "triple". cc Synonyms: [vector, v, vect]. cc vector An option in command "plane", to create a plane through cc a specified point with a specified normal vector. cc Synonyms: [vector, v, vect]. cc vector An option in command "variable", to create a variable cc with a value equal to the length of a specified cc vector. cc Synonyms: [vector, v, vect]. cg vector See "vector components", "vector conversion", cg "vector field", "vector operations", "vector sum". cg vector arrays Vectors may be created as arrays with subscripted cg names. See "subscript", "subscripted names", cg commands "do" and "enddo". cg vector components cg See "vector conversion". cg vector conversion cg Vectors and their bound points may be converted between cg different coordinate systems. cg See "coordinate conversion", "coordinate system", cg "invariant component", command "search". cg Angles may be in degrees or radians. See "angle". cg cg For vector "a", with components and bound point: cg Rectangular: (ax, ay, az ) at (x, y, z ), cg Cylindrical: (arc, ath, az ) at (rcyl, theta, z ), cg Spherical: (ars, ath, aph) at (rsph, theta, phi). cg cg Rectangular (x, y, z): cg ax = arc*costh - ath*sinth, cg ax = (ars*sinph + aph*cosph)*costh - ath*sinth, cg ay = arc*sinth + ath*costh, cg ay = (ars*sinph + aph*cosph)*sinth + ath*costh, cg az = ars*cosph - aph*sinph. cg cg Cylindrical (rcyl, theta, z): cg arc = (ax *x + ay *y ) / rcyl, cg arc = ax *costh + ay *sinth, cg arc = ars*sinth + aph*cosph, cg ath = (ay *x - ax *y ) / rcyl, cg ath = -ax *sinth + ay *costh, cg az = ars*cosph - aph*sinph. cg cg Spherical (rsph, theta, phi): cg ars = (ax*x + ay*y + az*z) / rsph cg ars = (ax *costh + ay *sinth)*sinph + az *cosph, cg ars = arc*sinph + az *cosph, cg ath = (ay *x - ax *y ) / rcyl, cg ath = -ax *sinth + ay *costh, cg aph = (((ax*x+ay*y)*z / rcyl) - az*rcyl) / rsph, cg aph = (ax *costh + ay *sinth)*cosph - az *sinph, cg aph = (z *arc - rcyl*az ) / rsph, cg aph = arc*cosph - az *sinph. cg cg Where: cg rcyl^2 = x^2 + y^2, cg rsph^2 = x^2 + y^2 + z^2 = rcyl^2 + z^2, cg theta = arctan (y / x), cg sinth = sin (theta) = y / rcyl, cg costh = cos (theta) = x / rcyl, cg phi = arctan (rcyl / z), cg sinph = sin (phi) = rcyl / rsph. cg cosph = cos (phi) = z / rsph, cg cg Note: to avoid numerical error and insure that cg all conversion are reversible, add 10^99 to ax, cg arc and ars, 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 vector field The equation for a quadric surface F(x,y,z) = 0, cg may also represent a linear vector field, with the cg x, y and z components of the vector at a particular cg point in space given by the partial derivatives of cg F with respect to x, y and z at that point. cg Each of these components is linear (at most) in cg x, y and z, and independent of any constant term cg in F. The vector field at every point in space is cg normal to the quadric surface that has the particular cg constant term required for the surface to pass cg through that point. cg See "quadric surface", "normal", command "arc". cg vector operations cg To find the length of a vector: cg cg vector VNAME cg cg Display vector VNAME. cg cg variable VARNAME vector VNAME cg cg Create variable VARNAME with the value of the cg length of vector VNAME. cg cg To find the angle between two vectors: cg cg dot VNAME1 VNAME2 cg cg Find the dot (inner) product of vectors VNAME1 and cg VNAME2, the included angle, in the current angle cg units, and its cosine. cg cg variable VARNAME angle VNAME1 VNAME2 cg cg Create variable VARNAME with the value of the angle cg between the two vectors VNAME1 and VNAME2, in the cg current angle units. See "angles". cg cg To find the dot product of two vectors: cg cg dot VNAME1 VNAME2 cg cg Find the dot (inner) product of vectors VNAME1 and cg VNAME2, the included angle, in the current angle cg units, and its cosine. 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 To find the cross product of two vectors: cg cg cross VNAME1 VNAME2 cg cg Find the vector which is the cross (outer) product cg of vectors VNAME1 and VNAME2. cg cg vector VNAME cross VNAME1 VNAME2 cg cg Create vector VNAME, the cross (outer) product of cg vectors VNAME1 and VNAME2. cg cg To find the scalar triple product of three vectors: cg cg variable VARNAME triple VNAME1 VNAME2 VNAME3 cg cg Create variable VARNAME with the value of the cg scalar triple product of the three vectors VNAME1, cg VNAME2, VNAME3: (VNAME1 cross VNAME2) dot VNAME3. cg cg To find the vector triple product of three vectors: cg cg vector VNAME triple VNAME1 VNAME2 VNAME3 cg cg Create vector VNAME with the value of the vector cg triple product of the three vectors VNAME1, VNAME2, cg VNAME3: VNAME1 cross (VNAME2 cross VNAME3). cg cg Note: (VNAME1 cross VNAME2) cross VNAME3 = cg VNAME3 cross (VNAME2 cross VNAME1). cg vector product See "cross product". cg vector sum To find an arbitrary vector sum, e.g.: cg VNAME = VA * A + VB * B + VC * C + ... cg with vector VNAME bound at point P0 = (X0, Y0, Z0), cg and where A, B, C, ... are scalar coefficients, and cg VA, VB, VC, ... are specified vectors, execute the cg following until done: cg cg vector VNAME sum VNAME1 VNAME2 [VMULT1 [VMULT2]] cg cg Create vector VNAME, the weighted sum of vectors cg VNAME1 and VNAME2: cg VMULT1 * VNAME1 + VMULT2 * VNAME2 cg where VMULT1 and VMULT2 default to 1. VNAME may be cg VNAME1, VNAME2 or any arbitrary name. cg vectors To create a family of vectors rotated or scaled from a cg base vector, use command "vector", options "rotate" cg or "scale". cg velocity In a display of data resulting from command cg "accelerate", the x, y, z components of the velocity cg vector on the parabolic trajectory of a particle cg undergoing uniform acceleration. cg cg If only the initial position "p", the magnitude |v| cg of the initial velocity "v", and the constant cg acceleration "a" of the parabolic trajectory of a cg particle undergoing uniform acceleration are known, cg along with a point "b" on the path, the vector cg components of 0, 1 or 2 initial velocities may be cg found as follows: cg pb = b - p cg pb^2 - (pb dot a + v^2)*t^2 + 0.25*a*t^4 = 0 cg Solve for the 0, 1 or 2 real roots t, and if any cg exist, the vector components of "v" are as follows: cg v = pb/t - 0.5*a*t cg If no real roots exist, point "b" is "out of range". cg velocity To randomly sample particle velocities from a cg relativistic Maxwellian gas: cg cg bin BINAME maxwell RATIO cg cg Create probability bin BINAME with a relativistic cg Maxwellian probability distribution, for a cg dimensionless ratio RATIO of gas temperature to cg particle rest mass energy. The random variable cg is the ratio of particle velocity to the velocity cg of light, v / c, ranging from 0 to 1. cg c = 299,792,458 m / s. cg 1 K = 8.617385E-5 eV (+/- 8.5 ppm), cg 1 amu = 9.3149432E8 eV (+/- 0.3 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 v / c. If specified, NSAMP must be a positive cg integer or integer variable. Only the first 26 cg samples will be displayed on the standard output cg medium ( normally the user's terminal. cg All will be displayed in the current output file. cg If two or more values are sampled, the total, the cg minimum and maximum, the mean and the standard cg deviation are displayed. Multiply each sampled cg value by c to get velocity. cg velocity See "velocity of light". cg velocity of light cg The velocity of light is exactly c = 299,792,458 m / s. cg verification The following command requires verification, when input cg is from the user's terminal: cg cg delete OBJTYPE all cg cg When asked for verification, you must type "yes" for cg all objects of type OBJTYPE to be deleted. cg If you have named an object "all" (BAD!), you must cg rename it before you can delete it. cg cg If input is from a file or from the input lines cg saved in memory, no verification is required, so cg TAKE CARE! cc vert An option in command "plot", to display or specify the cc vertical axis label, and the coordinate to be cc plotted on the vertical axis. cg vertex A vertex of the mesh or the mesh block occurs cg wherever each of the mesh indices k, l and m has cg either its minimum or maximum possible value. cg The mesh and the mesh block each have eight vertices. cg cg A vertex is a point where two or more lines meet, cg where three or more surfaces meet, or where a curve cg or surface has the greatest curvature. cg See "vertex weights". cg vertex An intersection of two or more lines. See "triangle", cg "polygon", "polyhedron", "Steiner vertex", cg "tetrahedron". cg cg Given a tetrahedron with the three vertex points P1, cg P2 and P3, to find the fourth vertex point P4 such cg that the edge from P1 to P4 makes angle ANG12 with cg the edge from P1 to P2, makes angle ANG13 with the cg edge from P1 to P3, and has length EDGE14, use the cg following commands: cg cg vector V12 point P1 P2 cg vector V13 point P1 P3 cg vector V14 angle ANG12 V12 ANG13 V13 cg copy point P1 P4 cg move point P4 V14 EDGE14 cg vertex See "vertex weights". cg vertex weights A point on a line or in a triangle, polygon or cg polyhedron may be specified by its vertex weights. cg The coordinates of the point are a weighted average cg of the coordinates of the vertices of the object. cg See "local coordinates", "fractional distance". cc vi Command to display and/or edit a file. Input must be cc from the user's terminal. A new window will open. cc cc Forms include: cc ccin help vi ccin vi [help] cc cc Display the command options. cc cc Use VI commands to display and or edit one or more cc files: cc cc :e FILENAME<return> cc cc Open file FILENAME cc cc (other VI commands) cc cc ZZ<return> cc cc Close current file opened by VI. Return to GEOM. cc cc Synonyms: [help, h]. cx vi To set the environment for UNIX text editor vi, cx to search for entries in this file: cx cx vi geom_base<return> cx cx Start vi, open this file. cx cx :set ic<return> cx cx Ignore case. cx cx :map q /^......<return> cx cx Map the keyboard key "q" to the string "/^......", cx to start the search at the 7th character of each cx line. cx cx qENTRY<return> cx cx Search for entry "ENTRY" in this file, starting in cx column 7 of each line. cx cx See UNIX command "grep". cc view A synonym for viewfactor. cg view See "project", "view factor". cg See command "plane", option "outline". cg view factor The geometric view factor of an area, relative to a cg differential surface element (DSE), is the fraction cg of the thermal radiation leaving the DSE intercepted cg by the area. The geometric view factor may be found cg by projecting the area onto a unit sphere centered at cg the DSE, then projecting that image onto a plane cg through the DSE. The geometric view factor is the cg ratio of the area of the final image to that of a cg unit circle, pi. To make a view factor projection, cg use of the following forms of command "project": cg cg project point PNAME viewfactor PVIEW VNORM cg project cluster CLNAME viewfactor PVIEW VNORM cg project point all viewfactor PVIEW VNORM cg cg Create a geometric view factor view, by projecting cg point PNAME, cluster CLNAME or all points onto a cg plane through point PVIEW, with normal vector cg VNORM, by first projecting them onto a unit sphere cg centered at PVIEW, then projecting them cg perpendicularly onto the plane, then translating cg and rotating the projected points into the plane cg z = 0, with the image of point PVIEW at the origin. cg The ratio of any area on the projection to the area cg of the unit circle, pi, is the differential cg geometric view factor of the area, as seen from the cg differential surface element at point PVIEW with cg normal vector VNORM. See "aliases". cc viewfactor An option in command "project", to make a geometric cc view factor projection. cc Synonyms: [viewfactor, view]. cc VINIT In command "accelerate", the name of the initial cc velocity vector on the parabolic trajectory of a cc uniformly accelerated particle. cc If trying to find VINIT when only its magnitude, the cc initial point, the constant acceleration and a cc second point on the parabolic trajectory are known, cc use any vector with the same magnitude, and option cc "point". cc VMAX In command "brick", the maximum value of the second cc coordinate, which may be y, theta or theta, cc for a rectangular, cylindrical or spherical cc coordinate system, respectively. cc Must not be less than VMIN. cc If theta, specify in the current angle units. cc If theta, must not exceed VMIN by more than 360 cc degrees. cc VMAX In command "slice", the maximum value of a coordinate cc for which points are to be found in a slice of a cc quadric surface. See "DV", "VMIN". cc VMAX In command "variable", a limit on the range of a cc randomly sampled variable value. See "VMIN". cc VMIN In command "brick", the minimum value of the second cc coordinate, which may be y, theta or theta, cc for a rectangular, cylindrical or spherical cc coordinate system, respectively. cc Must not be more than VMAX. cc If theta, specify in the current angle units. cc If theta, must not be more than 360 degrees less than cc VMAX. cc VMIN In command "slice", the minimum value of a coordinate cc for which points are to be found in a slice of a cc quadric surface. See "DV", "VMAX". cc VMIN In command "variable", a limit on the range of a cc randomly sampled variable value. See "VMAX". cc VMOVE The name of a vector used as a translation operator. cc Used in commands "axisym", "cone", "cylinder", cc "ellipsoid", "move", "quadric", "sphere". cc VMULT In command "move", multiplies the movement of a cc point. Defaults to 1. cc May be integer, floating point or a variable. cc Examples of uses: -1.0 to move in the opposite cc direction of the vector; 0.5 to move to the midpoint cc of a line corresponding to the vector. cc VMULT1 In command "vector", option "sum", the weight of the cc first vector to be added. See "VMULT2". cc VMULT2 In command "vector", option "sum", the weight of the cc second vector to be added. See "VMULT1". cc VNAME The name of a vector. May have up to 24 characters, cc including any subscripts, and may be ASCII, integer cc or floating point. cc No vector name may be "+", "-", "all", "h", "help", cc "list", "random" or "thru", begin with "!" or cc contain ";". cc Specified with command "vector". cc cc Also referred to as VNAME1, VNAME2, VNAME3, ... cc Also referred to as VACC, VAXIS, VINIT, VMOVE, VNORM. cc cc Vectors may appear in commands: cc accelerate, area, axisym, cone, copy, cross, cc cylinder, delete, disk, dot, ellipsoid, extrema, cc help, invert, last, list, move, operator, plane, cc point, project, quadric, reflect, rename, rotate, cc scale, search, sort, sphere, track, twist, variable, cc vector, volume, walk. cc VNORM The name of a vector normal to a plane. Used in cc commands "area", "disk", "plane", "project". cc void An option in command "mesh", to display unassigned cc mesh indices. cc vol A synonym for volume. cg volume Bricks and tetrahedrons have exact calculated volumes. cg Zone volumes may be specified or estimated by a cg Monte Carlo method. cg cg To display the volume of a tetrahedron: cg cg tetrahedron TETNAME cg cg Display the coordinates of the vertices and the cg volume of tetrahedron TETNAME. cg cg To display the volume of a brick: cg cg brick BRNAME cg Display the coordinates of the bounding surfaces, cg the edge lengths, the surface areas and the volume cg of brick BRNAME. cg cg To specify the volume of a zone: cg cg zone ZNAME volume ZVOL cg cg Specify the volume ZVOL of zone ZNAME. cg cg To estimate the volume of a zone: cg cg mcvol ZNAME BRNAME NSAMP cg cg Estimate the volume of any part of zone ZNAME that cg is in brick BRNAME, using NSAMP randomly sampled cg points in BRNAME. See command "mcvol". cg cg To display the volume of a parallelepiped, with the cg three edge vectors VNAME1, VNAME2 and VNAME3 at one cg vertex: cg cg variable VARNAME triple VNAME1 VNAME2 VNAME3 cg cg Create variable VARNAME with the value of the cg scalar triple product of the three vectors VNAME1, cg VNAME2, VNAME3: (VNAME1 cross VNAME2) dot VNAME3. cg cg See "volume of brick", "volume of parallelepiped", cg "volume of revolution", "volume of zone". cc volume Command to find the volume of revolution of a general cc polygon around an axis. cc cc Command "volume" relates to objects: point, symbol, cc vector. cc cc Forms include: cc ccin help volume ccin volume [help] cc cc Display the command options. cc ccin volume PAXIS VAXIS PNAME1 PNAME2 PNAME3 ... cc cc Find the volume of revolution around the axis cc through point PAXIS, with direction vector VAXIS, cc of the general polygon with from 3 to 36 vertices cc PNAME1, PNAME2, PNAME3, ... The vertex points need cc not be coplanar or in a plane through the axis of cc revolution. cc cc For a large number of points, e.g. P1 through P24, cc several volumes may be found and added, by beginning cc each point list with the first point, and repeating cc the last point of the previous group: cc cc volume PAXIS VAXIS P1 P2 P3 P4 P5 P6 P7 P8 cc volume PAXIS VAXIS P1 P8 P9 P10 P11 P12 P13 P14 cc volume PAXIS VAXIS P1 P14 P15 P16 P17 P18 P19 P20 cc volume PAXIS VAXIS P1 P20 P21 P22 P23 P24 cc cc Synonyms: [help, h], [volume, vol]. cc volume An option in command "zone", to specify the volume of cc the zone. cc Synonyms: [volume, vol]. cc volume An option in command "point", to make the intervals cc between mesh points equal volumes instead of equal cc intervals. cg volume See "volume element", "volume of brick", cg "volume of parallelpiped", "volume of revolution", cg "volume of tetrahedron", "volume of zone", cg "volume units". cg volume element A volume element in the mesh is defined by eight cg adjacent mesh points with two adjacent values of each cg of the three mesh indices, forming a hexahedron, cg bounded by two adjacent layers in each of the k, l cg and m directions of the mesh. cg See "point element", "line element", cg "surface element". cg volume element See "brick", "tetrahedron", "zone". cg volume of brick The volume VOL of a volume bounded by two fixed values cg of each of the coordinates of an orthogonal cg coordinate system (u, v, w) is given below. cg cg Cartesian coordinates (x,y,z): cg VOL = (x2 - x1) * (y2 - y1) * (z2 - z1) cg cg Cylindrical coordinates (rcyl, theta, z) cg (rcyl => 0, abs (theta2 - theta1) <= 2*pi: cg VOL = (rcyl2^2 - rcyl1^2) * cg (theta2 - theta1) * (Z2 - Z1) / 2 cg Angle theta = arctan (y / x), in radians, and cg pi = 3.141592653589793... cg cg Spherical coordinates (rsph, theta, phi): cg (rsph => 0, abs (theta2 - theta1) <= 2*pi, cg 0 <= phi <= pi): cg VOL = (rsph2^3 - rsph1^3) * cg (theta2 - theta1) * cg (cos (phi1) - cos (phi2)) / 3 cg cg See "brick". cg volume of parallelepiped cg The volume of a parallelepiped with the three edge cg vectors VNAME1, VNAME2 and VNAME3 sharing a common cg vertex, may be found with command: cg cg variable VARNAME triple VNAME1 VNAME2 VNAME3 cg cg Create variable VARNAME with the value of the cg scalar triple product of the three vectors VNAME1, cg VNAME2, VNAME3: (VNAME1 cross VNAME2) dot VNAME3. cg volume of revolution cg The volume of a body of revolution formed by rotating cg an area A, measured in a plane through the z axis, cg and with a centroid at radius rcyl, around the z axis cg though the azimuth angles from theta1 to theta2 cg (in radians), is given by: cg VOL = rcyl * A * (theta2 - theta1) cg cg To find the volume of revolution of an area formed cg by a general polygon, use command "volume". cg volume of tetrahedron cg The volume of a tetrahedron is given by: cg cg v = area * height / 3 cg cg where area is the area of any of the four triangular cg faces, and height is the distance of the opposite cg vertex point from that face, or cg cg v = abs ((v12 cross v13) dot v14) / 6 cg cg where v12, v13 and v14 are any three edge vectors cg sharing a common vertex. cg cg E.g., for vertex points p1 = (p1x, p1y, p1z), and cg p2 = (p2x, p2y, p2z), the edge vector is cg v12 = (v12x, v12y, v12z), where cg v12x = p2x - p1x, v12y = p2y - p1y, v12z = p2z - p1z. cg cg volume of zone The volume of a zone may be found in several ways. cg See commands "area", "volume", "mcvol", "zone". cg See "volume of brick". cg volume units See "conv.mac", "conversion factors". cc VRAN A discrete value of a random variable, assigned a cc relative probability in command "bin". cc VRANL A discrete value or the left extreme of a range of cc values of a random variable, assigned a relative cc probability in command "bin". See "VRANR". cc VRANR The right extreme of a range of values of a random cc variable, assigned a relative probability in command cc "bin". See "VRANL".
W-------------------------------------------------------------------------------
cc W In commands "point", "vector" or "plane", cc a coordinate in the z, z or phi direction, for a cc rectangular, cylindrical or spherical coordinate cc system, respectively. cc In command "operator", the name of an axis (x, y cc or z). cc May be integer, floating point or a variable. cc w A synonym for output. cg w Indicates a major axis of the current coordinate cg system. The third coordinate displayed by cg command "coordinate". cc W1 See "W1, W2, W3, W4". cc w1 See "w1, w2, w3, w4". cc W1, W2, W3, W4 In command "point", the vertex weights of a point in cc a triangle or a tetrahedron. The coordinates of the cc point are a weighted average of the coordinates of cc the vertex points, with weight factors W1, W2 and W3 cc (triangle) or W1, W2, W3 and W4 (tetrahedron). cg w1, w2, w3, w4 The vertex weights of a point in a triangle or a cg tetrahedron. The coordinates of the point are a cg weighted average of the coordinates of the vertex cg points, with weight factors w1, w2 and w3 (triangle) cg or w1, w2, w3 and w4 (tetrahedron). cc W2 See "W1, W2, W3, W4". cc w2 See "w1, w2, w3, w4". cc W3 See "W1, W2, W3, W4". cc w3 See "w1, w2, w3, w4". cc W4 See "W1, W2, W3, W4". cc w4 See "w1, w2, w3, w4". cg walk See "random walk". cc walk A command to randomly walk a point in 3-D space. cc See "random walk". cc cc Command "walk" relates to objects: point, symbol, cc variable, vector. cc cc Forms include: cc ccin help walk ccin walk [help] cc cc Display the command options. cc ccin walk PNAME [VNAME,random] DMEAN PATHMAX [DMAX|1E99] cc cc Randomly walk point PNAME, with initial direction cc vector [VNAME, random], mean mean free path DMEAN, cc until it walks a total path length PATHMAX, or cc reaches distance DMAX from its initial position, cc whichever occurs first. cc cc Note: if PATHMAX is reached first, the net distance cc from the initial position is stored, and if DISTMAX cc is reached first, the total path length is stored. cc cc The stored distance may be stored in variable cc VARNAME with command "variable VARNAME distance". cc cc The vector between the initial and final positions of cc point PNAME may be stored in vector VNAME with the cc command "vector VNAME proximal". cc cc Synonyms: [help, h], [proximal, prox], [random, ?]. cg wall See "Steiner wall". cc WANGLE In command "operator", option "serial", the angle cc (counterclockwise, with the axis pointed at the cc observer) of rotation around the w axis (x, y or z). cc May be integer, floating point or a variable. cg warning Warning messages are displayed when the code detects a cg condition which may not be incorrect, but may cause cg trouble, if not the condition desired by you (the cg user). cg web site The GEOM files and APT files are on web site cg http://nuclear.llnl.gov/CNP/apt/. cg wedge To create a zone partially bounded by two or more cg planes with parallel lines of intersection, use cg command "plane" with option "rotate", and command cg "zone". cg weight See "vertex weights", "local coordinates", cg "fractional distance". cc when Command to find the code date and the run date. cc cc Forms include: cc ccin help when ccin when help cc cc Display the command options. cc ccin when cc cc Display the date, hour and machine when the code cc was made, and if known, when the run started. cc cc Synonyms: [help, h]. cg where A point is either in or not in any given volume element cg (brick, tetrahedron or zone). cg A point may simultaneously be in two or more volume cg elements, if on a boundary surface shared by two or cg more volume elements. See "tol", "TOL". cg cg To find where a point PNAME is in the mesh, use cg command "point PNAME". cg cg To find where the points in cluster CLNAME are in the cg mesh, use command "cluster CLNAME". cc where Command to find if one point, one cluster of points, or cc all points are in one or all volume elements (bricks, cc tetrahedrons, zones). For bricks, the fractional cc distance across the brick is displayed. cc For tetrahedrons, the fractional distance from the cc opposite face to each vertex is displayed. cc For zones, F(x,y,z) is displayed for each surface. cc See commands "distance", "proximal", "side", "track". cc cc Command "where" relates to objects: brick, cluster, cc point, symbol, tetrahedron, zone. cc cc Forms include: cc ccin help where ccin where [help] cc cc Display the command options. cc ccin where point PNAME brick BRNAME ccin where point PNAME tetrahedron TETNAME ccin where point PNAME zone ZNAME cc cc Find out if point PNAME is in brick BRNAME, cc tetrahedron TETNAME or zone ZNAME, respectively. cc ccin where point PNAME brick all ccin where point PNAME tetrahedron all ccin where point PNAME zone all cc cc Find out if point PNAME is in any brick, cc tetrahedron or zone, respectively. cc ccin where point PNAME cc cc Find out if point PNAME is in any brick, cc tetrahedron or zone. cc ccin where cluster CLNAME brick BRNAME ccin where cluster CLNAME tetrahedron TETNAME ccin where cluster CLNAME zone ZNAME cc cc Find out if any of the points in cluster CLNAME are cc in brick BRNAME, tetrahedron TETNAME or zone ZNAME, cc respectively. cc ccin where cluster CLNAME brick all ccin where cluster CLNAME tetrahedron all ccin where cluster CLNAME zone all cc cc Find out if any of the points in cluster CLNAME are cc in any brick, tetrahedron or zone, respectively. cc ccin where cluster CLNAME cc cc Find out if any of the points in cluster CLNAME are cc in any brick, tetrahedron or zone. cc ccin where point all brick BRNAME ccin where point all tetrahedron TETNAME ccin where point all zone ZNAME cc cc Find out if any points are in brick BRNAME, cc tetrahedron TETNAME or zone ZNAME, respectively. cc ccin where point all brick all ccin where point all tetrahedron all ccin where point all zone all cc cc Find out if any points are in any brick, cc tetrahedron or zone, respectively. cc ccin where point all cc cc Find out if any points are in any brick, cc tetrahedron or zone. cc cc Synonyms: [brick, br], [cluster, cl], cc [help, h], [point, p, pnt, pt], cc [tetrahedron, tet, tetra], [zone, z, zn]. cg who To find the problem title (the first input line of the cg run), use command "id". cc Wien A synonym for wien. cg Wien See "Wien spectrum". cc wien An option in command "bin", to create a probability bin cc with a Wien probability distribution. cc Synonyms: [wien, Wien]. cg Wien spectrum A Wien spectrum is a type of probability bin or cg probability distribution function (pdf), which is an cg approximation to an equilibrium radiation spectrum, cg for which the pdf is cg p(x) = (1 / 6) * x^3 * exp (-x), 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 4, with cg a standard deviation of 2. A random sample of cg 100,000 values ranged from 0.13 to 20.7. cg 1 Hz = 4.1356692E-15 eV (+/- 0.3 ppm). cg See commands "bin", "pdf", "sample". cg Also see "Planck". cc WMAX In command "brick", the maximum value of the third cc coordinate, which may be z, z or phi, for a cc rectangular, cylindrical or spherical coordinate cc system, respectively. cc Must not be less than WMIN. cc If angle phi, specify in the current angle units. cc WMIN In command "brick", the minimum value of the third cc coordinate, which may be z, z or phi, for a cc rectangular, cylindrical or spherical coordinate cc system, respectively. cc Must not be more than WMAX. cc If angle phi, specify in the current angle units. cg word A word in an input line is a delimited ASCII character cg string. See "delimiter", "commands", "key words". cg See "machine words". cg word length See "machine words". cd work Directory for working files. cd Open YANA Cluster: ~edwards/work cd Archives: /users/u47/edwards cc wr A synonym for output. cc write A synonym for output. cg write See "output", "output file".
X-------------------------------------------------------------------------------
cc X A synonym for x. cc X In command "define", any single letter or character, cc (case ignored), for which the first line of any cc entry in geom_base beginning with that character cc will be displayed. cc X In command "roots", option "?", the real part of the cc variable at which the function is to be evaluated, cc and which is to be subjected to Newton's iteration cc to search for a value closer to a root. cc X The x coordinate of a point, in a Cartesian or cc rectangular coordinate system. May be integer, cc floating point or a variable. Specified cc with commands "point", "vector" or "plane". cc X = RCYL * cos (THETA) cc X = RSPH * cos (THETA) * sin (PHI) cc x An option in command "brick", to make surface of cc constant x the shared coordinate surfaces between cc adjacent bricks. cc x An option in command "quadric", to use a plane parallel cc to the plane x = 0. cc x An option in command "slice", to slice a quadric cc surface or a zone parallel to an X plane. cg x The major axis for measuring x coordinates in a cg Cartesian or rectangular coordinate system. cg Also refers to a plane perpendicular to the x axis. cg x See "x plane". cg x plane To create a plane at a constant value X of coordinate cg x in an (x, y, z) orthogonal coordinate system: cg cg plane plx = X 0 0 1 0 0 cg cg To create a family of equally spaced x planes, use cg command "plane", option "parallel". cg To create a family of planes with spacing forming a cg geometric series, use command "plane", option cg "scale". cg x' The axis of a quadric surface which will be rotated cg to the x axis to transform the quadric surface into cg standard form. The coordinate along the x' axis. cc X(n) In command "root", option "=", the nth real root cc (n = 1, N) of a polynomial equation of order N. cc X1 See "X1, X2, X3, X4". cc X1, X2, X3, X4 In command "roots", option "=", the real part of the cc first, second, third and fourth roots of a polynomial cc equation. See "Y1, Y2, Y3, Y4". cc X2 See "X1, X2, X3, X4". cc X3 See "X1, X2, X3, X4". cc X4 See "X1, X2, X3, X4". cc XMAX In command "root", the maximum trial value of x, cc to search for roots, extrema and inflection points of cc a polynomial equation by Newtonian iteration. cc Will be changed if not in the calculated range of x cc that includes all roots, extrema and inflection cc points. The value x = 0 will also be tested. cc See "DX", "XMIN". cc XMIN In command "root", the minimum trial value of x, cc to search for roots, extrema and inflection points of cc a polynomial equation by Newtonian iteration. cc Will be changed if not in the calculated range of x cc that includes all roots, extrema and inflection cc points. The value x = 0 will also be tested. cc See "DX", "XMAX". cg xn The x component of a normal vector. cg XNU The random variable sampled from a Planck or Wien cg spectrum. XNU has the same units as BTEMP cg (energy = Planck's constant * frequency, cg energy = Boltzmann's constant * temperature). cg See commands "bin", "pdf", "sample". cc XROOT In command "rootf", one root of a polynomial equation. cg xy Indicates a plane parallel to the x and y axes, at a cg constant value of z. cc xyz A synonym for cartesian. cc xyz See "(xyz)", "coordinate", "rectangular".
Y-------------------------------------------------------------------------------
cc Y A synonym for y. cc Y The y coordinate of a point, in a Cartesian or cc rectangular coordinate system. May be integer, cc floating point or a variable. Specified cc with commands "point", "vector" or "plane". cc Y = RCYL * sin (THETA) cc Y = RSPH * sin (THETA) * sin (PHI) cc Y In command "roots", option "?", the imaginary part of cc the variable at which the function is to be cc evaluated, and which is to be subjected to Newton's cc iteration to search for a value closer to a root. cc y A synonym for yes in command "output". cc y An option in command "brick", to make planes of cc constant y the shared coordinate surfaces between cc adjacent bricks. cc y An option in command "quadric", to use a plane parallel cc to the plane y = 0. cc y An option in command "slice", to slice a quadric cc surface or a zone parallel to a Y plane. cg y The major axis for measuring y coordinates in a cg Cartesian or rectangular coordinate system. cg Also refers to a plane perpendicular to the y axis. cg y See "y plane". cg y plane To create a plane at a constant value Y of coordinate cg y in an (x, y, z) orthogonal coordinate system: cg plane ply = 0 Y 0 0 1 0 cg cg To create a family of equally spaced y planes, use cg command "plane" with option "parallel". cg To create a family of planes with spacing forming a cg geometric series, use command "plane", option cg "scale". cg y' The axis of a quadric surface which will be rotated cg to the y axis to transform the quadric surface into cg standard form. The coordinate along the y' axis. cc Y1 See "Y1, Y2, Y3, Y4". cc Y1, Y2, Y3, Y4 In command "roots", option "=", the imaginary part of cc the first, second, third and fourth roots of a cc polynomial equation. See "X1, X2, X3, X4". cc Any complex roots must occur in conjugate pairs, cc after any real roots. cc Y2 See "Y1, Y2, Y3, Y4". cc Y3 See "Y1, Y2, Y3, Y4". cc Y4 See "Y1, Y2, Y3, Y4". cc yes Following command "output", if the specified output cc file already exists, and the current input is from cc the user's terminal, user response to allow GEOM to cc write over the existing output file. To append to cc the existing output file, the response is "add". cc Synonyms: [add, a]. cg yn The y component of a normal vector. cg yx Indicates a plane parallel to the x and y axes, at a cg constant value of z.
Z-------------------------------------------------------------------------------
cc Z A synonym for z.
cc Z The z coordinate of a point, in a Cartesian or
cc rectangular coordinate system. May be integer,
cc floating point or a variable. Specified
cc with commands "point", "vector" or "plane".
cc Z = RSPH * cos (PHI)
cc z A synonym for zone.
cc z An option in command "brick", to make planes of
cc constant a shared coordinate surfaces between
cc adjacent bricks.
cc z An option in command "quadric", to use a plane parallel
cc to the plane z = 0.
cc z An option in command "slice", to slice a quadric
cc surface or a zone parallel to a Z plane.
cg z The major axis for measuring z coordinates in a
cg Cartesian or rectangular coordinate system.
cg Also refers to a plane perpendicular to the z axis.
cg The main axis of a quadric surface in the standard
cg form, obtained by applying the principal axis
cg transformation.
cg z See "z plane".
cg z plane To create a plane at a constant value Z of coordinate
cg z in an (x, y, z) orthogonal coordinate system:
cg plane plz = 0 0 Z 0 0 1
cg
cg To create a family of equally spaced z planes, use
cg command "plane" with option "parallel".
cg To create a family of planes with spacing forming a
cg geometric series, use command "plane", option
cg "scale".
cg z' The axis of a quadric surface which will be rotated
cg to the z axis to transform the quadric surface into
cg standard form. The coordinate along the z' axis.
cc zero A synonym for 0 in command "big".
cg zero Either absolutely zero or extremely small relative to
cg some nonzero value. See "TOL".
cg zero A zero of a function. See "root", "roots".
cg zeta See "zeta function".
cg zeta function The zeta function of x is the sum over all
cg positive integers n of the terms 1 / n^x.
cg This may be calculated using command file zeta.mac
cg or test problem zeta.mac.test in ~/work/geom/test .
cn zeta.mac A macro file for finding the zeta function of any real
cn variable x. In ~/work/geom/test .
cn Used in test problem zeta.mac.test .
cc zn A synonym for zone.
cg zn The z component of a normal vector.
cc ZNAME The name of a zone. May have up to 24 characters,
cc including any subscripts, and may be ASCII, integer,
cc or floating point.
cc No zone name may be "+", "-", "all", "h", "help",
cc "list" or "thru", begin with "!" or contain ";".
cc Specified with command "zone".
cc Also referred to as ZNAME1, ZNAME2, ZNAME3, ...,
cc ZN(1), ZN(2), ...
cc
cc Zones may appear in commands:
cc copy, delete, distance, mcvol, point, rename, slice,
cc track, where, zone.
cc Also see commands: last, list, repack, search, sort.
cg zone A zone is a solid geometric object in 3-D space,
cg specified by the quadric surfaces bounding it, and by
cg the direction of exit from the zone, relative to the
cg normal vector at each bounding surface. The volume
cg of a zone may be estimated, using command "mcvol".
cg If the zone is a body of revolution of a
cg polygonal area around an axis, its volume may be
cg found with command "volume".
cg The maximum number of zones is now 1000.
cg The maximum number of quadric surfaces bounding a
cg zone is now 16.
cg
cg Zones are created by command "zone".
cg
cg To find if a point is in a zone, use command
cg "distance" or "where".
cg
cg To find the distances from a point to the quadric
cg surfaces bounding the zone, use command "distance".
cg
cg The following commands relate to zones:
cg cluster, copy, debug, delete, distance, help, last,
cg list, mcvol, point, rename, repack, search, slice,
cg sort, symbol, synonym, tables, track, where, zone.
cg
cg Also see "zone (cylindrical)", "zone (rectangular)",
cg "zone (spherical)".
cg
cg Also see "brick", "mesh", "tetrahedron".
cc zone An option in command "debug", to display GEOM internal
cc parameters and variables for zones bounded by
cc quadric surfaces: nzonem, nzones, nzquadm,
cc azone, nzquads, azquad, lazquad, azsign,
cc volz, volzerr, azones, lazones,
cc Synonyms: [zone, z, zn].
cc zone Command to display or create one or more zones.
cc Creating a zone replaces any existing zone having the
cc same name. The display for a zone includes the name
cc and description of each bounding surface, and the
cc estimated volume, if it has been found with command
cc "mcvol".
cc See "do loop use".
cc
cc Command "zone" relates to objects: axisym, cone,
cc cylinder, ellipsoid, hyperb, plane, quadric, sphere,
cc symbol, variable, zone.
cc
cc Forms include:
cc
ccin help zone
ccin zone [help]
cc
cc Display the command options.
cc
ccin zone [all,list]
cc
cc Display all zones (short display).
cc
ccin zone list ZNAME1 ZNAME2 ZNAME3 ...
cc
cc Display zones ZNAME1, ZNAME2, ZNAME3, ..., with or
cc without subscripts (short display).
cc
ccin zone ZNAME
cc
cc Display zone ZNAME (long display).
cc
ccin zone ZNAME volume ZVOL
cc
cc Specify the volume ZVOL of existing zone ZNAME.
cc
cc In the following commands, use "+" or "-" instead of
cc ZNAME or ZN(2) to form the name by incrementing or
cc decrementing the base name for zones.
cc See command "last", "increment names".
cc
ccin zone ZNAME = [+,-|+] Q1 [+,-|+] Q2 [+,-|+] Q3 ...
cc
cc Create zone ZNAME, bounded by the quadric surfaces
cc Q1, Q2, Q3, ... The sign arguments should be blank
cc or "+" if the direction out of the zone is in the
cc direction of the normal vector of the surface,
cc otherwise "-". Note that a "+" or "-" must be
cc separated by a field delimiter from the following
cc quadric surface name. Also note that cones have
cc two sections, requiring use of an additional
cc surface to restrict the boundary of the zone to one
cc of those two sections.
cc
ccin zone ZNAME & [+,-|+] Q1 [+,-|+] Q2 [+,-|+] Q3 ...
cc
cc Specify additional bounding surfaces for existing
cc zone ZNAME. Do not use "+" for ZNAME.
cc
ccin zone ZN(2) incr NUMZN INC ZN(1) IQ1 IQ2 ...
cc
cc Create a family of NUMZN zones ZN(2), ..., with
cc names incremented by INC characters, by
cc incrementing the names of the bounding surfaces of
cc the preceding zone by the number of characters
cc indicated by the integers IQ1, IQ2, ..., starting
cc from zone ZN(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 zone ZNAME family IZU AQU IQU NQU
cc
cc Create a family of NQU - 1 zones ZNAME, ..., with
cc names incremented by IZU characters, bounded by the
cc NQU quadric surfaces AQU, ..., with names
cc incremented by IQU characters, and ordered in the
cc direction of their normal vectors. For a
cc continuous sequence of zone names, use IZU = 1.
cc Note that the normal vector of a cone reverses
cc direction at a vertex angle of 90 degrees.
cc
cc This command can be replaced, using a do loop.
cc See "do loop use", commands "do", "enddo".
cc
ccin zone ZNAME fam IZU AQU IQU NQU fam IZV AQV IQV NQV
cc
cc Like the preceding command, but with each zone
cc expanded to a family of NQV - 1 zones, with names
cc incremented by IZV characters, bounded by the NQV
cc quadric surfaces AQV, ..., with names incremented
cc by IQV characters, and ordered in the direction of
cc their normal vectors. For a continuous sequence of
cc zone names, use IZU = 1 and IZV = NQU, and use
cc command "sort zone" after this command.
cc
cc This command can be replaced, using do loops.
cc See "do loop use", commands "do", "enddo".
cc
ccinz zone ZNAME fam IZU AQU IQU NQU fam IZV AQV IQV NQV
ccinz fam IZW AQW IQW NQW ! (all on one line)
cc
cc Like the preceding command, but with each zone
cc expanded to a family of NQW - 1 zones, with names
cc incremented by IZW characters, bounded by the NQW
cc quadric surfaces AQW, ..., with names incremented
cc by IQW characters, and ordered in the direction of
cc their normal vectors.
cc For a continuous sequence of zone names, use
cc IZU = 1, IZV = NQU and IZW = NQU * NQV, and use
cc command "sort zone" after this command.
cc
cc This command can be replaced, using do loops.
cc See "do loop use", commands "do", "enddo".
cc
cc Synonyms: [family, fam], [help, h],
cc [increment, incr], [volume, vol], [zone, z, zn].
cc zone An option in commands "copy", "cluster", "delete",
cc "last", "list", "rename", "repack" and "sort", to
cc perform the specified operation on one or more zones.
cc Synonyms: [zone, z, zn].
cg zone See "zone (cylindrical)", "zone (rectangular)",
cg "zone (spherical)".
cg zone (cylindrical)
cg A zone bounded by fixed values of the cylindrical
cg orthogonal coordinates rcyl, theta and z.
cg
cg To create zone zcyl in cylindrical coordinates,
cg bounded by RCYL1, RCYL2, THETA1, THETA2, Z1, Z2,
cg which may be integer or floating point values or
cg variables.
cg
cg coord cyl
cg point pcen = 0 0 0
cg vector vaxis = 0 0 1
cg cylinder cyl1 = RCYL1 pcen vaxis !(RCYL1 > 0)
cg cylinder cyl2 = RCYL2 pcen vaxis
cg plane plth1 = 1 THETA1 0 0 1 0
cg plane plth2 = 1 THETA2 0 0 1 0
cg plane plz1 = 0 0 Z1 0 0 1
cg plane plz2 = 0 0 Z2 0 0 1
cg zone zcyl = - cyl1 cyl2 - plth1 plth2 - plz1 plz2
cg
cg Delete cyl1 if RCYL1 = 0.
cg
cg Also see command "cylinder", options "concentric" and
cg "scale", command "plane", options "parallel",
cg "rotate" and "scale", and command "zone", option
cg "increment".
cg
cg See "area", "volume", "brick".
cg zone (rectangular)
cg A zone bounded by fixed values of the orthogonal
cg coordinates x, y and z.
cg
cg To create zone zxyz in rectangular coordinates,
cg bounded by X1, X2, Y1, Y2, Z1, Z2, which are integer
cg or floating point values or variables:
cg
cg coord rect
cg plane plx1 = X1 0 0 1 0 0
cg plane plx2 = X2 0 0 1 0 0
cg plane ply1 = 0 Y1 0 0 1 0
cg plane ply2 = 0 Y2 0 0 1 0
cg plane plz1 = 0 0 Z1 0 0 1
cg plane plz2 = 0 0 Z2 0 0 1
cg zone zxyz = - plx1 plx2 - ply1 ply2 - plz1 plz2
cg
cg Also see command "plane", options "parallel",
cg "rotate" and "scale", and command "zone", option
cg "increment".
cg
cg See "area", "volume", "brick".
cg zone (spherical) A zone bounded by fixed values of the spherical
cg orthogonal coordinates rsph, theta and phi.
cg
cg To create zone zsph in spherical coordinates,
cg bounded by RSPH1, RSPH2, THETA1, THETA2, PHI1, PHI2:
cg
cg cs sph
cg point pcen = 0 0 0
cg sphere sph1 = RSPH1 pcen (RSPH1 > 0)
cg sphere sph2 = RSPH2 pcen
cg plane plth1 = 1 THETA1 90 0 1 0
cg plane plth2 = 1 THETA2 90 0 1 0
cg cs xyz
cg vector vaxis = 0 0 1
cg cone cone1 angle pcen vaxis PHI1 (PHI1 > 0)
cg cone cone2 angle pcen vaxis PHI2
cg zone zsph = - sph1 sph2 - plth1 plth2 - cone1 cone2
cg
cg Delete sph1 if RSPH1 = 0.
cg Delete cone1 if PHI1 = 0.
cg
cg Also see command "sphere", options "concentric" and
cg "scale", command "plane", option "rotate", command
cg "cone", option "nest", and command "zone", option
cg "increment".
cg
cg See "area", "volume", "brick".
cg zone arrays Zones may be created as arrays with subscripted
cg names. See "subscript", "subscripted names",
cg commands "do" and "enddo".
cg zones A family of zones may be created with command "zone",
cg option "family".
cg A family of zones, bricks, tetrahedrons or triangles
cg may be created with command "zone", "brick",
cg "tetrahedron" or "triangle", respectively, with
cg option "increment".
cg zoning See "brick", "polyhedron", "tetrahedron", "triangle",
cg "zone", "ratio zoning".
cg See "k/l line zoning".
cc ZOOMULT In command "plot", option "zoom", a multiplier of the
cc current zoom factor for both axes or just the
cc horizontal or vertical axis. Use ZOOMULT = 0
cc to make the zoom factor 1 (no zoom).
cg ZR Indicates a plane parallel to the r and z axes, with
cg symmetry in the theta (azimuthal) direction.
cc ZVOL The volume of a zone. Used in command "zone".
cg zx Indicates a plane parallel to the x and z axes, at a
cg constant value of y.
cg zy Indicates a plane parallel to the y and z axes, at a
cg constant value of x.
cc zzz A synonym for lock.
cc [ Square brackets are used in the description of GEOM
cc commands, but not in the commands themselves, to
cc indicate an optional command argument, of one of the
cc forms:
cc
cc [a] Specify literal "a" or do not.
cc There is no default.
cc [A] Specify value "A" or do not.
cc There is no default.
cc [a|b] Specify literal "a", or do not, and let
cc the argument default to "b".
cc [A|B] Specify value "A", or do not, and let
cc the argument default to value "B".
cc [A|<] Specify value "A", or do not, and let
cc the argument default to the preceding
cc argument.
cc [a,b,c|d] Specify literal "a", "b" or "c", or do
cc not, and let the argument default to
cc literal "c".
cc [a [b]] or [a] [[b]]
cc Specify literal "A" or do not. Only if
cc literal "A" is specified, then specify
cc literal "B" or do not.
cc [A [B]] or [A] [[B]]
cc Specify value "A" or do not. Only if
cc value "A" is specified, then specify
cc value "B" or do not.
cc [[A|B] [[C|D]]] or [A|B] [[C|D]]
cc Specify value "A" or do not, and let
cc the argument default to value "B".
cc Only if value "A" is specified, then
cc specify value "C", or do not, and let
cc the argument default to value "D".
cc
cc Note: the character "|" is the "pipe" symbol.
cg [, ] The square bracket characters. No special use in GEOM.
cg [a,b,c,d] In a list of synonyms, indicates that "a", "b", "c"
cg and "d" are synonyms. In a command, indicates a set
cg of options, each of which has a different result.
cc [a,b,c|d] Indicates an optional command argument a, b or c,
cc which defaults to d.
cc [a] Indicates an optional command argument a, which
cc has no default, unless specified in the description
cc of the command. A lower case argument means a
cc literal, upper case means the user must provide
cc an integer or floating point value or the name of a
cc variable.
cc [A|<] Indicates an optional command argument a, which
cc defaults to the preceding argument.
cc [a|b] Indicates an optional command argument a, which
cc defaults to b.
cc ] Indicates end of an optional command argument.
cc See "[".
cc ^ An option in commands "big", "icalc" and "variable",
cc argument FUNCTION, to indicate raising to a power.
cc In command "big", BIGNAME ^ N means BIGNAME^N or
cc BIGNAME**N.
cc In command "icalc", M ^ N means M^N or M**N.
cc In command "variable", A ^ B means A^B or A**B.
cc Synonyms: [^, **].
cc ^ An option in command "variable VARNEW = K ^ L mod M",
cc to raise K to the power L, or if L has an order
cc in modulo M, to the power L mod J(M), where J(M)
cc is the Euler totient function of modulo M.
cf ^ Indicates raising to a power. In the expression
cf x^y, indicates that x is raised to the power y.
cf x^y = exp (y * ln (x)).
cf Synonyms: [^, **].
cg ^L The character for page restore. When a file is
cg printed, the character "^L" (ctrl-L) in column 1 of a
cg line will cause the printer to start a new page.
cg _ The underline character. No special use in GEOM.
cg {, } The curly bracket characters. No special use in GEOM.
cc | Separates optional value (on left) from default value
cc (on right), when inside square brackets.
cc May be referred to as the "pipe" symbol.
cg |...| When bracketing a quantity, indicates the absolute
cg value of the quantity. When preceded by a sign,
cg indicates the required sign of the bracketed
cg quantity.
cg Examples: - |QYY| indicates that QYY is negative.
cg ~ The "tilde". Indicates home directory, "~edwards"
cg for me. "~" for the user's own home directory.
cg ~ When sorting ASCII strings, the last character in the
cg set of characters available on a standard keyboard.
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