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

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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.

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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.

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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.

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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.

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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.

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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".

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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".

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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.

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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".

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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.

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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.

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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".

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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".

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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