apt_key APT KEY WORDS Author: Arthur L. Edwards (email: xenopsbird@comcast.net). Updated: 2006 August 4 14:10. . .and. Boolean operation "intersection" on two machine words aptbitb .comp. Boolean operation "complement" on a machine word aptbitb .eqv. Boolean operation "equivalence" on two machine words aptbitb .int. Boolean operation "intersection" on two machine words aptbitb .nand. Boolean operation "comp. .and." on two machine words aptbitb .nor. Boolean operation "nor" on two machine words aptbitb .not. Boolean operation "not" on two machine words aptbitb .or. Boolean operation "union" on two machine words aptbitb .shl. Boolean operation "shift-left" on a machine word aptbits .shr. Boolean operation "shift-right" on a machine word aptbits .un. Boolean operation "union" on two machine words aptbitb .xnor. Boolean operation "equivalence" on two machine words aptbitb .xor. Boolean operation "exclusive or" on two machine words aptbitb 0 1-D axis, distance between two points on aptvdil 1-D cylinder, intersection with a linear track aptrkcl 1-D major plane, intersection with a linear track aptrksl 1-D sphere, intersection with a linear track aptrkrl 2-D (see "major plane") 3 by 3 determinant, value of aptrips, aptdet3 3 by 3 matrix, determinant of aptdet3 4 by 4 determinant, value of aptdet4 4 by 4 matrix, determinant of aptdet4 A absolute precision, round-off floating point data to aptrnds, aptnint accelerated trajectory, parabolic, given points on vs time aptpart accelerated trajectory, parabolic, intersection with line aptparl accelerated trajectory, parabolic, intersection with plane aptparp, aptparq accelerated trajectory, parabolic, intersection with point aptparh, aptpars accelerated trajectory, parabolic, intersection with quadric aptparq accelerated trajectory, parabolic, path length vs time aptparb accelerated trajectory, parabolic, points on vs time aptparb, aptpart accelerated trajectory, parabolic, proximal point to line aptparl accelerated trajectory, parabolic, proximal point to plane aptparp accelerated trajectory, parabolic, proximal point to point aptpars accelerated trajectory, parabolic, proximal to quadric aptparx accelerated trajectory, parabolic, velocities vs time aptparb accelerated trajectory, time-dependent acceleration on aptacct acceleration on a parabolic trajectory, uniform aptparb, aptpart acceleration on a trajectory, time-dependent aptacct add two big integers (one digit per word), any number base aptbadd add vectors aptvsum, aptvadd add vectors (major plane) aptvadc adjust values of floating point data near 0 and 1 aptfdad, aptfdav adjust values of floating point data, by round-off aptrnds, aptnint adjust values of vector components aptvlim, aptvtol adjust values of vector components (major plane) aptvlic, aptvtoc aliased PDF bins, generate aptalin aliased PDF bins, randomly sample aptalsb, aptalst alphabetic characters, increment or decrement aptchin alphabetic characters, random, put into a character string aptchrn alphanumeric characters (see "character", "string") alphanumeric characters, random, put into a character string aptchrn alphanumeric characters, squeeze out of a character string aptchsq alternate geometric representations (see "line", "plane") altitudes of a triangle, orthocenter apttrig analyze a reflection operator to find axis aptrefq analyze a rotation operator to find rotation angle, axis aptrotq and, Boolean operation "intersection" on two machine words aptbitb angle (see also "cosine", "sine", "bisector") angle (see also "parallel", "perpendicular") angle and axis, rotate with a specified aptrota angle between unit vectors, cosine of aptvdos, aptvdot angle between unit vectors, cosine of (major plane) aptvdoc angle between unit vectors, sine of aptvaxb, aptvxun angle between unit vectors, sine of (major plane) aptvaxc, aptvxuc angle bisector aptbang, apttrig angle bisector (major plane) aptbanc angle bisectors of a triangle, inscribed circle apttrig angle interval, randomly sample direction vectors within aptscar angle of intersection of a track with a cone aptrkis angle of intersection of a track with a cylinder aptrkis angle of intersection of a track with a plane aptrkis angle of intersection of a track with a quadric surface aptrkis angle of intersection of a track with a sphere aptrkis angle of intersection of a track with an axisymmetric quadric aptrkis angle of intersection of a track with an ellipsoid aptrkis angle of rotation of a specified rotation operator aptrotq angle of rotation, operator for a specified axis and aptrota angle, conversion to degrees, grads, radians, arc deg, min, sec aptcang angle, cosine and sine of (major plane) aptvanc angle, cosine of, between two vectors aptvang angle, cosine of, between unit vectors aptvdos, aptvdot angle, cosine of, between unit vectors (major plane) aptvdoc angle, operator to rotate around an axis by a specified aptrota angle, rotate a vector by a specified (major plane) aptrotc angle, scattering, fixed, randomly sample azimuth angle for aptscan, aptscaz angle, scattering, fixed, randomly sample azimuth angle for aptscma, aptscmz angle, sine of, between two unit vectors aptvaxb, aptvxun angle, sine of, between two unit vectors (major plane) aptvaxc, aptvxuc angle, vector bisecting aptbang, apttrig angle, vector bisecting (major plane) aptbanc angles from two other lines, line at specified aptetrv angles from two other vectors, vector at specified aptetru, aptetrv angles of a triangle apttrig angles, rotation operator to rotate around major axes by 3 aptrots annular disk, randomly sample points on (any plane) aptlocd annulus of a circle, randomly sample points in aptlocd annulus of a conical frustrum, randomly sample points in aptscnv annulus of a cylinder, randomly sample points in aptlocy annulus of a sphere, randomly sample points in aptlocs approximate distance from a point to a quadric surface aptquad approximate distance from a point to a quadric surface (OBS) aptqprr, aptqprt arc degrees, minutes, seconds, convert an angle to aptcang arc minutes, seconds, degrees, convert an angle to aptcang arc seconds, degrees, minutes, convert an angle to aptcang area between two lines aptvaxb, aptvxun area between two lines (major plane) aptvaxc, aptvxuc area between two vectors aptvaxb, aptvxun area of a regular polygon aptpoly area of a regular polygon (major plane) aptpolc area of a triangle apttrid, apttrig area of a triangle aptvpln area of a triangle (major plane) apttric area of a triangle, extend two sides to double aptrext areas of faces of a brick (bounded by coordinate surfaces) aptbrkv areas of vertices of a quadrangle (major plane) aptqvac areas, equal, straight line that cuts a triangle into aptrabc, aptrcab areas, equal, straight line that cuts a triangle into aptrcut, aptrext arithmetic, modular, multiplicative inverse of an integer in aptmord arithmetic, order of an integer in modular aptmord array, delete rows containing duplicate values aptdeld array, maximum and minimum, positive values and indices aptmnlg array, maximum and minimum, values and indices aptmnlg, aptmnmx array, mean value and standard deviation of aptmean array, minimum and maximum, positive values and indices aptmnlg array, minimum and maximum, values and indices aptmnlg, aptmnmx array, randomly reorder an aptrand array, reverse the order of the digits of a big integer aptbrev array, sort, any direction or mode (one column) aptsort array, sort, any direction, mode, or number of columns aptsorm array, type integer*1, contract bits from into machine words aptbitp array, type integer*1, expand bits from machine words into an aptbite ascending order, reorder data (multiple columns) aptsorm ascending order, reorder data (one column) aptsort ASCII characters (see "character", "string") ASCII characters, increment or decrement aptchin ASCII characters, move from one string to another aptchmv ASCII characters, replace with other characters aptchrp ASCII characters, translate binary integer numbers into aptchia ASCII characters, translate into integer (oct, dec, hex) aptchai ASCII characters, translate into integer and floating point aptchtp ASCII characters, translate into integer and floating point aptffip, aptchap ASCII characters, translate into integer and floating point aptsubs, aptsubm ASCII strings longer than eight characters, parse aptchtp ASCII strings longer than eight characters, parse aptffip, aptchap ASCII strings longer than eight characters, parse aptsubs, aptsubm ASCII subscript, form subscripted variable from a stem and aptsuba, aptsubb ASCII subscript, form subscripted variable from a stem and aptsuba, aptsubb average coordinates, edge lengths, face areas of a brick aptbrkv average value of an array aptmean axes and center of symmetry of a quadric surface aptaxis axes of two circular cylinders, proximal points on, parallel? aptcycy axes, randomly sample direction vectors relative to aptscav axes, rotation operator for 3 angles around 3 major aptrots axial position, rotate points around axis depending on aptwist axis and angle, rotate with a specified aptrota axis and radius, equation of a cylinder with a specified aptclis, aptcyis axis intercepts of a quadric surface, major and symmetry aptqupt axis of a cylinder, distance from the center of a sphere aptspcy axis of a cylinder, proximal point on, to a sphere aptspcy axis of rotation of a specified rotation operator aptrotq axis of rotation, operator for a specified angle and aptrota axis, major, equation of a cone on aptcnis, aptcois axis, major, equation of a cylinder on aptcyis axis, major, equation of a hyperboloid on aptelis axis, major, equation of an ellipsoid on aptelis, aptesis axis, major, randomly sample directions at fixed angle from aptscma, aptscmz axis, project toward aptprcy, aptprsp axis, randomly sample direction vectors at fixed angle from aptscan, aptscaz axis, randomly sample direction vectors at fixed angle from aptscma, aptscmz axis, randomly sample direction vectors relative to aptscap, aptscar axis, randomly sample direction vectors relative to aptscav axis, rotate points around, depending on axial position aptwist axis, rotate points around, depending on radial position aptwirl axis, rotation operator for a specified angle and aptrota axis, scale cylindrically around a specified aptsclr axis, sphere on a specified, and through two specified points aptspha axis, z, equation of a cone on, through two specified points aptcniz axisymmetric coordinates, effects of linear motion in aptmove, aptmvcy axisymmetric coordinates, intersection of a linear track in aptrkcy axisymmetric surface (see also "quadric surface") axisymmetric surface, distance from a plane to, intersection aptplqu axisymmetric surface, intersection of linear track with aptrkcy, aptrkis azimuth angle, randomly sample, for a fixed scatter angle aptscan, aptscaz azimuth angle, randomly sample, for a fixed scatter angle aptscma, aptscmz B backslash, use as an escape character aptfbrk base, find the value of a big integer in a specified number aptbasb base, find the value of a decimal number in a specified number aptbase basic matrix operations (see "matrix", "product") basic symmetry operations (see "reflect", "rotate") basic symmetry operations (see "reflection", "rotation") basic symmetry operations (see also "inversion", "translation") basic symmetry operations (see also "invert", "translate") basic vector operations (see "sum", "difference", "product") big integer (any base), convert from decimal integer into a aptdtob big integer (any base), convert from floating point into a aptftob, aptltob big integer (any base), convert into decimal integer aptbtod big integer (any base), convert into floating point aptbtod big integer (any base), count the occurrences of each digit in aptbsum big integer (any base), find a power of a aptbpow big integer (any base), find a root of a aptbrtn big integer (any base), find in any other number base aptbasb big integer (any base), find the exponential of a aptbexp big integer (any base), find the factorial of a aptbfac big integer (any base), find the log (base 10) of a aptbtod big integer (any base), find the prime factors of a aptpfab big integer (any base), reverse the order of the digits of a aptbrev big integers (any base), find the difference of two aptbsub big integers (any base), find the modulus of two aptbmod big integers (any base), find the product of two aptbmul big integers (any base), find the quotient of two aptbdiv big integers (any base), find the remainder of division of two aptbdiv big integers (any base), find the sum of two aptbadd big integers, nonzero, concatenate the digits of two aptbcat bin values, equal-probability PDF, randomly sample apteqxs, aptrans binary Boolean operations "and", "int", "nand", "or", "un" aptbitb binary Boolean operations "nor", "xor", "xnor", "eqv" aptbitb binary search table look-up, ascending or descending order aptlook bins, aliased PDF, generate aptalin bins, aliased PDF, randomly sample aptalsb, aptalst bins, cumulative PDF, generate aptcump bins, cumulative PDF, randomly sample aptcums, aptcumt bins, equal-probability PDF, generate apteqin, apteqxn bins, equal-probability PDF, randomly sample apteqsb bins, randomly sample, aliased aptalsb, aptalst bins, randomly sample, cumulative probability aptcums, aptcumt bins, randomly sample, equal-probability apteqsb bins, specified, randomly sample from linear PDFs in aptalsl bins, specified, randomly sample from uniform PDFs in aptalsh bisector of a line (major plane) aptpblc bisector of a vertex angle of a quadrangle (major plane) aptbanc bisector of a vertex angle of a quadrangle, vector aptbang bisector of a vertex angle of a triangle, vector aptbang, apttrig bisector of a vertex angle of a triangle, vector (major plane) aptbanc bisector of an angle aptbang, apttrig bisector of an angle (major plane) aptbanc bisector of edge of a quadrangle, perpendicular (major plane) aptpblc bisector of edge of a quadrangle, perpendicular plane aptpbln bisector of edge of a triangle, perpendicular (major plane) aptpblc bisector of edge of a triangle, perpendicular plane aptpbln bit operations "comp", "not", "and", "int", "nand", "or", "un" aptbitb bit operations "nor", "xor", "xnor", "eqv" aptbitb bit string, find in a machine word aptbitf bit string, move from one machine word to another aptbitm bit string, put random bits into aptbitr bits from a type integer*1 array, contract into machine words aptbitp bits, randomly sample and put into a bit string aptbitr bits, shift right or left, wrap-around or pull aptbits blank-delimited (see "data fields", "delimited") blank-delimited character string, parse into symbols aptfbrk blank-delimited character string, parse into symbols aptffip, aptchap blank-delimited character string, read and parse into symbols aptfbrk blank-delimited character string, read and parse into symbols aptffip, aptchap blanks, imbedded, translate data fields with aptchtp blanks, imbedded, translate data fields with aptsubs, aptsubm blanks, multiple, reduce to a single blank in character string aptblsq blanks, remove leading and/or trailing from a character string aptrmbl blanks, squeeze out of a character string aptchsq body, geometric (see "cylinder", "sphere", "polyhedron") body, geometric (see also "ellipsoid") Boolean operations "and", "int", 'nand", "or", "un", "nor" aptbitb Boolean operations "shift-right" and "shift-left" aptbits Boolean operations "xor", "xnor", "eqv", "comp" and "not" aptbitb boomerang quadrangle, test for (major plane) aptqvac bowtied quadrangle, test for (major plane) aptqvac brackets, use as field delimiters (square [] or curly {}) aptfbrk brick, bounded by surfaces of the coordinate system aptbrkn, aptbrkp brick, bounded by surfaces of the coordinate system aptbrks, aptbrkv brick, find a point with given local coordinates in aptbrkp brick, geometric data for a aptbrkv brick, randomly sample points in a aptbrks brick, see if a point is in, and find distances to surfaces aptbrkn brick, see if a point is in, and find local coordinates aptbrkn C calculate (implied in any entry without a verb) cartesian coordinates, transform to or from aptcsys, aptcsyv cartesian coordinates, volume bounded by surfaces of aptbrkn, aptbrkp cartesian coordinates, volume bounded by surfaces of aptbrks, aptbrkv center and edges of a quadrangle, fit a quadric surface to aptqhyp center and radius of circle through 3 points (major plane) aptcirc center and radius of circumscribed circle of a triangle apttrig center and radius of inscribed circle of a triangle apttrig center and radius of sphere on an axis, and through 2 points aptspha center and radius of sphere through four specified points aptsphp center of circle tangent to two specified tangent circles aptcirl, aptcirk center of sphere tangent to three specified tangent spheres aptsphl, aptsphk center of symmetry, axes and type of a quadric surface aptaxis center, radius and orientation of circle through 3 points aptcirp centers and radii of circles tangent to 3 tangent circles aptcirk centers and radii of spheres tangent to 4 tangent spheres aptsphk centers of circles tangent to three specified tangent circles aptcirk centers of curvature at a point on a quadric surface, extremes aptcris centers of spheres tangent to four specified tangent spheres aptsphl, aptsphk centroid of a brick (bounded by coordinate surfaces) aptbrkv centroid of a brick, coordinates, lengths and areas through aptbrkv centroid of a tetrahedron apttetd, aptrans centroid of a triangle apttrid, apttrig centroid of a triangle (major plane) apttric centroid of a volume bounded by coordinate surfaces aptbrkp, aptbrkv character data, sort, in either direction aptsorm character data, sort, in either direction (one column) aptsort character set, randomly sample characters from aptchrn character string into integer words, copy digits from a aptatob character string, convert from integer to (oct, dec, hex) aptchia character string, convert to integer (oct, dec, hex) aptchai character string, convert to integer and floating point aptchtp character string, convert to integer and floating point aptffip, aptchap character string, convert to integer and floating point aptsubs, aptsubm character string, delimited, find in another character string aptchfs character string, delimited, find in another character string aptquot, aptfbrk character string, find in another character string aptchfp character string, form from a stem and ASCII subscript aptsuba, aptsubb character string, form from a stem and ASCII subscripts aptsubb character string, form from a stem and integer subscripts aptsubi character string, increment or decrement characters in aptchin character string, longer than eight characters, parse aptchtp character string, longer than eight characters, parse aptffip, aptchap character string, longer than eight characters, parse aptsubs, aptsubm character string, move from one location to another aptchmv character string, put random characters in aptchrn character string, reduce multiple blanks to a single blank in aptblsq character string, remove leading and/or trailing blanks from a aptrmbl character string, replace characters in aptchrp character string, replace in a character string aptchmv character string, squeeze out a character aptchsq character string, store ASCII integer character into aptchia character string, translate hexadeimal into a decimal number apthexd character string, translate into a stem and subscript aptsubr character string, translate into a stem and subscripts, more aptsubm character string, translate into integer (oct, dec, hex) aptchai character string, translate into integer and floating point aptchtp character string, translate into integer and floating point aptffip, aptchap character string, translate into integer and floating point aptsubs, aptsubm character, delimiter, use to separate data fields aptffip, aptchap character, delimiter, use to separate data fields aptquot, aptfbrk character, squeeze out of a character string aptchsq characteristics of quadric curves (conic sections) aptplqu characters, duplicate, find in Sudoku puzzle row, column, box aptsudu characters, increment or decrement in a character string aptchin characters, move from one location to another aptchmv characters, non-graphic, find in a character string and replace aptchip characters, random, put into a character string aptchrn characters, replace in a character string aptchrp characters, translate ASCII (oct, dec, hex) into integer aptchai characters, translate integer into ASCII (oct, dec, hex) aptchia chemistry, 1999 fundamental constants, in cm-g-sh-keV units aptconl chemistry, 1999 fundamental constants, in SI units aptconm Chinese Remainder Theorem Problem inverse aptcrtr Chinese Remainder Theorem Problem solution aptcrts circle of intersection of a plane and a quadric surface aptplqu circle of intersection of a plane and a sphere aptplqu circle of intersection of a plane and a sphere aptplsp, aptplis circle of intersection of two spheres aptplqu circle of intersection of two spheres aptspsp circle tangent to a circle? (major plane) aptcinc circle tangent to three specified tangent circles, radius of aptcirt, aptcirk circle tangent to two specified tangent circles, center of aptcirl, aptcirk circle through 3 points, center and radius of (major plane) aptcirc circle through 3 points, center, radius and orientation of aptcirp circle, annulus of, randomly sample points inside aptlocd circle, distance from a point to a aptwhis circle, intersection with a line (major plane) aptlnic circle, intersection with another circle (major plane) aptcinc circle, line tangent to? (major plane) aptlnic circle, project points onto (major plane) aptproc circle, radially project points onto (major plane) aptproc circle, randomly sample points inside aptlocd circle, randomly sample points inside annulus of aptlocd circle, randomly sample points on (any plane) aptlocd circles coincident (major plane)? aptcinc circles tangent to three specified tangent circles, radii of aptcirt, aptcirk circles, intersection of two (major plane) aptcinc circular annulus, randomly sample points in aptlocd circular cylinder and a plane, distance between, intersection aptplcy, aptplis circular cylinder and a plane, distance between, intersection aptplqu circular cylinder and a sphere, distance between, intersection aptspcy circular cylinder, distance from a plane to a, intersection aptplcy, aptplis circular cylinder, distance from a plane to a, intersection aptplqu circular cylinder, distance from a sphere to a aptspcy circular cylinder, intersection with a plane, distance aptplcy, aptplis circular cylinder, intersection with a sphere, distance aptspcy circular cylinder, proximal line to a plane, distance aptplcy, aptplis circular cylinder, proximal line to a plane, distance aptplqu circular cylinder, proximal point to a sphere, distance aptspcy circular cylinder, proximal point to another circular cylinder aptcycy circular cylinders, distance between (apart or overlapping) aptcycy circular cylinders, distance between axes aptcycy circular cylinders, proximal points on axes, surfaces aptcycy circumscribed circle of a regular polygon aptpoly circumscribed circle of a regular polygon (major plane) aptpolc circumscribed circle of a triangle, perpendicular bisectors apttrig classify a quadrangle shape (major plane) aptqvac classify a quadric surface, and center of symmetry and axes aptaxis classify a quadric surface, and center of symmetry and extrema aptqext coaxial, see if two circular cylinders are aptcycy coefficients of continued fraction, find, given value aptcofr coincident circles (major plane)? aptcinc coincident circular cylinders? aptcycy coincident lines (major plane)? aptlnlc coincident lines? aptlnln coincident planes? aptplpl, aptplis coincident points (major axis)? aptvdil coincident points (major plane)? aptvdic coincident points? aptdist, aptvdis coincident spheres? aptspsp colinear points (major plane)? apttric colinear points? aptpfit colinear points? apttrid, aptvpln colon, use as a field delimiter (:) aptfbrk colon, use as a field delimiter (:) aptffip, aptchap columnar data, delete rows containing duplicate values aptdeld columnar data, sort, any direction or mode (one column) aptsort columnar data, sort, any direction, mode or # of columns aptsorm combinations and permutation of N things taken M at a time aptbpac combinations and permutation of N things taken M at a time aptcomb, aptperm combinations of N things taken M at a time aptcomb, aptcomf combinations of N things taken M at a time, number of aptcomf combine a stem and an ASCII subscript into a character string aptsuba, aptsubb combine a stem and ASCII subscripts into a character string aptsubb combine a stem and integer subscripts into a character string aptsubi combined symmetry operations (see "matrix", "operator") comma, use as a field delimiter (,) aptfbrk comma, use as a field delimiter (,) aptffip, aptchap common divisor of a set of integers, find the greatest aptcmad common multiple of a set of integers, find the least aptcmad comp, Boolean operation "complement" on a machine word aptbitb complement of a machine word (Boolean operation "comp.") aptbitb complex roots of a cubic equation, and real roots aptcubs complex roots of a quadratic equation, and real roots aptqrts, aptqrtv complex roots of a quartic polynomial equation, and real roots aptquar components of vectors, limit range of values (major plane) aptvtoc components of vectors, limit range of values of aptvtol components of vectors, lower limits on values aptvlim components of vectors, lower limits on values (major plane) aptvlic components, vector, in a plane aptvplp components, vector, parallel and perpendicular to a vector aptvpap components, vector, parallel to another vector (major plane) aptvcvc concatenate the digits of two nonzero big integers aptbcat concentric, see if two circular cylinders are aptcycy concentric, see if two spheres are aptspsp cone (see also "quadric surface") cone symmetric around a major axis, equation of aptcois, aptcnis cone symmetric around the z axis, equation of aptcniz cone, distance from a plane to, intersection aptplqu, aptplis cone, distance from a point to a aptwhis cone, elliptic, distance from a plane to an aptplqu cone, equation of, on a major axis, given two points aptcnis cone, equation of, on a major axis, given vertex and angle aptcois cone, equation of, on the z axis, given two points aptcniz cone, intersection with a linear track, normal, angle aptrkcy, aptrkis cone, intersection with a major plane aptplis cone, right circular, randomly sample points in aptscnv cone, right circular, randomly sample points in annulus aptscnv cone, right circular, randomly sample points in frustrum aptscnv cone, right circular, randomly sample points on surface aptscns, aptscnv conic sections (see "quadric surface") conic sections, characteristics of quadric curves aptplqu conjugate axis of a hyperbola or hyperbolic cylinder aptplqu constants, 1999 values of the fundamental aptconl, aptconm continued fraction, find coefficients of, given value aptcofr continued fraction, find value of, given coefficients aptcofv contract bits from a type integer*1 array into machine words aptbitp control characters, find in a character string and replace aptchip Control-V, use as an escape character (^V) aptfbrk convergents of a continued fraction, partial sums aptcofr, aptcofv conversion factors for energy units, 1999 values of aptconl, aptconm convert a big integer (any base) into a decimal integer aptbtod convert a big integer (any base) into a floating point aptbtod convert a character string to integer (oct, dec, hex) aptchai convert a character string to integer and floating point aptchtp convert a character string to integer and floating point aptffip, aptchap convert a character string to integer and floating point aptsubs, aptsubm convert an angle to degrees, grads, radians, arc deg, min, sec aptcang convert an integer into ASCII characters (oct, dec, hex) aptchia convert an integer into ASCII characters (oct, dec, hex) aptchia convert coordinate systems aptcsys, aptcsyv convert from a decimal integer into a big integer (any base) aptdtob convert from floating point into a big integer (any base) aptftob, aptltob convex quadrangle, test for aptqvac coordinate differences aptdist, aptvdis coordinate differences (major plane) aptvdic coordinate differences in groups of points aptdirt, aptprim coordinate system, cartesian, volume bounded by surfaces of aptbrkn, aptbrkp coordinate system, cartesian, volume bounded by surfaces of aptbrks, aptbrkv coordinate system, cylindrical, volume bounded by surfaces of aptbrkn, aptbrkp coordinate system, cylindrical, volume bounded by surfaces of aptbrks, aptbrkv coordinate system, spherical, volume bounded by surfaces of aptbrkn, aptbrkp coordinate system, spherical, volume bounded by surfaces of aptbrks, aptbrkv coordinate system, transform aptcsys, aptcsyv coordinates of a brick, minimum, centroid and maximum aptbrkv coordinates, linearly combine aptvadd, aptvsum coordinates, linearly combine (major plane) aptvadc, aptvsuc coplanar vector triple? aptrips copy digits from a character string into integer words aptatob cosine and sine of an angle (major plane) aptvanc cosine of an angle aptvang cosine of angle between unit vectors aptvdos, aptvdot cosine of angle between unit vectors (major plane) aptvdoc cosine of vertex angle of a quadrangle aptvang cosine of vertex angle of a quadrangle (major plane) aptvanc cosine of vertex angle of a triangle aptvang cosine of vertex angle of a triangle (major plane) aptvanc cosine-power PDF, randomly sample direction vectors from aptscap, aptscar cosine-power PDF, randomly sample direction vectors from aptscav crack free-format input lines, read and aptffip, aptchap crack free-format input lines, with ASCII words of any length aptffip, aptchap cross product of two vectors aptvaxb, aptvxun cross product of two vectors (major plane) aptvaxc, aptvxuc crystal structures, unique directions and distances in aptdirt, aptprim cube, generate aptcube cubic equation, real and complex roots of aptcubs cumulative PDF bins, randomly sample aptcums, aptcumt cumulative PDF, generate aptcump curly brackets, use as field delimiters {} aptfbrk curvature at a point on a quadric surface, extreme values of aptcris curve, parabolic, produced by uniform acceleration aptparb curve, produced by time-dependent acceleration aptacct curve, quadric, minimum and maximum points on (major plane) aptqexc, aptqext curves of intersection between planes and quadric surfaces aptplqu curves, quadric, characteristics of aptplqu cut a triangle into two parts with equal areas, perimeters aptrabc, aptrcab cut a triangle into two parts with equal areas, perimeters aptrcut, aptrext cut through a quadric surface with a major plane aptplis cylinder (see also "quadric surface") cylinder symmetric around a major axis, equation of aptcyis cylinder with a specified axis and radius, equation of aptclis, aptcyis cylinder, circular, distance from a plane to a, intersection aptplcy, aptplis cylinder, circular, distance from a plane to a, intersection aptplqu cylinder, circular, distance from a sphere to a, intersection aptspcy cylinder, circular, intersection with a plane, distance aptplcy, aptplis cylinder, circular, intersection with a sphere, distance aptspcy cylinder, circular, proximal line to a plane, distance aptplcy, aptplis cylinder, circular, proximal line to a plane, distance aptplqu cylinder, circular, proximal point to a sphere, distance aptspcy cylinder, circular, proximal point to another circular cyl aptcycy cylinder, distance from a point to a aptwhis cylinder, elliptic, distance from a plane to aptplqu cylinder, equation of, symmetric around a major axis aptcyis cylinder, equation of, with a specified axis and radius aptclis, aptcyis cylinder, hyperbolic, distance from a plane to aptplqu cylinder, intersection with a linear track (1-D) aptrkcl cylinder, intersection with a linear track, normal, angle aptrkcy, aptrkis cylinder, intersection with a major plane aptplis cylinder, project onto, options aptprcy cylinder, project points onto, options aptprcy cylinder, randomly sample points in aptlocy cylinder, randomly sample points in annulus of aptlocy cylinder, randomly sample points on aptlocy cylinders, circular, distance between (apart or overlapping) aptcycy cylinders, circular, distance between axes aptcycy cylinders, circular, proximal points on axes, surfaces aptcycy cylindrical annulus, randomly sample points in aptlocy cylindrical coordinates, direction between two points in aptdist cylindrical coordinates, distance between two points in aptdist cylindrical coordinates, intersection of linear track in aptrkcy, aptrkis cylindrical coordinates, intersection of linear track in (1-D) aptrkcl cylindrical coordinates, move a point in aptmove, aptmvcy cylindrical coordinates, transform to or from aptcsys, aptcsyv cylindrical coordinates, volume bounded by surfaces of aptbrkn, aptbrkp cylindrical coordinates, volume bounded by surfaces of aptbrks, aptbrkv cylindrical quadric surface tangent to another quadric surface aptqupr cylindrically project points onto a sphere aptprsp cylindrically scale a set of points or vectors around an axis aptsclr D data fields, delimited, separate a character string into aptffip, aptchap data fields, delimited, separate a character string into aptquot, aptfbrk data look-up in a table (binary search) aptlook data range, test for (any mode) aptword data values between two specified limits (any mode)? aptword data, floating point, round-off of aptrnds, aptnint data, reorder in any direction or mode aptsorm data, reorder in any direction or mode (one column) aptsort data, reorder in random order aptrand decimal characters, ASCII, translate an integer into aptchia decimal characters, ASCII, translate into an integer aptchai decimal characters, ASCII, translate into an integer aptchtp decimal digits from a character string, copy into integer words aptatob decimal fraction, reciprocal integer series for a aptiris decimal integer, convert to big integer (any base) from a aptdtob decimal integer, convert a big integer (any base) into aptbtod decimal number in a specified number base, find the value of a aptbase decimal number, integer plus reciprocal integer series for a aptiris decimal number, translate hexadecimal string into apthexd decimal number, translate into hexadecimal string apthexa decrement or increment characters in a character string aptchin degrees, convert an angle to aptcang degrees, minutes, seconds, convert an angle to aptcang delete duplicate values from a sorted or unsorted array aptdeld delimited character string (see "symbol") delimited character string, find in another string aptchfs delimited character string, find in another string aptquot, aptfbrk delimited character strings, convert mode of aptchap, aptchtp delimited character strings, convert to integer (oct, dec, hex) aptchai delimited character strings, read and convert mode of aptffip, aptchap delimited fields, separate a character string into aptffip, aptchap delimited fields, separate a character string into aptquot, aptfbrk delimiter, use of escape character to ignore aptquot, aptfbrk delimiter, use of to separate and translate data fields aptffip, aptchap delimiter, use of to separate and translate data fields aptquot, aptfbrk derivatives of a polynomial equation, evaluate at a set of args aptpole derivatives of a polynomial equation, real roots of aptpolr descending order, reorder data aptsorm descending order, reorder data (one column) aptsort determinant of 3 by 3 matrix aptrips, aptdet3 determinant of 4 by 4 matrix aptdet4 deviation from the mean, standard aptmean difference between coordinates in groups of points aptdirt, aptprim difference between vectors or coordinates aptdist, aptvdis difference between vectors or coordinates (major plane) aptvdic difference of two big integers (any base), find the aptbsub diffraction pattern, crystal directions that determine it aptdirt, aptprim digit in a big integer, find the number of occurrences of each aptbsum digit, squeeze out of a character string aptchsq digits of a big integer array, reverse the order of aptbrev digits of two nonzero big integers, concatenate the aptbcat digits, increment or decrement in a character string aptchin digits, replace in a character string aptchrp direction between two points aptdist, aptvdis direction between two points (major plane) aptvdic direction vector, randomly sample, uniform (scalar) aptsciz direction vectors in a specified plane, randomly sample aptscad direction vectors, randomly sample at a fixed scatter angle aptscan, aptscaz direction vectors, randomly sample at a fixed scatter angle aptscma, aptscmz direction vectors, randomly sample in a specified plane aptscad direction vectors, randomly sample within an angle interval aptscar direction vectors, randomly sample, cosine-power PDF aptscap, aptscar direction vectors, randomly sample, cosine-power PDF aptscav direction vectors, randomly sample, isotropic aptscap, aptscat direction vectors, randomly sample, isotropic aptscav direction vectors, randomly sample, isotropic, limits aptscar direction vectors, randomly sample, uniform aptscap, aptscat direction vectors, randomly sample, uniform aptscav direction vectors, randomly sample, uniform, limits aptscar direction, effect of linear motion on, any coordinate system aptmove direction, effect of linear motion on, cylindrical coordinates aptmvcy direction, effect of linear motion on, spherical coordinates aptmove, aptmovs direction, extrema of a quadric curve in a specified aptqexv direction, motion with distance and, any coordinates aptmove direction, motion with distance and, cylindrical coordinates aptmove, aptmvcy direction, motion with distance and, spherical coordinates aptmove, aptmovs directions between pairs of points, unique in groups of points aptdirt, aptprim directions of extreme curvature at a point on a quadric surface aptcris directions, unique, in regular arrays of points aptdirt, aptprim distance along a parabolic trajectory vs time aptparb distance along a uniformly accelerated trajectory vs time aptparb distance between the axes of two circular cylinders aptcycy distance between two circular cylinders, or overlap aptcycy distance between two lines aptlnln distance between two lines (major plane) aptlnlc distance between two parallel planes aptplpl, aptplis distance between two points on a major axis aptvdil distance between two points, and direction aptdist, aptvdis distance between two points, and direction (major plane) aptvdic distance between two points, and direction in groups of points aptdirt distance from a circular cylinder to a plane, intersection aptplcy, aptplis distance from a circular cylinder to a sphere, intersection aptspcy distance from a circular cylinder to another circular cylinder aptcycy distance from a line to a line (major plane) aptlnlc distance from a line to a line, direction, proximal points aptlnln distance from a line to a plane aptlnpl, aptlntr distance from a line to a point aptptln distance from a line to a point (major plane) aptptlc distance from a nonplanar quadrangle to a point aptquad distance from a parabolic trajectory to a line aptparl distance from a parabolic trajectory to a plane aptparp distance from a parabolic trajectory to a point aptpars distance from a plane to a circular cylinder, intersection aptplcy, aptplis distance from a plane to a circular cylinder, intersection aptplqu distance from a plane to a cone, intersection aptplqu distance from a plane to a line aptlnpl, aptlntr distance from a plane to a quadric surface, intersection aptplqu distance from a plane to a sphere, intersection aptplsp, aptplis distance from a plane to an axisymmetric quadric surface aptplqu distance from a plane to an ellipsoid, intersection aptplqu distance from a point to a circle aptwhis distance from a point to a circular hyperboloid aptwhis distance from a point to a cone aptwhis distance from a point to a cylinder aptwhis distance from a point to a hyperbolic paraboloid aptwhis distance from a point to a hyperbolic paraboloid (approx) (OBS) aptqprr, aptqprt distance from a point to a hyperbolic paraboloid (approx) (OBS) aptquad distance from a point to a line aptptln distance from a point to a line (major plane) aptptlc distance from a point to a nonplanar quadrangle aptquad distance from a point to a pair of intersecting planes aptwhis distance from a point to a pair of parallel planes aptwhis distance from a point to a paraboloid aptwhis distance from a point to a plane aptptpl, aptwhis distance from a point to a quadric surface aptwhis distance from a point to a quadric surface (approx) (OBS) aptqprr, aptqprt distance from a point to a quadric surface, exact or approx aptquad distance from a point to a sphere aptwhis distance from a point to a triangular face aptript, apttrip distance from a point to an ellipsoid aptwhis distance from a point to an elliptic hyperboloid aptwhis distance from a point to an elliptic paraboloid aptwhis distance from a point to plane of a triangle aptript, apttrip distance from a quadrangle surface to a point aptquad distance from a sphere to a circular cylinder, intersection aptspcy distance from a sphere to another sphere, intersection aptspsp distance from a uniformly accelerated trajectory to a line aptparl distance from a uniformly accelerated trajectory to a plane aptparp distance from a uniformly accelerated trajectory to a point aptpars distance from plane of a triangle to the origin aptpfit distance, fractional, limit range of value of aptfdad, aptfdav distance, motion with direction and, any coordinates aptmove distance, motion with direction and, cylindrical coordinates aptmove, aptmvcy distance, motion with direction and, spherical coordinates aptmove, aptmovs distances between points, and directions in groups of points aptdirt distances from a point to edges of a quadrangle (major plane) aptqdic, aptqinc distances from a point to edges of a tetrahedron aptetrn distances from a point to edges of a triangle aptripl, apttrip distances from a point to edges of a triangle (major plane) apttinc distances from a point to faces of a tetrahedron aptetrn distances from a point to surfaces of a brick aptbrkn distances from Steiner vertex of a triangle to its vertices aptsver divide a spherical pentagon into 5 spherical triangles aptpent divide big integers (arrays of digits in specified number base) aptbdiv divisor of a set of integers, find the greatest common aptcmad divisors, remainders for several, find integer when given aptcrts divisors, remainders for several, find when given integer aptcrtr dodecahedron, generate a regular aptdode dot product of vectors aptvdos, aptvdot dot product of vectors (major plane) aptvdoc double area of a triangle by extending two sides aptrext double quotes, use as a field delimiter ("), with an escape aptquot drop file, destroy or save on termination after an error aptnerr duplicate characters, find in row, column, box of Sudoku puzzle aptsudu duplicate values, delete from a sorted or unsorted array aptdeld E eccentricity of a hyperbola or hyperbolic cylinder aptplqu eccentricity of an ellipse or elliptic cylinder aptplqu edge lengths of a brick (bounded by coordinate surfaces) aptbrkv edge lengths of a regular polygon aptpoly edge lengths of a regular polygon (major plane) aptpolc edge lengths of a triangle apttrid, apttrig edge lengths of a triangle (major plane) apttric, aptpolc edge lengths, three, find if form a real triangle aptrich edge of a quadrangle, perpendicular bisector of aptpbln edge of a quadrangle, perpendicular bisector of (major plane) aptpblc edge of a tetrahedron at specified angles from two other edges aptetrv edge of a triangle, perpendicular bisector (major plane) aptpblc edge of a triangle, perpendicular bisector of aptpbln, apttrig edge vectors of a triangle, lengths apttrig edge weights (see "local coordinates") edges of a cube, coordinates of ends of aptcube edges of a quadrangle, distances from a point (major plane) aptqdic, aptqinc edges of a regular dodecahedron, coordinates of ends of aptdode edges of a regular icosahedron, coordinates of ends of apticos edges of a regular octahedron, coordinates of ends of aptocta edges of a regular polygon, lengths of aptpoly edges of a regular polygon, lengths of (major plane) aptpolc edges of a regular tetrahedron, coordinates of ends of apttetr edges of a tetrahedron, distances from a point to aptetrn edges of a triangle, distances from a point to (major plane) apttinc edges of a triangle, distances from a point to the aptripl, apttrip edges of a triangle, extend two to double area aptrext edges of a triangle, midpoints apttrig edges, center of a quadrangle, fit a quadric surface to aptqhyp edges, project equilateral triangle onto triangle with given aptripr effect of linear motion on position and direction, any coords aptmove effect of linear motion on position and direction, cylindrical aptmove, aptmvcy effect of linear motion on position and direction, spherical aptmove, aptmovs eliminate duplicate values or rows from a table aptdeld ellipse (see "circle", "quadric curve") ellipse of intersection between a plane and a quadric surface aptplqu ellipse, eccentricity, major and minor axis aptplqu ellipsoid (see also "quadric surface") ellipsoid at the origin, aligned with the axes, equation of aptesis ellipsoid symmetric around major axis, equation of aptelis ellipsoid, distance from a plane to, intersection aptplqu, aptplis ellipsoid, distance from a point to an aptwhis ellipsoid, equation of, given three semiaxes aptesis ellipsoid, equation of, on major axis, given center, 2 points aptelis ellipsoid, intersection with a linear track, normal, angle aptrkis ellipsoid, intersection with a major plane aptplis elliptic cylinder, eccentricity, major and minor axis aptplqu elliptic paraboloid, distance from a point to aptwhis energy unit conversion factors, 1999 values of aptconl, aptconm energy, randomly sample, from a relativistic Maxwellian PDF aptmaxw equal areas and perimeters, cut a triangle into two parts with aptrabc, aptrcab equal areas and perimeters, cut a triangle into two parts with aptrcut, aptrext equal-probability PDF bin values, randomly sample apteqxs, aptrans equal-probability PDF bins, randomly sample apteqsb equal-probability PDF, generate apteqin, apteqxn equation of a cone on a major axis, given two points aptcnis equation of a cone on a major axis, given vertex and angle aptcois equation of a cone on the z axis, given two points aptcniz equation of a cylinder on a major axis, given the radius aptcyis equation of a cylinder with a specified axis and radius aptclis, aptcyis equation of a hyperboloid of two sheets on a major axis aptelis equation of a plane through three points aptpfit equation of a quadric surface after a matrix operation aptrois equation of a quadric surface after a specified translation apttris equation of a quadric surface, extrema on parabolic trajectory aptparx equation of a sphere at the origin, given radius aptesis equation of an ellipsoid aligned with the major axes aptesis equation of an ellipsoid at the origin, given three semiaxes aptesis equation of an ellipsoid on a major axis aptelis, aptesis equation, cubic, real and complex roots of a aptcubs equation, polynomial, evaluate at a set of arguments aptpole equation, polynomial, real roots using Newtonian iteration aptnewt equation, polynomial, real roots, extrema, inflection points aptpolr equation, polynomial, reduce order by factoring out root aptpolf equation, quadratic, real or complex roots of a aptqrts, aptqrtv equation, quartic polynomial, real and complex roots of a aptquar equations, solve a set of linear aptsolv equilateral projection of triangle with specified edges aptripr equilateral projection of triangle with specified vertices aptripp equilateral triangle, projection onto specified triangle aptripp, aptripr equipartition of a triangle by a straight line cut aptrabc, aptrcab equipartition of a triangle by a straight line cut aptrcut, aptrext equivalence of two machine words, Boolean "xnor" or "eqv" aptbitb eqv, Boolean operation "equivalence" on two machine words aptbitb error messages, write to specified IOCs, and return or exit aptnerr escape character, use to ignore a delimiter character aptquot, aptfbrk estimated distance from a point to a quadric surface aptquad estimated distance from a point to a quadric surface (OBS) aptqprr, aptqprt Euler's totient function for arithmetic modulo an integer aptpfac evaluate a polynomial equation at a set of arguments aptpole exclusive or of two machine words (Boolean operation .xor.) aptbitb expand bits from machine words into a type integer*1 array aptbite exponential of a big integer (any base), find the aptbexp exponential PDF, randomly sample values from aptexpl, aptexps extend two edges of a triangle to double area aptrext extrapolate from two points, any combo of linear/log axes aptintp extrema of a polynomial equation, and roots, inflection points aptpolr extrema of a polynomial equation, using evaluation aptpole extrema of a quadric curve (major plane) aptqexc, aptqext extrema of a quadric surface equation on a parabolic trajectory aptparx extrema of a quadric surface in a specified direction aptqexv extrema of a quadric surface, and axis intercept points aptqupt extrema of a quadric surface, and type and symmetry center aptqext extreme curvatures at a point on a quadric surface aptcris F face, triangular of tetrahedron, distances from a point aptetrn face, triangular, distance from a point to a apttrip face, triangular, point on which side? apttrip face, triangular, proximal point apttrip faces of a cube, generate aptcube faces of a regular dodecahedron, generate aptdode faces of a regular icosahedron, generate apticos faces of a regular octagon, generate aptocta faces of a regular tetrahedron, generate apttetr faces of a tetrahedron, distances of a point from aptetrn factor a root out of a polynomial equation, reduce its order aptpolf factorial function, find all prime factors of a aptpfft factorial function, in integer and real form aptfact factorial of a big integer (any base), find the aptbfac factors of a factorial function, find all prime aptpfft factors of a big integer (any base), find all prime aptpfab factors of an integer, find all prime, and totient function aptpfac factors of an integer, find all unexponentiated aptafac factors, irreducible proportion geometric, and prime numbers aptprim factors, prime, for irreducible proportions in geometric arrays aptprim factors, prime, of a big integer (any base) aptpfab factors, prime, of an integer, and Euler's totient function aptpfac factors, unexponentiated, of an integer aptafac families of quadric surfaces, quadric surface where two ortho aptqper family of quadric surfaces, extreme curvatures at a point aptcris family of quadric surfaces, plane tangent to at a point aptqnor family of quadric surfaces, vector normal to at a point aptqnor fields, delimited, separate a character string into aptffip, aptchap fields, delimited, separate a character string into aptquot, aptfbrk fields, delimited, translate into integer (oct, dec, hex) aptchai fields, delimited, translate into integer and floating point aptchtp fields, delimited, translate into integer and floating point aptffip, aptchap fields, delimited, translate into integer and floating point aptsubs, aptsubm figure, geometric (see "circle", triangle", quadrangle") figure, geometric (see also "polygon") file names longer than eight characters, parse from input aptchtp file names longer than eight characters, parse from input aptffip, aptchap fill a bit string with random bits aptbitr fill a character string with random characters aptchrn fill a type integer*1 array with bits from machine words aptbite fill machine words with bits from a type integer*1 array aptbitp find (implied in any entry without a verb) find a big integer (any base) in any other number base aptbasb find a bit string in a machine word aptbitf find a character string in another character string aptchfp find a character string in another character string aptquot, aptfbrk find a delimited character string in another character string aptchfs find a delimited character string in another character string aptquot, aptfbrk find a power of a big integer (any base) aptbpow find a root of a big integer (any base) aptbrtn find all prime factors of a factorial function aptpfft find all prime factors of a big integer (any base) aptpfab find all prime factors of an integer aptpfac, aptpfab find all unexponentiated factors of an integer aptafac find an integer given remainders for several divisors aptcrts find duplicate characters in Sudoku puzzle row, column, box aptsudu find Euler's totient function for arithmetic modulo an integer aptpfac find non-graphic characters in a character string and replace aptchip find remainders for several divisors when given integer aptcrtr find the difference of two big integers (any base) aptbsub find the exponential of a big integer (any base) aptbexp find the factorial of a big integer (any base) aptbfac find the greatest common divisor and least common multiple aptcmad find the greatest common divisor of a set of integers aptcmad find the least common multiple of a set of integers aptcmad find the log (base 10) of a big integer (any base) aptbtod find the modulus of a a big integer (any base) aptbmod find the number of occurrences of each digit in a big integer aptbsum find the prime factors of a big integer (any base) aptpfab find the product of two big integers (any base) aptbmul find the quotient of two big integers (any base) aptbdiv find the remainder of division of two big integers (any base) aptbdiv find the sum of two big integers (any base) aptbadd find the value of a big integer in a specified number base aptbasb find the value of a decimal number in a specified number base aptbase first derivative of a polynomial equation, evaluate at args aptpole first derivative of a polynomial equation, real roots of aptpolr fit a plane to three points aptpfit fit a quadric surface to edges, center of a quadrangle aptqhyp fit a quadric surface to nine points aptqfit fit a sphere to an axis and two specifed points on its surface aptspha fit a sphere to four specifed points on its surface aptsphp fixed scattering angle, randomly sample azimuth angle for aptscan, aptscaz fixed scattering angle, randomly sample azimuth angle for aptscma, aptscmz floating point data, round-off aptrnds, aptnint floating point data, sort (one column) aptsort floating point data, sort, in either direction aptsorm floating point values, randomly sample from an interval aptrfps, aptrfpt floating point, convert to a big integer (any base) from a aptftob, aptltob floating point, convert from a big integer (any base) into a aptbtod, aptltob form a real triangle, find if three edge lengths can aptrich four points, sphere through aptsphp fourth-order polonomial, real and complex roots of aptquar fraction, continued, find coefficients of, give value aptcofr fraction, continued, find value of, give coefficients aptcofv fraction, decimal, reciprocal integer series for a aptiris fraction, rational, find for a continued fraction, given coeff aptcofv fractional distance, limit range of value of aptfdad, aptfdav fractional distances of triangle equipartition cut line aptrabc, aptrcab fractional distances of triangle equipartition cut line aptrcut, aptrext free-format input lines, read and parse using field delimiters aptffip, aptchap free-format input lines, read and parse using field delimiters aptquot, aptfbrk frequencies, randomly sample from a Planck or Wien spectrum aptxnup frustrum of a cone, randomly sample points on aptscns, aptscnv frustrum of an annulus of a cone, randomly sample points in aptscnv function for arithmetic modulo an integer, find Euler's totient aptpfac function, find all prime factors of a factorial aptpfft function, quadric surface, value at a specified point aptqval function, value of a quadric surface, at a point aptwhis fundamental constants, 1999 values of aptconl, aptconm fuzzy geometry (see "adjust", "limit", "range", "round-off") G generate a cube aptcube generate a cumulative PDF from tabulated data aptcump generate a PDF to randomly sample from, aliased aptalin generate a PDF to randomly sample from, cumulative aptcump generate a PDF to randomly sample from, equal-probability apteqin, apteqxn generate a regular dodecahedron aptdode generate a regular hexahedron (cube) aptcube generate a regular icosahedron apticos generate a regular octahedron aptocta generate a regular polygon aptpoly generate a regular polygon (major plane) aptpolc generate a regular tetrahedron apttetr generate an aliased PDF from tabulated data aptalin generate an equal-probability PDF from tabulated data apteqin, apteqxn generate bins to randomly sample from, aliased bins aptalin generate bins to randomly sample from, cumulative aptcump generate bins to randomly sample from, equal-probabilitility apteqin, apteqxn generate combinations and permutations of M out of N things aptbpac generate combinations and permutations of M out of N things aptcomb, aptperm geometric factors, irreducible proportion, and prime numbers aptprim geometric optics (see "ray tracing") geometric series, find ratio given first term, sum, length aptgrat geometry, fuzzy (see "adjust", "limit", "range", "round-off") grads, convert an angle to aptcang grapestakes, distance between stakes in unique directions aptdirt grapestakes, unique directions of rows in a regular array of aptdirt, aptprim greatest common divisor of a set of integers, find the aptcmad group separated into smaller groups, number of ways aptcomg groups of points, unique directions and their minimum distances aptdirt, aptprim H hexadecimal characters, ASCII, translate an integer into aptchia hexadecimal characters, ASCII, translate into an integer aptchai hexadecimal string, translate a decimal number into apthexa hexadecimal string, translate into a decimal number apthexd histogram PDF, generate bins to randomly sample, aliased aptalin histogram PDF, generate bins to randomly sample, cumulative aptcump histogram PDF, generate bins to randomly sample, equal-prob apteqin, apteqxn histogram PDF, randomly sample bins, aliased aptalsb, aptalst histogram PDF, randomly sample bins, cumulative aptcums, aptcumt histogram PDF, randomly sample from aliased bins aptalsh, aptalsl histogram PDF, randomly sample from cumulative PDF bins aptalsh, aptalsl histogram PDF, randomly sample from equal-probability bins apteqsb, apteqxs horizon, perspective projection with a specified aptpers hyperbola (see also "quadric curve") hyperbola of intersection between a plane and a quadric surface aptplqu hyperbola, eccentricity, transverse and conjugate axes aptplqu hyperbola, intersection with a line (major plane) aptrkcy hyperbolic cylinder, distance from a point to a aptwhis hyperbolic cylinder, eccentricity, transverse, conjugate axes aptplqu hyperbolic paraboloid, distance from a plane to a aptplqu hyperbolic paraboloid, distance from a point to a aptwhis hyperbolic paraboloid, distance from a point to a (approx) aptquad hyperbolic paraboloid, distance from a point to a (approx)(OBS) aptqprr, aptqprt hyperbolic paraboloid, fit to edges, center of a quadrangle aptqhyp hyperboloid (see also "quadric surface") hyperboloid of one sheet, distance from a point to a aptwhis hyperboloid of one sheet, distance from a point to a (approx) aptqprr, aptqprt hyperboloid of two sheets symmetric around major axis aptelis hyperboloid of two sheets, distance from a point to a aptwhis hyperboloid of two sheets, distance from a point to a (approx) aptqprr, aptqprt hyperboloid, circular, distance from a point to aptwhis hyperboloid, elliptical, distance from a point to aptwhis hyperboloid, elliptical, distance from a point to (approx)(OBS) aptqprr, aptqprt I icosahedron, generate a regular apticos ignore delimiter by use of an escape character aptquot, aptfbrk imaginary roots of a cubic polynomial aptcubs imaginary roots of a quadratic polynomial aptqrts, aptqrtv imaginary roots of a quartic polynomial equation aptquar imbedded blanks, squeeze out of a character string aptchsq imbedded blanks, translate data fields with aptchtp imbedded blanks, translate data fields with aptsubs, aptsubm increment or decrement characters in a character string aptchin indices and values of minimum and maximum of an array aptmnlg, aptmnmx indices and values of positive minimum and maximum of an array aptmnlg inflection points of a polynomial equation, and roots, extrema aptpolr inflection points of a polynomial equation, using evaluation aptpole initial position and velocity of a parabolic trajectory aptpart initialize data to randomly sample aliased bins aptalin initialize data to randomly sample cumulative aptcump initialize data to randomly sample equal-probability apteqin, apteqxn input lines, read and parse using field delimiters aptffip, aptchap input lines, read and parse using field delimiters aptquot, aptfbrk inscribed circle of a regular polygon aptpoly inscribed circle of a regular polygon (major plane) aptpolc inscribed circle of a triangle, angle bisectors apttrig inside or outside a brick, a point, and distances? aptbrkn inside or outside a quadrangle, a point (major plane)? aptqinc inside or outside a quadric surface? aptqval, aptwhis inside or outside a tetrahedron, a point? aptetrn inside or outside a triangle, a point (major plane)? apttinc inside or outside a triangle, a point? apttrip int, Boolean operation "intersection" on two machine words aptbitb integer array, reverse the order of an aptbrev integer data, sort, in either direction aptsorm integer data, sort, in either direction (one column) aptsort integer for which remainders for several divisors given, find aptcrts integer multiple of a factor, round off number to, if close aptnint integer plus reciprocal integer series for a decimal number aptiris integer subscripts, form subscripted variable from a stem and aptsubi integer words, copy digits from a character string into aptatob integer*1 array, expand bits from machine words into a type aptbite integer, big (any base), convert from decimal integer into a aptdtob integer, big (any base), convert from floating point into a aptftob, aptltob integer, big (any base), convert into decimal integer aptbtod integer, big (any base), convert into floating point aptbtod integer, big (any base), count the occurrences of each digit in aptbsum integer, big (any base), find a power of a aptbpow integer, big (any base), find a root of a aptbrtn integer, big (any base), find in any other number base aptbasb integer, big (any base), find the exponential of a aptbexp integer, big (any base), find the factorial of a aptbfac integer, big (any base), find the log (base 10) of a aptbtod integer, big (any base), find the prime factors of a aptpfab integer, big (any base), reverse the order of the digits of a aptbrev integer, find all prime factors and Euler's totient function of aptpfac integer, find all unexponentiated factors aptafac integer, find Euler's totient function for arithmetic modulo an aptpfac integer, find remainders, given several divisors aptcrtr integer, find the number of occurrences of each digit in a big aptbsum integer, multiplicative inverse of, in modular arithmetic aptmord integer, order of an, in modular arithmetic aptmord integer, randomly sample from a sequence of integers aptrins integer, round off floating point number to nearest, if close aptnint integer, squeeze out of a character string aptchsq integer, translate from ASCII octal, decimal or hexadecimal aptchai integer, translate into ASCII octal, decimal or hexadecimal aptchia integers, big (any base), find the difference of two aptbsub integers, big (any base), find the modulus of two aptbmod integers, big (any base), find the product of two aptbmul integers, big (any base), find the quotient of two aptbdiv integers, big (any base), find the remainder of division of two aptbdiv integers, big (any base), find the sum of two aptbadd integers, concatenate the digits of two nonzero big aptbcat integers, find the greatest common divisor of a set of aptcmad integers, find the least common multiple of a set of aptcmad integers, increment or decrement in a character string aptchin integers, randomly sample with any degree of striation aptrint integers, replace in a character string aptchrp intercepts on major and symmetry axes of a quadric surface aptqupt interpolate between two points, any combo of linear/log axes aptintp intersecting planes, distance from a point to a pair of aptwhis intersection of a circle with a line (major plane) aptlnic intersection of a cone with a linear track, normal, angle aptrkis intersection of a cone with a major plane aptplis intersection of a cylinder with a linear track (1-D) aptrkcl intersection of a cylinder with a linear track, normal, angle aptrkcy, aptrkis intersection of a cylinder with a major plane aptplis intersection of a cylinder with a plane, distance aptplcy intersection of a hyperbola with a line (major plane) aptrkcy intersection of a line with a circle (major plane) aptlnic intersection of a line with a hyperbola (major plane) aptrkcy intersection of a line with a line (major plane) aptlnlc intersection of a line with a plane aptlnpl intersection of a line with a plane through a triangle aptlntr intersection of a linear track with a cone, normal, angle aptrkcy, aptrkis intersection of a linear track with a cylinder (1-D) aptrkcl intersection of a linear track with a cylinder, normal, angle aptrkcy, aptrkis intersection of a linear track with a line (major plane) aptrklc intersection of a linear track with a major plane aptrksl intersection of a linear track with a plane aptrkpl intersection of a linear track with a quadric surface, normal aptrkis intersection of a linear track with a sphere (1-D) aptrkrl intersection of a linear track with a sphere, normal, angle aptrkis intersection of a linear track with a triangle aptrkpl intersection of a linear track with an ellipsoid, normal, angle aptrkis intersection of a major plane with a cone aptplis intersection of a major plane with a cylinder aptplis intersection of a major plane with a quadric surface aptplis intersection of a major plane with an ellipsoid aptplis intersection of a parabolic trajectory with a line aptparl intersection of a parabolic trajectory with a plane aptparp, aptparq intersection of a parabolic trajectory with a point aptparh, aptpars intersection of a parabolic trajectory with a quadric surface aptparq intersection of a plane with a circular cylinder, distance aptplcy, aptplis intersection of a plane with a circular cylinder, distance aptplqu intersection of a plane with a cone, distance aptplqu intersection of a plane with a line aptlnpl, aptlntr intersection of a plane with a quadric surface, distance aptplqu intersection of a plane with a sphere (none, point or circle?) aptplqu intersection of a plane with a sphere (none, point or circle?) aptplsp, aptplis intersection of a plane with an axisymmetric surface, distance aptplqu intersection of a plane with an ellipsoid, distance aptplqu intersection of a quadric surface with a major plane aptplis intersection of a quadric surface with a plane, distance aptplqu intersection of a quadric surface with another quadric surface aptintq intersection of a sphere with a linear track (1-D) aptrkrl intersection of a sphere with a linear track, normal, angle aptrkis intersection of a sphere with a major plane aptplis intersection of a sphere with a plane (none, point or circle?) aptplsp, aptplis intersection of a triangle with a line aptlntr intersection of a uniformly accelerated track with a line aptparl intersection of a uniformly accelerated track with a plane aptparp, aptparq intersection of a uniformly accelerated track with a point aptparh, aptpars intersection of a uniformly accelerated track with a quadric aptparq intersection of an ellipsoid with a linear track, normal, angle aptrkis intersection of an ellipsoid with a major plane aptplis intersection of three quadric surfaces, triple point of aptripq intersection of triangle altitudes, medians, angle bisectors apttrig intersection of triangle perpendicular bisectors of edges apttrig intersection of two circles (major plane) aptcinc intersection of two lines aptlnln intersection of two machine words (Boolean operation .and.) aptbitb intersection of two machine words (Boolean operation .int.) aptbitb intersection of two planes, line of aptplpl, aptplis intersection of two quadric surfaces, points on the curve of aptintq intersection of two spheres (none, point or circle?) aptspsp interval, uniformly randomly sample in (floating point) aptrfps, aptrfpt interval, uniformly randomly sample in (integer) aptrins, aptrint invariant point (see "inversion", "reflection", "rotation") invariant point (see also "scale", "scaling") inverse of aptrabc, aptrcab, aptrcut aptrext inverse, multiplicative, of an integer in modular arithmetic aptmord inversion operator, to invert a set of points or vectors aptinvp, aptsclu invert a point or vector (major plane) aptinvc invert a point or vector, and operator to aptinvp, aptsclu invert a quadric surface aptrois invert a square matrix, any size aptsolv irrational number, find rational approximations for aptcofr irreducible proportion geometric factors, prime numbers and aptprim isotropic (see also "uniform") isotropically randomly sample a direction vector (scalar) aptsciz, aptscmz isotropically randomly sample direction vectors aptscan, aptscma isotropically randomly sample direction vectors aptscap, aptscat isotropically randomly sample direction vectors aptscav isotropically randomly sample direction vectors in a plane aptscad isotropically scale a set of points or vectors aptsclu iteration, Newtonian, to find real roots of polynomial equation aptnewt J K kissing circle, center of one tangent to two others aptcirl, aptcirk kissing circles, radii of two, each tangent to three others aptcirt, aptcirk kissing sphere, center of one tangent to three others aptsphl, aptsphk kissing spheres, radii of two, each tangent to four others aptspht, aptsphk L latus rectum of a hyperbola or hyperbolic cylinder aptplqu latus rectum of a parabola or parabolic cylinder aptplqu leading and/or trailing blanks, remove from a character string aptrmbl least common multiple of a set of integers, find the aptcmad left-shift bits in a machine word, wrap-around or pull aptbits left-shift characters in a character string aptchmv length of a triangle equipartion cut line aptrabc, aptrcab length of a triangle equipartion cut line aptrcut, aptrext length of a vector aptvunb, aptvusz lengths of regular polygon edges aptpoly lengths of regular polygon edges (major plane) aptpolc lengths of triangle edges aptpoly lengths of triangle edges apttrid, apttrig lengths of triangle edges (major plane) apttric, aptpolc lengths of triangle edges, find if three can form real triangle aptrich letter, squeeze out of a character string aptchsq letters, increment or decrement in a character string aptchin letters, replace in a character string aptchrp limit range of values aptfdad, aptfdav limit range of values of vector components aptvtol limit range of values of vector components (major plane) aptvtoc limits, data values between two specified (any mode)? aptword limits, lower, on values of vector components aptvlim limits, lower, on values of vector components (major plane) aptvlic line (see also "track", "vector") line at specified angles from two other lines aptetrv line between two points, length of (major plane) aptvdic line between two points, length of and vector along aptdirt line between two points, length of and vector along aptdist, aptvdis line between two points, vector along (major plane) aptvdic line if intersection of a plane and a quadric surface aptplqu line in a plane? aptlnpl, aptlntr line of intersection of two planes aptplpl, aptplis line of tangency of a plane and a quadric surface aptplqu line perpendicular to another line (major plane) aptvplc line tangent to a circle (major plane)? aptlnic line, distance form a parabolic trajectory, and proximal point aptparl line, distance from a parabolic trajectory to aptparl line, distance from a plane to a aptlnpl, aptlntr line, distance from a point to a aptptln line, distance from a point to a (major plane) aptptlc line, distance from a uniformly accelerated trajectory to aptparl line, intersection with a circle (major plane) aptlnic line, intersection with a hyperbola (major plane) aptrkcy line, intersection with a line (major plane) aptlnlc line, intersection with a linear track (major plane) aptrklc line, intersection with a parabolic trajectory aptparl line, intersection with a plane aptlnpl, aptlntr line, intersection with a uniformly accelerated trajectory aptparl line, perpendicular bisector of aptpbln line, perpendicular bisector of (major plane) aptpblc line, perspective projection of points onto (major plane) aptperc line, point on (major plane)? aptptlc line, point on? aptptln line, project points onto (major plane) aptperc line, proximal point to a parabolic trajectory aptparl line, proximal point to a uniformly accelerated trajectory aptparl line, randomly sample points on aptspar, aptbrks line, reflect from (major plane) aptrefc line, straight, that cuts triangle into equal areas, perimeters aptrabc, aptrcab line, straight, that cuts triangle into equal areas, perimeters aptrcut, aptrext line, vector parallel to aptvdis line, vector parallel to (major plane) aptvdic linear equations, solve a set of aptsolv linear motion effect on position and direction, any coords aptmove linear motion effect on position and direction, cylindrical aptmove, aptmvcy linear motion effect on position and direction, spherical aptmove, aptmovs linear PDF, randomly sample values from aptslid, aptsliv linear PDFs, randomly sample in specified bins aptalsl linear projection of a quadric surface aptqupr linear projection of points aptprop linear projection of points (major plane) aptperc linear track, intersection with a cylinder (1-D) aptrkcl linear track, intersection with a line (major plane) aptrklc linear track, intersection with a major plane aptrksl linear track, intersection with a plane aptrkpl linear track, intersection with a quadric surface, normal aptrkis linear track, intersection with a sphere (1-D) aptrkrl linear track, intersection with a triangle aptrkpl linear track, intersection with an axisymmetric surface aptrkcy, aptrkis linear-linear two-point interpolation or extrapolation aptintp linear-log two-point interpolation or extrapolation aptintp linearly combine vectors or coordinates aptvadd, aptvsum linearly combine vectors or coordinates (major plane) aptvadc, aptvsuc linearly scale a set of points or vectors aptscll linearly scale a set of points or vectors (major plane) aptsclc lines between points, lengths of and vectors along aptdirt lines coincident or parallel (major plane)? aptlnlc lines coincident or parallel? aptlnln lines of intersection of a plane and a quadric surface aptplqu lines parallel or coincident (major plane)? aptlnlc lines parallel or coincident? aptlnln lines, distance between aptlnln lines, distance between (major plane) aptlnlc local coordinates in a brick, point with aptbrkp local coordinates in a quadrangle, point with aptquwt, aptetrw local coordinates in a tetrahedron, point with aptetrw local coordinates in a triangle, point with aptrlop local coordinates in a triangle, point with (major plane) aptrloc local coordinates of a point in a quadrangle (major plane) aptfdqc, aptqfdc local coordinates of a point in a tetrahedron aptetrl, aptetrn local coordinates of a point in a triangle aptripl, apttrip local coordinates of a point in a triangle (major plane) aptripc log (base 10) of real number into big integer, translate from aptltob log (base 10) of a big integer (any base), find the aptbtod log-linear two-point interpolation or extrapolation aptintp log-log two-point interpolation or extrapolation aptintp log-scale minimum and maximum of an array, values and indices aptmnlg long ASCII strings, parse from free-format input lines aptchtp long ASCII strings, parse from free-format input lines aptffip, aptchap look-up value in a table, binary search aptlook Lorentz transformation aptscll Lorentz transformation (major plane) aptsclc lower case characters, randomly sample from aptchrn lower limits on values of vector components aptvlim lower limits on values of vector components (major plane) aptvlic M machine word, Boolean "and", "int", "nand", "or", "un" aptbitb machine word, Boolean "nor", "xor", "xnor", "eqv" aptbitb machine word, complement of aptbitb machine word, find bit string in aptbitf machine word, find character string in aptchfp machine word, find delimited character string in aptchfs machine word, find delimited character string in aptquot, aptfbrk machine word, put random bits into aptbitr machine word, shift right or left, with wrap-around or pull aptbits machine words, equivalence, complement, nor, not of two aptbitb machine words, intersection, union, exclusive or, of two aptbitb magnitude of a vector aptvunb, aptvusz major axes (x, y, z, u, v, w, r) aptvdil major axes, intercepts of a quadric surface on aptqupt major axes, rotation operator to rotate by 3 angles around aptrots major axis, distance between two points on aptvdil major axis, equation of a cone on aptcois, aptcnis major axis, equation of a cylinder on aptcyis major axis, equation of a hyperboloid of two sheets on aptelis major axis, equation of an ellipsoid on aptelis, aptesis major axis, points coincident? aptvdil major axis, randomly sample directions at fixed angle from aptscma, aptscmz major axis, test for points being coincident aptvdil major plane (see apt subroutines ending with "c") major plane, add vectors aptvadc major plane, adjust values of vector components aptvlic, aptvtoc major plane, angle between two unit vectors, cosine of aptvdoc major plane, angle between two unit vectors, sine of aptvaxc, aptvxuc major plane, angle between two vectors, sine and cosine of aptvanc major plane, angle bisector aptbanc major plane, angle, cosine and sine of aptvanc major plane, angle, cosine of, between unit vectors aptvdoc major plane, angle, rotate a vector by a specified aptrotc major plane, angle, sine of, between two unit vectors aptvaxc, aptvxuc major plane, angle, vector bisecting aptbanc major plane, area between two lines aptvaxc, aptvxuc major plane, area of a regular polygon aptpolc major plane, area, edge lengths, centroid of a triangle apttric major plane, areas of vertices of a quadrangle aptqvac major plane, bisector of a line aptpblc major plane, bisector of a vertex angle of a quadrangle aptbanc major plane, bisector of a vertex angle of a triangle, vector aptbanc major plane, bisector of an angle in aptbanc major plane, bisector of edge of a quadrangle, perpendicular aptpblc major plane, bisector of edge of a triangle, perpendicular aptpblc major plane, boomerang quadrangle, test for aptqvac major plane, bowtied quadrangle, test for aptqvac major plane, centroid of a triangle apttric major plane, centroid, area, edge lengths of a triangle apttric major plane, circle tangent to a circle? aptcinc major plane, circle through 3 specified points in aptcirc major plane, circle, intersection with a line aptlnic major plane, circle, intersection with another circle aptcinc major plane, circle, line tangent to? aptlnic major plane, circle, radially project points onto aptproc major plane, circles coincident? aptcinc major plane, circles tangent to each other, radii, centers aptcirk major plane, circles tangent to each other, radii, centers aptcirl, aptcirt major plane, circles, intersection of two aptcinc major plane, circumscribed circle of a regular polygon aptpolc major plane, classify a quadrangle shape aptqvac major plane, coincident circles? aptcinc major plane, coincident lines? aptlnlc major plane, coincident points? aptvdic major plane, components of vector parallel to another vector aptvcvc major plane, components of vectors, limit range of values aptvtoc major plane, components of vectors, lower limits on values aptvlic major plane, components, vector, parallel to another vector aptvcvc major plane, coordinate differences aptvdic major plane, coordinates, linearly combine aptvadc, aptvsuc major plane, cosine and sine of an angle between two vectors aptvanc major plane, cosine of angle between unit vectors aptvdoc major plane, cosine of vertex angle of a quadrangle aptvanc major plane, cosine of vertex angle of a triangle aptvanc major plane, cross product of two vectors aptvaxc, aptvxuc major plane, direction vector and distance between two points aptvdic major plane, distance between two lines aptlnlc major plane, distance between two points aptvdic major plane, distance from a line to a line aptlnlc major plane, distance from a line to a point aptptlc major plane, distance from a point to a line aptptlc major plane, distances from a point to edges of a quadrangle aptqdic, aptqinc major plane, distances from a point to edges of a triangle apttinc major plane, distances from a point to quadrangle edges in aptqdic, aptqinc major plane, distances from a point to triangle edges in aptqdic, apttinc major plane, dot product of two vectors aptvdoc major plane, edge lengths of a regular polygon aptpolc major plane, edge lengths of a triangle apttric, aptpolc major plane, edge lengths, area, centroid of a triangle apttric major plane, edge of a quadrangle, perpendicular bisector of aptpblc major plane, edge of a triangle, perpendicular bisector aptpblc major plane, edges of a quadrangle, distances from a point aptqdic, aptqinc major plane, edges of a regular polygon, lengths of aptpolc major plane, edges of a triangle, distances from a point apttinc major plane, generate a regular polygon, properties aptpolc major plane, hyperbola, intersection with a line aptrkcy major plane, inscribed circle of a regular polygon aptpolc major plane, inside or outside a quadrangle, a point? aptqinc major plane, inside or outside a triangle, a point? apttinc major plane, intersection of a circle with a line in aptlnic major plane, intersection of a hyperbola with a line aptrkcy major plane, intersection of a line with a circle in aptlnic major plane, intersection of a line with a hyperbola aptrkcy major plane, intersection of a line with a line aptlnlc major plane, intersection of a linear track with a line in aptrklc major plane, intersection of two circles in aptcinc major plane, intersection of two lines in aptlnlc major plane, intersection with a cone aptplis major plane, intersection with a cylinder aptplis, aptplcy major plane, intersection with a linear track aptrksl major plane, intersection with a quadric surface aptplis major plane, intersection with a sphere aptplis, aptplsp major plane, intersection with an ellipsoid aptplis major plane, invert a point or vector in aptinvc major plane, lengths of regular polygon edges aptpolc major plane, lengths of triangle edges apttric, aptpolc major plane, limit range of values of vector components aptvtoc major plane, limits, lower, on values of vector components aptvlic major plane, line between two points, vector and length aptvdic major plane, line perpendicular to another line aptvplc major plane, line tangent to a circle? aptlnic major plane, line, distance from a point to a aptptlc major plane, line, intersection with a circle aptlnic major plane, line, intersection with a hyperbola aptrkcy major plane, line, intersection with a line aptlnlc major plane, line, intersection with a linear track aptrklc major plane, line, perpendicular bisector of aptpblc major plane, line, perspective projection of points onto aptperc major plane, line, point on? aptptlc major plane, line, reflect from aptrefc major plane, line, vector parallel to aptvdic major plane, linear track, intersection with a line aptrklc major plane, linearly combine vectors or coordinates aptvadc, aptvsuc major plane, linearly scale a set of points or vectors aptsclc major plane, lines coincident or parallel? aptlnlc major plane, lines, distance between aptlnlc major plane, local coordinates in a triangle, point with aptrloc major plane, local coordinates of a point in a quadrangle in aptfdqc, aptqfdc major plane, local coordinates of a point in a triangle in aptripc major plane, Lorentz transformation aptsclc major plane, lower limits on values of vector components aptvlic major plane, minimum and maximum points of a quadric curve aptqexc, aptqext major plane, nearest point on a line, to a point aptptlc major plane, normalize a vector product to a unit vector aptvxuc major plane, normalize a vector to a unit vector aptvuac, aptvubc major plane, on a line, a point? aptptlc major plane, origin, translate to a specified point apttrac major plane, orthogonal vectors? aptvdoc major plane, parallel lines? aptlnlc major plane, parallel vectors? aptvaxc, aptvxuc major plane, parallelogram area at a vertex aptvaxc, aptvxuc major plane, perpendicular bisector of a line in aptpblc major plane, perpendicular bisector of edge of a quadrangle aptpblc major plane, perpendicular bisector of edge of a triangle aptpblc major plane, perpendicular to a line, the line aptvplc major plane, perpendicular vectors? aptvdoc major plane, perspective projection of points onto a line aptperc major plane, point in a triangle with given local coordinates aptrloc major plane, point in a triangle, local coordinates of aptripc major plane, point inside or outside a quadrangle? aptqinc major plane, point inside or outside a triangle? apttinc major plane, point on a line? aptptlc major plane, point with given distance, direction from a point aptvadc, aptvsuc major plane, point with given local coordinates in a triangle aptrloc major plane, point, distance from a line to a aptptlc major plane, point, distances from edges of a quadrangle to a aptqdic, aptqinc major plane, point, distances from edges of a triangle to a apttinc major plane, point, invert a aptinvc major plane, point, linearly scale aptsclc major plane, point, local coordinates in a triangle aptripc major plane, point, proximal on a line, to a point aptptlc major plane, point, reflect from a line aptrefc major plane, point, translate through space apttrac major plane, points coincident? aptvdic major plane, points in a quadrangle, randomly sample aptqloc major plane, points in a triangle, randomly sample aptqloc, apttloc major plane, points, direction and distance between two aptvdic major plane, points, linearly scale aptsclc major plane, points, perspective projection onto a line aptperc major plane, points, radially project onto a circle aptproc major plane, points, randomly sample, in a quadrangle aptqloc major plane, points, randomly sample, in a triangle aptqloc, apttloc major plane, points, vector between two aptvdic major plane, polygon, regular, generation and properties of aptpolc major plane, product of two vectors, cross or vector aptvaxc, aptvxuc major plane, product of two vectors, dot or scalar aptvdoc major plane, project onto a line, perspective aptperc major plane, project points onto a circle, radially aptproc major plane, project points onto a perspective view line aptperc major plane, properties of regular polygons aptpolc major plane, proximal point on a line, to a point aptptlc major plane, quadrangle edges, distances from a point to aptqdic, aptqinc major plane, quadrangle vertex angle, sine and cosine of aptvanc major plane, quadrangle vertex areas and shape type aptqvac major plane, quadrangle, cosine and sine of a vertex angle aptvanc major plane, quadrangle, distances from a point to edges of aptqdic, aptqinc major plane, quadrangle, local coordinates of a point in aptfdqc, aptqfdc major plane, quadrangle, perpendicular bisector of an edge aptpblc major plane, quadrangle, point in, local coordinates of aptfdqc, aptqfdc major plane, quadrangle, point inside or outside? aptqinc major plane, quadrangle, randomly sample points in aptqloc major plane, quadrangle, sine and cosine of a vertex angle aptvanc major plane, quadrangle, vector bisecting a vertex angle aptbanc major plane, radially project points onto a circle aptproc major plane, radius of circumscribed circle of regular polygon aptpolc major plane, radius of inscribed circle of a regular polygon aptpolc major plane, randomly sample points in a quadrangle in aptqloc major plane, randomly sample points in a triangle in apttloc, aptqloc major plane, range of values of vector components, limit aptvtoc major plane, reflect a point or vector from a line aptrefc major plane, reflect from a plane perpendicular to a aptrefl major plane, regular polygon generation and properties aptpolc major plane, rotate a vector by a specified angle aptrotc major plane, scalar product of two vectors aptvdoc major plane, scale linearly, points or vectors aptsclc major plane, shape type of a quadrangle (bowtie or boomerang?) aptqvac major plane, sine and cosine of an angle between two vectors aptvanc major plane, sine of angle between two unit vectors aptvaxc, aptvxuc major plane, sine of vertex angle of a quadrangle aptvanc major plane, sine of vertex angle of a triangle aptvanc major plane, sum of vectors, with specified coefficients aptvadc, aptvsuc major plane, tangent circles? aptcinc major plane, tangent, between line and circle aptlnic major plane, test for circle being tangent to a line aptlnic major plane, test for circles being coincident aptcinc major plane, test for circles being tangent aptcinc major plane, test for line being tangent to a circle aptlnic major plane, test for lines being coincident or parallel aptlnlc major plane, test for parallel vectors aptvaxc, aptvxuc major plane, test for point being inside a quadrangle aptqinc major plane, test for point being inside a triangle apttinc major plane, test for point being on a line aptptlc major plane, test for points being coincident aptvdic major plane, test for quadrangle convex, boomerang or bowtie aptqvac major plane, test for vectors being parallel aptvaxc, aptvxuc major plane, test for vectors being perpendicular aptvdoc major plane, track, linear, intersection with a line aptrklc major plane, translate points in a apttrac major plane, triangle area, centroid, edge lengths apttric major plane, triangle edges, distances from a point to aptqdic, aptqinc major plane, triangle edges, distances from a point to apttinc major plane, triangle vertex angle, sine and cosine of aptvanc major plane, triangle, bisector of a vertex angle aptbanc major plane, triangle, cosine of a vertex angle aptvanc major plane, triangle, distances from a point to edges of aptqdic, aptqinc major plane, triangle, distances from a point to edges of apttinc major plane, triangle, local coordinates of a point in aptripc major plane, triangle, perpendicular bisector of an edge aptpblc major plane, triangle, point in, local coordinates of aptripc major plane, triangle, point inside or outside? apttinc major plane, triangle, point with given local coordinates aptrloc major plane, triangle, randomly sample points in aptqloc, apttloc major plane, triangle, sine of a vertex angle aptvanc major plane, triangle, vector bisecting a vertex angle aptbanc major plane, two lines coincident or parallel? aptlnlc major plane, two lines, distance or intersection between aptlnlc major plane, two points coincident? aptvdic major plane, two points, distance and vector between aptvdic major plane, two vectors parallel? aptvaxc, aptvxuc major plane, two vectors perpendicular? aptvdoc major plane, uniformly randomly sample points in a quadrangle aptqloc major plane, uniformly randomly sample points in a triangle apttloc major plane, unit vector perpendicular to two other vectors aptvxuc major plane, unit vector, normalize aptvuac, aptvubc major plane, unit vectors, cosine of angle between aptvdoc major plane, unit vectors, sine of angle between two aptvaxc, aptvxuc major plane, values, limit range of, vector components aptvtoc major plane, values, lower limits on vector components aptvlic major plane, vector between two points aptvdic major plane, vector bisecting a vertex angle of a quadrangle aptbanc major plane, vector bisecting a vertex angle of a triangle aptbanc major plane, vector components parallel to another vector aptvcvc major plane, vector components, limit range of values of aptvtoc major plane, vector components, lower limits on values of aptvlic major plane, vector components, parallel to another vector aptvcvc major plane, vector cross product aptvaxc, aptvxuc major plane, vector differences aptvdic major plane, vector direction and distance between two points aptvdic major plane, vector dot product aptvdoc major plane, vector parallel to a line in aptvdic major plane, vector perpendicular to a line aptvplc major plane, vector product of two vectors aptvaxc, aptvxuc major plane, vector scalar product aptvdoc major plane, vector sum with specified coefficients aptvadc, aptvsuc major plane, vector, invert a aptinvc major plane, vector, linearly scale aptsclc major plane, vector, normalize to a unit vector aptvuac, aptvubc major plane, vector, reflect a aptrefc major plane, vector, rotate by a specified angle aptrotc major plane, vectors parallel? aptvaxc, aptvxuc major plane, vectors perpendicular? aptvdoc major plane, vectors, linearly combine aptvadc, aptvsuc major plane, vertex angle of a quadrangle, bisecting vector aptbanc major plane, vertex angle of a quadrangle, sine and cosine of aptvanc major plane, vertex angle of a triangle, bisecting vector aptbanc major plane, vertex angle of a triangle, sine and cosine of aptvanc major plane, vertex areas and shape type of a quadrangle aptqvac major plane, vertices of a regular polygon aptpolc major plane, view point, perspective projection with specified aptperc major plane, weighted sum of two vectors aptvadc, aptvsuc major planes (x-y, y-z, z-x, u-v, v-w, w-x, z-r) manipulate bits aptbitb, aptbite manipulate bits (more) aptbitf, aptbitm manipulate bits (more) aptbitp, aptbitr manipulate bits (more) aptbits manipulate characters and character strings aptchai manipulate characters and character strings (more) aptchfp manipulate characters and character strings (more) aptchfs, aptchia manipulate characters and character strings (more) aptchin, aptchmv manipulate characters and character strings (more) aptchrn, aptchrp manipulate characters and character strings (more) aptchsq manipulate characters and character strings (more) aptchtp manipulate characters and character strings (more) aptffip, aptchap manipulate characters and character strings (more) aptffip, aptchap manipulate characters and character strings (more) aptsubr manipulate characters and character strings (more) aptsubs, aptsubm masking operations "and", "int", "nand", "or", "un" aptbitb masking operations "nor", "xor", "xnor", "eqv" aptbitb masking operations ("comp", "not" aptbitb masking operations ("shl". "shr") aptbits matching bit string, find in a machine word aptbitf matching character string, find (delimited) in another string aptchfs matching character string, find in a character string aptchfp mathematics (see "equation") matrices, product of two square aptmprd matrix multiply square matrices aptmprd matrix operation on a point or vector aptmopv matrix operation on a quadric surface aptrois matrix operator (see "operator") matrix product of two square matrices aptmprd matrix, 3 by 3, determinant of aptdet3 matrix, 4 by 4, determinant of aptdet4 matrix, invert square aptsolv maximum and minimum coordinates of a brick, and centroid aptbrkv maximum and minimum of an array, positive values and indices aptmnlg maximum and minimum of an array, values and indices aptmnlg, aptmnmx maximum and minimum points of a polynomial equation aptpole, aptpolr maximum and minimum points of a quadric curve (major plane) aptqexc, aptqext maximum and minimum points of a quadric surface aptqupt maximum and minimum points of a quadric surface, and type aptqext, aptqexv Maxwellian PDF, relativistic, randomly sample values from aptmaxw mean value of an array aptmean medians of a triangle, centroid apttrig mensuration (see "angle", "area", "length", "volume") message, error, write to specified IOCs, and return or exit aptnerr midpoints of edges of a triangle apttrig minimum and maximum coordinates of a brick, and centroid aptbrkv minimum and maximum of an array, positive values and indices aptmnlg minimum and maximum of an array, values and indices aptmnlg, aptmnmx minimum and maximum points of a polynomial equation aptpole, aptpolr minimum and maximum points of a quadric curve (major plane) aptqexc, aptqext minimum and maximum points of a quadric surface aptqupt minimum and maximum points of a quadric surface, and type aptqext, aptqexv minimum distance between a point and a nonplanar quadrangle aptquad minimum distance between a point and a plane aptptpl minimum distance between points for each unique direction aptdirt minimum distance between two lines aptlnln minutes, seconds, degrees, convert an angle to aptcang mix data randomly aptrand mode, convert a character string to floating point aptchtp mode, convert a character string to floating point aptffip, aptchap mode, convert a character string to floating point aptsubs, aptsubm mode, convert a character string to integer (decimal) aptchtp mode, convert a character string to integer (decimal) aptffip, aptchap mode, convert a character string to integer (decimal) aptsubs, aptsubm mode, convert a character string to integer (oct, dec, hex) aptchai mode, convert from integer to character (oct, dec, hex) aptchia mode, sort data in any (character, integer, fl pt) aptsorm mode, sort data in any (one column) aptsort modular arithmetic, multiplicative inverse of an integer in aptmord modular arithmetic, order of an integer in aptmord modulus of a big integer (any base), find the aptbmod modulo an integer, find Euler's totient function for arithmetic aptpfac Monte Carlo (see "randomly sample", "PDF", "probability") move a bit string from one machine word to another aptbitm move a character string from one location to another aptchmv move a point any distance in any direction, any coordinates aptmove move a point any distance in any direction, cylindrical aptmove, aptmvcy move a point any distance in any direction, spherical aptmove, aptmovs mult big integers (arrays of digits in specified number base) aptbmul multiple blanks, reduce to a single blank in character string aptblsq multiple of a set of integers, find the least common aptcmad multiple quotes, treatment when used as a delimiter aptquot multiple-column sort of data arrays, any direction or mode aptsorm multiple-subscripted variables, separate stem and subscripts aptsubm multiplicative inverse of an integer in modular arithmetic aptmord N n-dimensional square matrices, product of two aptmprd n-dimensional square matrix, inverse of aptsolv name, longer than eight characters, parse from input lines aptffip, aptchap nearest integer multiple of a factor, round off number to aptnint nearest point on a line, to a point aptptln nearest point on a line, to a point (major plane) aptptlc nearest point on a nonplanar quadrangle, to a point aptquad nearest point on a parabolic trajectory, to a line aptparl nearest point on a parabolic trajectory, to a plane aptparp nearest point on a parabolic trajectory, to a point aptpars nearest point on a plane, to a point aptptpl nearest point on a quadric surface, to a point aptwhis nearest point on a quadric surface, to a point (approx) aptquad nearest point on a quadric surface, to a point (approx) (OBS) aptqprr, aptqprt nearest point on a triangular face, to a point apttrip nearest point on a uniformly accelerated trajectory, to a line aptparl nearest point on a uniformly accelerated trajectory, to a plane aptparp nearest point on a uniformly accelerated trajectory, to a point aptpars nearest triple point of three quadric surfaces to a given point aptripq Newtonian iteration, to find real root of a polynomial equation aptnewt non-ASCII characters, find in a character string and replace aptchip non-graphic characters, find in a character string and replace aptchip non-printing characters, find in a character string and replace aptchip nor of two machine words (Boolean operation .nor.) aptbitb nor, Boolean operation "nor" on two machine words aptbitb normal (see also "orthogonal", "perpendicular") Normal PDF, randomly sample values from aptnorm normal vector at intersection of a line with a quadric surface aptrkis normal vector of a quadric surface function at a point aptcris normal vector of a quadric surface function at a point aptwhis, aptqnor normal vector to plane of a triangle aptpfit normal vector to plane of a triangle aptvpln, apttrig normalize a vector product to a unit vector aptvxun normalize a vector product to unit vector (major plane) aptvxuc normalize to a unit vector aptvuna, aptvunb normalize to a unit vector (major plane) aptvuac, aptvubc normalize to a unit vector (scalar) aptvunz, aptvusz not of two machine words (Boolean operation .not.) aptbitb not, Boolean operation "not" on two machine words aptbitb null characters, find in a character string and replace aptchip number (see "data") number base, find the value of a big integer in a specified aptbasb number base, find the value of a decimal number in a specified aptbase number in a specified number base, find the value of a decimal aptbase number of combinations of N things taken M at a time aptcomf, aptcomb number of permutations of N things taken M at a time aptbpac number of permutations of N things taken M at a time aptperf, aptperm number of ways to separate a group into smaller groups aptcomg number, squeeze out of a character string aptchsq numbers, increment or decrement in a character string aptchin numbers, prime, and irreducible proportion geometric factors aptprim numbers, replace in a character string aptchrp numeric character, squeeze out of a character string aptchsq numeric characters, random, put into a character string aptchrn O occurrences of each digit in a big integer, find the number of aptbsum octahedron, generate a regular aptocta octal characters, ASCII, translate an integer into aptchia octal characters, ASCII, translate into an integer aptchai on a line, a point (major plane)? aptptlc on a line, a point? aptptln one-dimensional axis, distance between two points on aptvdil one-dimensional cylinder, intersection with a linear track aptrkcl one-dimensional major plane, intersection with a linear track aptrksl one-dimensional sphere, intersection with a linear track aptrkrl operations "and", "int", "nand", "or", "un", do on words aptbitb operations "comp", "not", do on words aptbitb operations "nor", "xor", "xnor", "eqv", do on words aptbitb operator, product of, with a vector aptmopv operator, product of, with another operator aptmprd operator, reflection aptrefs operator, reflection, analyze, to find axis aptrefq operator, rotation (see apt subroutines beginning with aptrot) operator, rotation, analyze, to find axis, angle aptrotq operator, rotation, for 3 angles around 3 major axes aptrots operator, rotation, for a specified angle and axis aptrota operator, rotation, to make a plane parallel to another plane aptrotp, aptrott operator, rotation, to make one vector parallel to another aptrott, aptrotv operator, to cylindrically scale a set of points or vectors aptsclr operator, to invert a set of points or vectors aptinvp, aptsclu operator, to isotropically scale a set of points or vectors aptsclu operator, to linearly scale a set of points or vectors aptscll operator, to scale cylindrically around a specified axis aptsclr operator, to scale isotropically, a set of points or vectors aptsclu operator, to scale linearly aptscll operator, to scale uniformly, a set of points or vectors aptsclu operator, to uniformly scale a set of points or vectors aptsclu operators, multiply two matrix aptmprd or, Boolean operation "or" = "union" on two machine words aptbitb order data, any direction or mode (multiple columns) aptsorm order data, any direction or mode (one column) aptsort order data, in random order aptrand order of a polynomial equation, reduce be factoring out root aptpolf order of an integer array, reverse the aptbrev order of an integer in modular arithmetic aptmord order of the digits of a big integer (any base), reverse the aptbrev origin, translate to a specified point apttran origin, translate to a specified point (major plane) apttrac orthocenter of a triangle (intersection of altitudes) apttrig orthogonal (see also "normal", "perpendicular") orthogonal quadric surface families, quadric surface where aptqper orthogonal vector triple? aptripv orthogonal vectors (major plane)? aptvdoc orthogonal vectors? aptvdos, aptvdot outline of a quadric surface, project in a specified direction aptqupr overlap between a circular cylinder and a plane aptplcy, aptplis overlap between a circular cylinder and a sphere aptspcy overlap between a plane and a circular cylinder aptplcy, aptplis overlap between a plane and a sphere aptplsp, aptplis overlap between a sphere and a circular cylinder aptspcy overlap between a sphere and a plane aptplsp, aptplis overlap between two circular cylinders, or distance apart aptcycy overlap between two spheres, or distance apart aptspsp P pack bits from a type integer*1 array into machine words aptbitp pair of points, vector and distance between aptdist pairs of points, unique directions and minimum distances beween aptdirt parabola (see "quadric surface") parabola of intersection of a plane and a quadric surface aptplqu parabolic cylinder, distance from a plane to a aptplqu parabolic cylinder, distance from a point to a aptwhis parabolic trajectory, given points on vs time aptpart parabolic trajectory, intersection with a line aptparl parabolic trajectory, intersection with a plane aptparp, aptparq parabolic trajectory, intersection with a point aptparh, aptpars parabolic trajectory, intersection with a quadric surface aptparq parabolic trajectory, path length vs time aptparb parabolic trajectory, points on vs time aptparb, aptpart parabolic trajectory, proximal point to line aptparl parabolic trajectory, proximal point to plane aptparp parabolic trajectory, proximal point to point aptpars parabolic trajectory, proximal point to quadric surface aptparx parabolic trajectory, velocities on vs time aptparb paraboloid, circular, distance from a plane to a aptplqu paraboloid, distance from a point to a aptwhis paraboloid, elliptic, distance from a plane to aptplqu paraboloid, elliptic, distance from a point to aptwhis paraboloid, hyperbolic, distance from a plane to a aptplqu paraboloid, hyperbolic, distance from a point to a aptwhis paraboloid, hyperbolic, distance from a point to a (approx) aptqprr, aptqprt paraboloid, hyperbolic, fit to edges, center of a quadrangle aptqhyp parallel axes of two circular cylinders? aptcycy parallel lines (major plane)? aptlnlc parallel lines? aptlnln parallel planes, distance from a point to a pair of aptwhis parallel planes? aptplpl, aptplis parallel projection of points onto a plane aptprop parallel projection of points onto a sphere aptprsp parallel to another vector, component of a vector aptvpap parallel vectors (major plane)? aptvaxc, aptvxuc parallel vectors? aptvaxb, aptvxun parallel, rotation operator to make two planes aptrotp, aptrott parallel, rotation operator to make two vectors aptrott, aptrotv parallelogram area at a vertex aptvaxb, aptvxun parallelogram area at a vertex (major plane) aptvaxc, aptvxuc parallelogram, randomly sample points in (any plane) aptspar, aptbrks parallelopiped, randomly sample points in aptspar, aptbrks parentheses, use as field delimiters () aptfbrk parse free-format input lines into ASCII strings, any length aptffip, aptchap parse free-format input lines into ASCII, integer, flt-point aptchtp parse free-format input lines into ASCII, integer, flt-point aptffip, aptchap partial sums, convergents of a continued fraction aptcofr, aptcofv particle trajectory with a time-dependent acceleration aptacct particle, parabolic trajectory of uniformly accelerated aptparb, aptpart path length on a parabolic trajectory to a proximal point aptparl, aptparp path length on a parabolic trajectory to a proximal point aptpars, aptparx path length on a parabolic trajectory to an intersection aptparh, aptparl path length on a parabolic trajectory to an intersection aptparq path length on a parabolic trajectory vs time aptparb path length on a uniformly accelerated trajectory vs time aptparb path of particle with a time-dependent acceleration aptacct path, parabolic, intersection or proximal point with line aptparl path, parabolic, intersection or proximal point with plane aptparp, aptparq path, parabolic, intersection or proximal point with point aptparh, aptpars path, parabolic, intersection with quadric surface aptparq path, parabolic, position and velocity vs time aptparb, aptpart path, parabolic, proximal point on line aptparl path, parabolic, proximal point on plane aptparp path, parabolic, proximal point to a quadric surface aptparx path, parabolic, proximal point to point aptpars pattern matching (see "character string", "bit string") PDF bins, randomly sample, aliased aptalsb, aptalst PDF bins, randomly sample, cumulative aptcums, aptcumt PDF bins, randomly sample, equal-probability apteqsb PDF, aliased bins, generate aptalin PDF, aliased bins, randomly sample aptalsb, aptalst PDF, aliased bins, randomly sample values from specified aptalsh, aptalsl PDF, cosine-power (cos**p) aptscap, aptscar PDF, cosine-power (cos**p) aptscav PDF, cumulative bins, generate aptcump PDF, cumulative bins, randomly sample aptcums, aptcumt PDF, equal-probablity bins, generate apteqin, apteqxn PDF, equal-probablity bins, randomly sample apteqsb PDF, equal-probablity bins, randomly sample values from apteqxs PDF, exponential aptexpl, aptexps PDF, generate, from tabulated data, aliased aptalin PDF, generate, from tabulated data, cumulative aptcump PDF, generate, from tabulated data, equal-probability apteqin, apteqxn PDF, histogram, aliased, generate bins aptalin PDF, histogram, aliased, randomly sample bins aptalsb, aptalst PDF, histogram, aliased, randomly sample values from bins aptalsh, aptalsl PDF, histogram, cumulative, generate bins aptcump PDF, histogram, cumulative, randomly sample bins aptcums, aptcumt PDF, histogram, equal-probability, generate bins apteqin, apteqxn PDF, histogram, equal-probability, randomly sample bins apteqsb PDF, histogram, equal-probability, randomly sample values apteqxs PDF, linear, randomly sample from aptslid, aptsliv PDF, linear, randomly sample in specified bins aptalsl PDF, Normal, randomly sample from aptnorm PDF, Planck or Wien spectrum, randomly sample from aptxnup PDF, power-law, randomly sample from aptspod PDF, relativistic Maxwellian, randomly sample from aptmaxw PDF, tabulated, aliased, generate bins aptalin PDF, tabulated, aliased, randomly sample bins aptalsb, aptalst PDF, tabulated, aliased, randomly sample values from bins aptalsh, aptalsl PDF, tabulated, cumulative, generate bins aptcump PDF, tabulated, cumulative, randomly sample bins aptcums, aptcumt PDF, tabulated, equal-probability, generate bins apteqin, apteqxn PDF, tabulated, equal-probability, randomly sample bins apteqsb PDF, tabulated, equal-probability, randomly sample values apteqxs PDF, uniform, randomly sample a direction vector (scalar) aptsciz, aptscma PDF, uniform, randomly sample bins apteqsb, aptrans PDF, uniform, randomly sample direction vectors aptscan, aptscar PDF, uniform, randomly sample direction vectors aptscat, aptscma pentagon, spherical, divide into 5 spherical triangles aptpent perimeters, equal, straight line that cuts a triangle to get aptrabc, aptrcab perimeters, equal, straight line that cuts a triangle to get aptrcut, aptrext permutations and combinations of M things taken N at a time aptbpac permutations and combinations of M things taken N at a time aptperf, aptcomf permutations and combinations of M things taken N at a time aptperm, aptcomb permutations of N things taken M at a time aptbpac permutations of N things taken M at a time aptperm, aptperf permutations of N things taken M at a time, number of aptbpac permutations of N things taken M at a time, number of aptperf perpendicular (see also "normal", "orthogonal") perpendicular bisector of a line (major plane) aptpblc perpendicular bisector of edge of a quadrangle aptpbln perpendicular bisector of edge of a quadrangle (major plane) aptpblc perpendicular bisector of edge of a triangle aptpbln, apttrig perpendicular bisector of edge of a triangle (major plane) aptpblc perpendicular bisectors of triangle edges, circumscribed circle apttrig perpendicular to a line, the bisecting plane aptpbln perpendicular to a line, the line (major plane) aptvplc perpendicular to a plane, the vector aptvpln, aptpfit perpendicular to another vector, component of a vector aptvpap perpendicular vectors (major plane)? aptvdoc perpendicular vectors? aptvdos, aptvdot perpendiculars from triangle vertices to opposite sides apttrig perspective projection of points onto a line (major plane) aptperc perspective projection of points onto a plane aptpers perspective view plane, project points onto aptpers physics, 1999 fundamental constants, in cm-g-sh-keV units aptconl physics, 1999 fundamental constants, in SI units aptconm piecewise linear tabulated PDF, aliased bins for aptalin piecewise linear tabulated PDF, cumulative bins for aptcump piecewise linear tabulated PDF, equal-probability bins for apteqin, apteqxn Planck or Wien spectrum, randomly sample values from aptxnup plane (see also "major plane") plane (see also "quadrangle", "surface", "triangle")) plane and vector, rotate onto another plane and vector aptrott plane of a triangle, distance of point from aptript, apttrip plane of a triangle, vector normal to aptvpln, aptpfit plane perpendicular to a major plane, reflect from aptrefl plane perpendicularly bisecting a line aptpbln plane perpendicularly bisecting a quadrangle edge aptpbln plane perpendicularly bisecting a triangle edge aptpbln plane tangent to a family of quadric surfaces at a point aptqnor plane tangent to a sphere aptsphl, aptsphk plane tangent to a sphere? aptplsp, aptplis plane through three points, vector normal to aptvpln, aptpfit plane, components of a vector on aptvplp plane, direction vectors in, randomly sample aptscad plane, distance from a circular cylinder to, intersection aptplqu plane, distance from a circular cylinder to, intersection aptplsp, aptplis plane, distance from a cone to, intersection aptplqu plane, distance from a line to a aptlnpl, aptlntr plane, distance from a parabolic trajectory to aptparp plane, distance from a point to a aptptpl, aptwhis plane, distance from a quadric surface to, intersection aptplqu plane, distance from a sphere to, intersection aptplqu plane, distance from a sphere to, intersection aptplsp, aptplis plane, distance from a uniformly accelerated trajectory to aptparp plane, distance from an axisymmetric quadric surface to aptplqu plane, distance from an ellipsoid to, intersection aptplqu plane, distance from another plane to, intersection aptplpl, aptplis plane, distance from another plane to, intersection aptplqu plane, fit to three points aptpfit, aptqhyp plane, intersection with a circular cylinder, distance aptplcy, aptplis plane, intersection with a line aptlnpl, aptlntr plane, intersection with a linear track aptrkpl plane, intersection with a parabolic trajectory aptparp, aptparq plane, intersection with a sphere (none, point or circle?) aptplsp, aptplis plane, intersection with a uniformly accelerated trajectory aptparp, aptparq plane, line in? aptlnpl, aptlntr plane, major (see also "major plane") plane, major, intersection with a linear track aptrksl plane, parallel projection of points onto aptprop plane, perspective projection of points onto aptpers plane, point on which side? aptptpl plane, proximal line to a circular cylinder, distance aptplcy, aptplis plane, proximal point to a parabolic trajectory aptparp plane, proximal point to a sphere, distance aptplsp, aptplis plane, proximal point to a uniformly accelerated trajectory aptparp plane, proximal point to another plane, distance aptplpl, aptplis plane, proximal point to another plane, distance aptplqu plane, r-z, equation of a cone through two points in aptcnis plane, r-z, equation of an ellipsoid through two points in aptelis plane, r-z, intersection of a 3-D linear track with a line in aptrkcy plane, r-z, move a point in, with a specified direction aptmove, aptmvcy plane, randomly sample direction vectors in a specified aptscad plane, reflect a point or vector from aptrefl, aptrefs plane, reflect a point or vector from (major plane) aptrefc plane, reflection, vector normal to aptrefq plane, rotation operator to make parallel to another plane aptrotp, aptrott plane, vector perpendicular to aptvpln, aptpfit plane, view factor projection of points onto aptradv plane, z-r, equation of a cone through two points in aptcnis, aptcniz plane, z-r, equation of an ellipsoid through two points in aptelis plane, z-r, intersection of a 3-D linear track with a line in aptrkcy plane, z-r, move a point in, with a specified direction aptmove, aptmvcy planes coincident or parallel? aptplpl, aptplis planes parallel or coincident? aptplpl, aptplis planes, intersection of two, line of aptplpl, aptplis planes, major (x-y, y-z, z-x, u-v, v-w, w-u, z-r) apt***c planes, volume bounded by, in cartesian coordinate system aptbrkn, aptbrkp planes, volume bounded by, in cartesian coordinate system aptbrks, aptbrkv point in a brick with given local coordinates aptbrkp point in a brick, local coordinates of, distances from surfaces aptbrkn point in a quadrangle nearest another point aptquad point in a quadrangle with given local coordinates aptquwt, aptetrw point in a quadrangle, local coordinates of aptfdqc, aptqfdc point in a tetrahedron with given local coordinates aptetrw point in a tetrahedron, local coordinates of aptetrl point in a triangle nearest nonplanar point apttrip point in a triangle with given local coordinates aptrlop point in a triangle with given local coordinates (major plane) aptrloc point in a triangle, local coordinates of aptripl, apttrip point in a triangle, local coordinates of (major plane) aptripc point inside or outside a brick, distances from surfaces aptbrkn point inside or outside a quadrangle (major plane)? aptqinc point inside or outside a quadric surface? aptwhis point inside or outside a tetrahedron? aptetrn point inside or outside a triangle (major plane)? apttinc point inside or outside a triangle? apttrip point of intersection of three quadric surfaces aptripq point of tangency of a plane and a quadric surface aptplqu point of tangency of a plane and a sphere aptplqu point of tangency of a plane and a sphere aptplsp, aptplis point of tangency of a plane and an ellipsoid aptplqu point of tangency of two spheres aptspsp point on a line (major plane)? aptptlc point on a line? aptptln, aptspcy point on a parabolic trajectory vs time aptparb point on a quadric surface nearest another point aptwhis point on a quadric surface nearest another point (approx) aptquad point on a quadric surface nearest another point (approx) (OBS) aptqprr, aptpqrt point on a quadric surface, extreme curvatures at a aptcris point on a trajection of a particle with time-dependent accel aptacct point on a uniformly accelerated trajectory vs time aptparb point with given distance and direction from a point aptvadd, aptvsum point with given distance, direction from point (major plane) aptvadc, aptvsuc point with given local coordinates in a brick aptbrkp point with given local coordinates in a quadrangle aptquwt, aptetrw point with given local coordinates in a tetrahedron aptetrw point with given local coordinates in a triangle aptrlop point with given local coordinates in a triangle (major plane) aptrloc point, cylindrically scale, relative to a specified axis aptsclr point, distance from a line to a aptptln point, distance from a line to a (major plane) aptptlc point, distance from a nonplanar quadrangle to a aptquad point, distance from a nonplanar quadrangle to a aptquad point, distance from a parabolic trajectory to aptpars point, distance from a plane of a triangle to a aptript, apttrip point, distance from a plane to a aptptpl point, distance from a quadric surface, and proximal point aptwhis, aptquad point, distance from a quadric surface, and proximal point(OBS) aptqprr, aptqprt point, distance from a triangular face to a aptript, apttrip point, distances from edges of a quadrangle to a (major plane) aptqdic, aptqinc point, distances from edges of a triangle to a aptripl, apttrip point, distances from edges of a triangle to a (major plane) apttinc point, distances from the faces of a tetrahedron to a aptetrn point, intersection with a parabolic trajectory aptparh, aptpars point, invariant (see "inversion", "reflection", "rotation") point, invariant (see also "scale", "scaling") point, invert (major plane) aptinvc point, invert, and operator to aptinvp, aptsclu point, isotropically scale, and operator to aptsclu point, linearly scale (major plane) aptsclc point, linearly scale, relative to a specified plane aptscll point, local coordinates in a brick, distances from surfaces aptbrkn point, local coordinates in a tetrahedron aptetrl point, local coordinates in a triangle aptripl, apttrip point, local coordinates in a triangle (major plane) aptripc point, matrix operation on a aptmopv point, proximal on a line aptptln point, proximal on a line (major plane) aptptlc point, proximal on a nonplanar quadrangle aptquad point, proximal on a parabolic trajectory, to a point aptpars point, proximal on a plane aptptpl point, proximal on a triangular face apttrip point, reflect a, analyze the operator to aptrefq point, reflect from a line (major plane) aptrefc point, reflect from a plane of aptrefl, aptrefs point, rotate (see also "operator", "rotate") point, rotate around an axis by a specified angle aptrota, aptmopv point, rotate around major axes by specified angles aptrots, aptmopv point, translate through space apttran point, translate through space (major plane) apttrac point, uniformly scale, and operator to aptsclu point, value of a quadric surface function at a specified aptqval points coincident (major axis)? aptvdil points coincident (major plane)? aptvdic points coincident? aptdist, aptvdis points colinear (major plane)? apttric points colinear? aptpfit points colinear? apttrid, aptvpln points in a brick, randomly sample aptbrks points in a circle, randomly sample (any plane) aptlocd, aptbrks points in a circular annulus, randomly sample aptlocd, aptbrks points in a cone, randomly sample aptscnv, aptbrks points in a conical annulus, randomly sample aptscnv, aptbrks points in a conical frustrum, randomly sample aptscnv, aptbrks points in a cylinder, randomly sample aptlocy, aptbrks points in a cylindrical annulus, randomly sample aptlocy, aptbrks points in a parallelopiped, randomly sample aptspar, aptbrks points in a quadrangle, randomly sample (major plane) aptqloc points in a sphere, randomly sample aptlocs, aptbrks points in a spherical annulus, randomly sample aptlocs, aptbrks points in a tetrahedron, randomly sample aptetrp points in a triangle, randomly sample apttlod points in a triangle, randomly sample (major plane) aptqloc, apttloc points of intersection of two quadric surfaces aptintq points on a circle, randomly sample (any plane) aptlocd, aptbrks points on a cone, randomly sample aptscns, aptscnv points on a cylinder, randomly sample aptlocy, aptscns points on a cylinder, randomly sample aptscnv, aptbrks points on a disk, randomly sample (any plane) aptlocd, aptbrks points on a linear or parabolic trajectory vs time aptparb points on a parabolic trajectory vs time aptparb, aptpart points on a quadric surface, extrema and axis intercepts aptqupt points on a sphere, randomly sample aptlocs, aptbrks points on a time-dependent accelerated trajectory aptacct points on a uniformly accelerated trajectory vs time aptparb points on an annular disk, randomly sample (any plane) aptlocd, aptbrks points, 2, center and radius of sphere through, on an axis aptspha points, 3, center and radius of circle through (major plane) aptcirc points, 3, center, orientation and radius of circle through aptcirp points, 3, plane through, equation of aptpfit points, 4, center and radius of sphere through aptsphp points, 4, quadric surface through quadrangle edges and vertex aptqhyp points, 9, quadric surface through aptqfit points, cylindrically scale, relative to a specified axis aptsclr points, direction and distance between two aptdist, aptvdis points, direction and distance between two (major plane) aptvdic points, distance between two (major axis) aptvdil points, invert (see "operator", "inversion") points, linearly scale (major plane) aptsclc points, linearly scale, relative to a specified plane aptscll points, parallel projection onto a plane aptprop points, perspective projection onto a line (major plane) aptperc points, perspective projection onto a plane aptpers points, project onto a cylinder, options aptprcy points, project onto a line (major plane) aptperc points, project onto a plane, parallel aptprop points, project onto a plane, perspective aptpers points, project onto a plane, view factor aptradv points, project onto a sphere, options aptprsp points, project onto a sphere, radial aptpros points, proximal, of a plane and a quadric surface aptplqu points, proximal, of a sphere and a cylinder aptspcy points, proximal, of two cylinders aptcycy points, proximal, of two lines aptlnln points, proximal, of two quadric surfaces aptintq points, proximal, of two spheres aptspsp points, radially project onto a circle (major plane) aptproc points, radially project onto a sphere aptpros, aptprsp points, randomly sample, in a circle aptlocd, aptbrks points, randomly sample, in a circular annulus aptlocd, aptbrks points, randomly sample, in a cylinder aptlocy, aptbrks points, randomly sample, in a cylindrical annulus aptlocy, aptbrks points, randomly sample, in a parallelogram (any plane) aptspar, aptbrks points, randomly sample, in a parallelopiped aptspar, aptbrks points, randomly sample, in a quadrangle (major plane) aptqloc points, randomly sample, in a sphere aptlocs, aptbrks points, randomly sample, in a spherical annulus aptlocs, aptbrks points, randomly sample, in a tetrahedron aptetrp points, randomly sample, in a triangle apttlod points, randomly sample, in a triangle (major plane) aptqloc, apttloc points, randomly sample, on a circle (any plane) aptlocd, aptbrks points, randomly sample, on a line aptspar, aptbrks points, randomly sample, on an annular disk (any plane) aptlocd, aptbrks points, reflect (see "operator", "reflection") points, rotate (see "operator", "rotation") points, rotate around an axis depending on axial position aptwist points, rotate around an axis depending on radial position aptwirl points, scale (see "operator", "scaling") points, transform the coordinate system of aptcsys, aptcsyv points, translate through space apttran points, translate through space (major plane) apttrac points, unique directions and their minimum distances between aptdirt points, vector between two aptdist, aptvdis points, vector between two (major plane) aptvdic points, vector normal to plane through three aptvpln, aptpfit points, vectors between, unique, and their minimum lengths aptdirt points, view factor projection onto a plane aptradv polygon (see also "triangle", "quadrangle") polygon, regular, generate aptpoly polygon, regular, generate (major plane) aptpolc polygon, regular, properties of aptpoly polygon, regular, properties of (major plane) aptpolc polyhedron (see also "cube", "dodecahedron", "icosahedron") polyhedron (see also "octahedron", "tetrahedron") polyhedron, regular, generate, cube aptcube polyhedron, regular, generate, dodecahedron aptdode polyhedron, regular, generate, icosahedron apticos polyhedron, regular, generate, octahedron aptocta polyhedron, regular, generate, tetrahedron apttetr polynomial equation, evaluate at a set of arguments aptpole polynomial equation, reduce order by factoring out root aptpolf polynomial roots, fourth order, real and complex (quartic) aptquar polynomial roots, real, and extrema, inflection points aptpolr polynomial roots, real, using Newtonian iteration aptnewt polynomial roots, second order, real or complex (quadratic) aptqrts, aptqrtv polynomial roots, third order, real and complex (cubic) aptcubs position, effect of linear motion on, any coordinate system aptmove position, effect of linear motion on, cylindrical coordinates aptmove, aptmvcy position, effect of linear motion on, spherical coordinates aptmove, aptmovs positions and times on a parabolic trajectory aptparb, aptpart positions and times on a time-dependent accelerated trajectory aptacct power of a big integer (any base), find a aptbpow power-law PDF, randomly sample values from aptspod precision, round-off floating point data to specified aptrnds, aptnint prime factors of a factorial function, find all aptpfft prime factors of a big integer (any base), find the aptpfab prime factors of an integer, find all aptpfac, aptpfab principal axis transformation of a quadric surface aptaxis probability distribution function (see "PDF") product of a matrix and a vector aptmopv product of three vectors, scalar aptrips product of three vectors, vector aptripv product of two big integers (any base), find the aptbmul product of two matrices aptmprd product of two vectors, cross or vector aptvaxb, aptvxun product of two vectors, cross or vector (major plane) aptvaxc, aptvxuc product of two vectors, dot or scalar aptvdos, aptvdot product of two vectors, dot or scalar (major plane) aptvdoc project a quadric surface in a specified direction aptqupr project a vector onto a plane aptvplp project an equilateral triangle onto a specified triangle aptripp, aptripr project onto a circle (major plane) aptproc project onto a cylinder, options aptprcy project onto a line, perspective (major plane) aptperc project onto a plane, parallel aptprop project onto a plane, perspective aptpers project onto a plane, view factor aptradv project onto a sphere, options aptprsp project onto a sphere, radial aptpros, aptprsp project points onto a circle, radial (major plane) aptproc project points onto a cylinder, options aptprcy project points onto a line, perspective (major plane) aptperc project points onto a plane, parallel aptprop project points onto a plane, perspective aptpers project points onto a plane, view factor aptradv project points onto a sphere, options aptprsp project points onto a sphere, radial aptpros, aptprsp properties of regular polygons aptpoly properties of regular polygons (major plane) aptpolc proportion, irreducible, geometric factors, and prime numbers aptprim proximal line on a circular cylinder, to a plane aptplcy, aptplis proximal line on a circular cylinder, to a plane aptplqu proximal line on a cone, to a plane, distance aptplqu proximal line on a plane, to a circular cylinder aptplcy, aptplis proximal line on a plane, to a circular cylinder aptplqu proximal line on a plane, to a cone aptplqu proximal line on a plane, to a hyperbolic paraboloid aptplqu proximal lines on a circular cylinder and a plane aptplcy, aptplis proximal lines on a plane and a circular cylinder aptplcy, aptplis proximal lines on a plane and a circular cylinder aptplqu proximal lines on a plane and a cone aptplqu proximal point in a quadrangle, to another point aptquad proximal point in a triangle, to non-planar point apttrip proximal point on a circular cylinder, to a plane aptplqu proximal point on a circular cylinder, to a sphere aptspcy proximal point on a cone, to a point aptwhis proximal point on a cylinder, to a point aptwhis proximal point on a line, to a point aptptln proximal point on a line, to a point (major plane) aptptlc proximal point on a nonplanar quadrangle, to a point aptquad proximal point on a parabolic trajectory, to line aptparl proximal point on a parabolic trajectory, to plane aptparp proximal point on a parabolic trajectory, to point aptpars proximal point on a paraboloid, to a point aptwhis proximal point on a plane, to a point aptptpl proximal point on a plane, to a sphere aptplsp, aptplis proximal point on a quadric surface, to a plane aptplqu proximal point on a quadric surface, to a point aptwhis, aptquad proximal point on a quadric surface, to a point (approx) (OBS) aptqprr, aptqprt proximal point on a sphere, to a circular cylinder aptspcy proximal point on a sphere, to a plane aptplqu proximal point on a sphere, to a plane aptplsp, aptplis proximal point on a sphere, to a point aptwhis proximal point on a sphere, to another sphere aptspsp proximal point on a triangular face, to a point apttrip proximal point on a uniformly accelerated trajectory, to line aptparl proximal point on a uniformly accelerated trajectory, to plane aptparp proximal point on a uniformly accelerated trajectory, to point aptpars proximal point on an axisymmetric surface, to a plane aptplqu proximal point on an ellipsoid, to a plane, distance aptplqu proximal point on the axis of a circular cylinder, to a sphere aptspcy proximal points on a circular cylinder and a sphere aptspcy proximal points on a parabolic trajectory, to a quadric surface aptparx proximal points on a plane and a quadric surface aptplqu proximal points on a plane and a sphere aptplqu proximal points on a plane and a sphere aptplsp, aptplis proximal points on a plane and an axisymmetric surface aptplqu proximal points on a plane and an ellipsoid aptplqu proximal points on a sphere and a circular cylinder aptspcy proximal points on a sphere and a plane aptspcy proximal points on the axes of two circular cylinders aptcycy proximal points on two circular cylinders aptcycy proximal points on two quadric surfaces aptintq pull bits right or left from one machine word into another aptbits puzzle, solve a sudoku aptsudo, aptsudx Q quadrangle (see also "surface", "triangle") quadrangle edges, distances from a point to (major plane) aptqdic, aptqinc quadrangle vertex angle, cosine of aptvang quadrangle vertex angle, sine and cosine of (major plane) aptvanc quadrangle vertex areas and shape type (major plane) aptqvac quadrangle, cosine and sine of a vertex angle (major plane) aptvanc quadrangle, cosine of a vertex angle aptvang quadrangle, distance from a point to a aptquad quadrangle, distances from a point to edges of (major plane) aptqdic, aptqinc quadrangle, local coordinates of a point in (major plane) aptfdqc, aptqfdc quadrangle, nonplanar, distance from a point to a aptquad quadrangle, perpendicular bisector of an edge aptpbln quadrangle, perpendicular bisector of an edge (major plane) aptpblc quadrangle, point in nearest another point aptquad quadrangle, point in, local coordinates of (major plane) aptfdqc, aptqfdc quadrangle, point inside or outside (major plane)? aptqinc quadrangle, point with given local coordinates aptquwt, aptetrw quadrangle, proximal point to a point aptquad quadrangle, quadric surface fit to edges and ceneter of a aptqhyp quadrangle, randomly sample points in (major plane) aptqloc quadrangle, sine and cosine of a vertex angle (major plane) aptvanc quadrangle, vector bisecting a vertex angle aptbang quadrangle, vector bisecting a vertex angle (major plane) aptbanc quadratic equation, real or complex roots of a aptqrts, aptqrtv quadric curve (see "circle", "ellipse", "hyperbola") quadric curve (see also "parabola") quadric curve, minimum and maximum points on (major plane) aptqexc, aptqext quadric surface (see also "cone", "cylinder", "ellipsoid") quadric surface (see also "sphere", "hyperboloid") quadric surface equation, extrema on a parabolic trajectory aptparx quadric surface function, value at a specified point aptqval quadric surface where two quadric surfaces families are ortho aptqper quadric surface, center of symmetry of aptaxis quadric surface, classify a aptaxis quadric surface, distance from a plane to, intersection aptplqu, aptplis quadric surface, distance from a point to a aptwhis, aptquad quadric surface, distance from a point to a (approx) (OBS) aptqprr, aptqprt quadric surface, extrema and axis intercept points aptqupt quadric surface, extrema and type and symmetry center aptqext quadric surface, extrema in a specified direction aptqexv quadric surface, extreme curvatures at a point, centers aptcris quadric surface, fit to edges and center of a quadrangle aptqhyp quadric surface, fit to nine points aptqfit quadric surface, intersection with a linear track aptrkis quadric surface, intersection with a major plane aptplis quadric surface, intersection with a parabolic trajectory aptparq quadric surface, invert a aptrois quadric surface, minimum, maximum and saddle points on aptqext, aptqexv quadric surface, normal vector and tangent plane at a point aptqnor quadric surface, principal axis transformation of aptaxis quadric surface, project in a specified direction aptqupr quadric surface, reflect a aptrois quadric surface, rotate a aptaxis, aptrois quadric surface, scale a aptrois quadric surface, side on which a point is on, distance aptwhis, aptquad quadric surface, side on which a point is on, distance (approx) aptqprt, aptqprr quadric surface, standard form of aptaxis quadric surface, symmetry operations on aptrois quadric surface, translate through space aptaxis, apttris quadric surfaces, intersections or proximal points of aptintq quadric surfaces, triple point of intersection of aptripq quartic polynomial equation, real and complex roots of aptquar quotes, use as field delimiters, single and double aptquot quotient of two big integers (any base), find the aptbdiv R r-z plane, equation of a cone through two points in aptcnis r-z plane, equation of an ellipsoid through two points in aptelis r-z plane, intersection of a 3-D linear track with a line in aptrkcy r-z plane, move a point in, with a specified direction aptmove, aptmvcy radial position, rotate points around axis depending on aptwirl radially project points onto a circle (major plane) aptproc radially project points onto a cylinder aptprcy radially project points onto a sphere aptpros, aptprsp radians, convert an angle to aptcang radii and centers of circles tangent to 3 tangent circles aptcirk radii and centers of spheres tangent to 4 tangent spheres aptsphk radii of circles tangent to three specified tangent circles aptcirt, aptcirk radii of spheres tangent to four specified tangent spheres aptspht, aptsphk radius of circle tangent to three specified tangent circles aptcirt, aptcirk radius of circumscribed circle of a regular polygon aptpoly radius of circumscribed circle of a triangle apttrig radius of circumscribed circle of regular polygon (major plane) aptpolc radius of curvature at a point on a quadric surface, extreme aptcris radius of inscribed circle of a regular polygon aptpoly radius of inscribed circle of a regular polygon (major plane) aptpolc radius of inscribed circle of a triangle apttrig radius of sphere tangent to four specified tangent spheres aptspht, aptsphk radius, center of circle through 3 points (major plane) aptcirc radius, center of sphere on an axis, and through 2 points aptspha radius, center of sphere through 4 specified points aptsphp radius, center, orientation of circle through 3 points aptcirp random characters, put into a character string aptchrn random characters, put into a character string aptchrn randomly reorder data aptrand randomly sample a direction vector, uniform aptsciz randomly sample a floating point value from a range of values aptrfps randomly sample aliased PDF bins aptalsb, aptalst randomly sample an integer from a sequence of integers aptrins randomly sample azimuth angle for a fixed scatter angle aptscan, aptscaz randomly sample azimuth angle for a fixed scatter angle aptscma, aptscmz randomly sample bins, aliased PDF aptalsb, aptalst randomly sample bins, cumulative PDF aptcums, aptcumt randomly sample bins, equal-probability PDF apteqsb randomly sample bits, and put into a machine word aptbitr randomly sample characters from a character set aptchrn randomly sample characters, and put into a character string aptchrn randomly sample cumulative PDF bins aptcums, aptcumt randomly sample direction vectors in a specified plane aptscad randomly sample direction vectors within an angle interval aptscar randomly sample direction vectors, cosine-power PDF aptscap, aptscar randomly sample direction vectors, cosine-power PDF aptscav randomly sample direction vectors, fixed angle from axis aptscan, aptscaz randomly sample direction vectors, fixed angle from axis aptscma, aptscmz randomly sample direction vectors, uniform aptscat randomly sample equal-probability PDF bin values apteqxs, aptrans randomly sample equal-probability PDF bins apteqsb randomly sample floating point values from an interval aptrfpt randomly sample frequencies from a Planck or Wien spectrum aptxnup randomly sample from a cumulative PDF, bins aptcums, aptcumt randomly sample from a cumulative PDF, generate bins for aptcump randomly sample from a histogram PDF, aliased bin generation aptalin randomly sample from a histogram PDF, aliased bin values aptalsh, aptalsl randomly sample from a histogram PDF, aliased bins aptalsb, aptalst randomly sample from a histogram PDF, cumulative bin generation aptcump randomly sample from a histogram PDF, cumulative bins aptcums, aptcumt randomly sample from a histogram PDF, equal-probability bins apteqsb, apteqxs randomly sample from a histogram PDF, equal-probability init apteqin, apteqxn randomly sample from a Planck or Wien PDF or spectrum aptxnup randomly sample from a relativistic Maxwellian PDF aptmaxw randomly sample from a tabulated PDF, aliased bin generation aptalin randomly sample from a tabulated PDF, aliased bin values aptalsh, aptalsl randomly sample from a tabulated PDF, aliased bins aptalsb, aptalst randomly sample from a tabulated PDF, cumulative bin generation aptcump randomly sample from a tabulated PDF, cumulative bins aptcums, aptcumt randomly sample from a tabulated PDF, equal-probability bins apteqsb, apteqxs randomly sample from a tabulated PDF, equal-probability init apteqin, apteqxn randomly sample from an aliased PDF, bins aptalsb, aptalst randomly sample from an aliased PDF, generate bins aptalin randomly sample from an aliased PDF, values from specified bins aptalsh, aptalsl randomly sample from an equal-probability PDF, bins apteqsb randomly sample from an equal-probability PDF, initialize apteqin, apteqxn randomly sample from an equal-probability PDF, values apteqxs, aptrans randomly sample from an exponential PDF aptexpl, aptexps randomly sample from linear PDFs in specified bins aptalsl randomly sample from uniform PDFs in specified bins aptalsh randomly sample integers, with any degree of striation aptrint randomly sample points in a circle aptlocd, aptbrks randomly sample points in a circular annulus aptlocd, aptbrks randomly sample points in a cone aptscnv, aptbrks randomly sample points in a conical annulus aptscnv, aptbrks randomly sample points in a conical frustrum aptscnv, aptbrks randomly sample points in a cylinder aptlocy, aptbrks randomly sample points in a cylindrical annulus aptlocy, aptbrks randomly sample points in a parallelogram (any plane) aptspar, aptbrks randomly sample points in a parallelopiped aptspar, aptbrks randomly sample points in a quadrangle (major plane) aptqloc randomly sample points in a sphere aptlocs, aptbrks randomly sample points in a spherical annulus aptlocs, aptbrks randomly sample points in a tetrahedron aptetrp randomly sample points in a triangle apttlod randomly sample points in a triangle (major plane) aptqloc, apttloc randomly sample points on a circle (any plane) aptlocd, aptbrks randomly sample points on a cone aptscns, aptscnv randomly sample points on a cylinder aptlocy, aptbrks randomly sample points on a sphere aptlocs, aptbrks randomly sample points on an annular disk (any plane) aptlocd, aptbrks randomly sample values from a linear PDF aptslid, aptsliv randomly sample values from a Normal PDF aptnorm randomly sample values from a Planck or Wien PDF aptxnup randomly sample values from a power-law PDF aptspod randomly sample values from a relativistic Maxwellian PDF aptmaxw randomly sample values from a set of equal-probability bins apteqxs randomly sample values from an exponential PDF aptexpl, aptexps randomly sample values from specified bins, linear PDF aptalsl randomly sample values from specified bins, uniform PDF aptalsh randomly sample, generate PDF to, aliased bins aptalin randomly sample, generate PDF to, cumulative bins aptcump randomly sample, generate PDF to, equal-probability bins apteqin, apteqxn range of values of vector components, limit aptvtol range of values of vector components, limit (major plane) aptvtoc range, data values in a specified (any mode)? aptword ratio, find for geom series, given first term, sum, length aptgrat rational approximations of an irrational number aptcofr, aptcofv rational fraction approximation of an irrational number aptcofr rational fractions, value, find for a continued fraction aptcofv ray tracing (see "distance", "intersection", "side", "track") read and parse free-format input lines aptffip, aptchap real roots of a cubic equation, and complex roots aptcubs real roots of a polynomial equation using Newtonian iteration aptnewt real roots of a polynomial equation, and extrema, infl pts aptpolr real roots of a quadratic equation, and complex roots aptqrts, aptqrtv real roots of a quartic polynomial equation, and complex roots aptquar real triangle, find if three edge lengths can form aptrich reciprocal integer series for a decimal fraction aptiris rectangular coordinates, volume bounded by surfaces of aptbrkn, aptbrkp rectangular coordinates, volume bounded by surfaces of aptbrks, aptbrkv reduce an array or table to unique values or lines aptdeld reduce multiple blanks to a single blank in character string aptblsq reduce order of a polynomial equation by factoring out a root aptpolf reflect a point or vector from a line (major plane) aptrefc reflect a point or vector from a plane aptrefl, aptrefs reflect a point or vector, analyze the operator to aptrefq reflect a quadric surface aptrois reflect from a plane normal to a major plane aptrefl reflect from a plane, of points or vectors aptrefs reflection operator, analyze, to find axis aptrefq reflection operator, to reflect from a plane aptrefs reflection plane, vector normal to aptrefq regular arrays, distinct directions in aptdirt, aptprim regular dodecahedron, generate aptdode regular hexahedron, generate (a cube) aptcube regular icosahedron, generate apticos regular octahedron, generate aptocta regular polygon, generation and properties of aptpoly regular polygon, generation and properties of (major plane) aptpolc regular polyhedron, generate, cube aptcube regular polyhedron, generate, dodecahedron aptdode regular polyhedron, generate, icosahedron apticos regular polyhedron, generate, octahedron aptocta regular polyhedron, generate, tetrahedron apttetr regular tetrahedron, generate apttetr relative precision, round-off floating point data to aptrnds relativistic Maxwellian PDF, randomly sample values from aptmaxw remainder of division of two big integers (any base), find the aptbdiv remainders for several divisors, find integer when given aptcrts remainders for several divisors, find when given integer aptcrts remove leading and/or trailing blanks from a character string aptrmbl reorder data, any direction or mode (multiple columns) aptsorm reorder data, any direction or mode (one column) aptsort reorder data, in random order aptrand replace a bit string with random bits aptbitr replace a character string with random characters aptchrn replace characters in a character string aptchrp, aptchip replace non-graphic characters in a character string aptchip replace part of a character string with another string aptchmv replacement, randomly sample without aptrans reverse the order of an integer array aptbrev right-fill a character string with blanks aptchsq right-shift bits in a machine word, wrap-around or pull aptbits right-shift characters in a character string aptchmv root of a big integer (any base), find a aptbrtn root of a polynomial equation, factor out, reduce order aptpolf roots of a cubic equation, real and complex aptcubs roots of a polynomial equation, real, and extrema, infl pts aptpolr roots of a polynomial equation, real, using evaluation aptpole roots of a polynomial equation, real, using Newtonian iteration aptnewt roots of a quadratic equation, real or complex aptqrts, aptqrtv roots of a quartic polynomial equation, real and complex aptquar rotate (see also "operator") rotate a point or vector aptmopv, aptrot* rotate a quadric surface aptrois rotate a vector by a specified angle (major plane) aptrotc rotate points around an axis depending on axial position aptwist rotate points around an axis depending on radial position aptwirl rotation (see also "operator") rotation angle and axis of a specified rotation operator aptrotq rotation angle, operator for a specified axis and aptrota rotation angle, vary with position along rotation axis aptwist rotation angle, vary with radius from rotation axis aptwirl rotation axis and angle of a specified rotation operator aptrotq rotation axis, operator for a specified angle and aptrota rotation operator for 3 angles around 3 major axes aptrots rotation operator for a specified axis and angle aptrota rotation operator to make a plane parallel to another plane aptrotp, aptrott rotation operator to make a vector parallel to another vector aptrott, aptrotv rotation operator, analyze, to find axis, angle aptrotq round-off floating point data aptrnds, aptnint rows containing duplicate values, delete from a 2-D array aptdeld rows formed by a reqular array of points, like a vineyard aptdirt, aptprim rows in unique directions, minimum spacing in a regular array aptdirt, aptprim S saddle points, minima and maxima on a quadric surface aptqext, aptqexv sample (see "randomly sample") scalar product of three vectors aptrips scalar product of two vectors aptvdos, aptvdot scalar product of two vectors (major plane) aptvdoc scale a quadric surface aptrois scale cylindrically, points or vectors, relative to an axis aptsclr scale isotropically, points or vectors, and operator to aptsclu scale linearly, points or vectors (major plane) aptsclc scale linearly, points or vectors, relative to a plane aptscll scale uniformly, points or vectors, and operator to aptsclu scaling operator, to cylindrically scale points or vectors aptsclr scaling operator, to isotropically scale points or vectors aptsclu scaling operator, to linearly scale points or vectors aptscll scaling operator, to uniformly scale points or vectors aptsclu scattering angle, randomly sample azimuth angle for fixed aptscan, aptscaz scattering angle, randomly sample azimuth angle for fixed aptscma, aptscmz search for a bit string in a machine word aptbitf search for a character string in another character string aptchfp search for a delimited character string in another string aptchfs search for a delimited character string in another string aptquot, aptfbrk second derivative of a polynomial equation, evaluate at args aptpole second derivative of a polynomial equation, real roots of aptpolr second-order polonomial, real or complex roots of a aptqrts, aptqrtv second-order surface (see quadric surface) seconds, degrees, minutes, convert an angle to aptcang semiaxes, three, equation of an ellipsoid with aptesis separate a character string into a stem and subscript, more aptsubr separate a character string into a stem and subscripts, more aptsubm separate a character string into delimited fields aptffip, aptchap separate a character string into delimited fields aptquot, aptfbrk separate a group into smaller groups, number of ways aptcomg series, geometric, find ratio given first term, sum, length aptgrat series, integer plus reciprocal integer, for a decimal number aptiris series, reciprocal integer, for a decimal fraction aptiris set of integers, find the greatest common divisor of a aptcmad set of integers, find the least common multiple of a aptcmad shape of a quadric surface at a point, curvatures aptcris shape type of a quadrangle (bowtie or boomerang?) (major plane) aptqvac sheets, one or two (see "hyperboloid") shift bits right or left in a machine word, wrap-around or pull aptbits shift characters left or right in a character string aptchmv SI units, 1999 values of the fundamental constants in aptconm side of a quadric surface a point is on aptwhis sine and cosine of an angle (major plane) aptvanc sine of angle between two unit vectors aptvaxb, aptvxun sine of angle between two unit vectors (major plane) aptvaxc, aptvxuc sine of vertex angle of a quadrangle (major plane) aptvanc sine of vertex angle of a triangle (major plane) aptvanc single quotes, use as a field delimiter ('), with an escape aptquot single blank, reduce multiple blanks in a character string to aptblsq slash, use as a field delimiter (/) aptfbrk slash, use as a field delimiter (/) aptffip, aptchap slice through a quadric surface with a major plane aptplis solve a set of linear equations aptsolv solve a sudoku puzzle aptsudo, aptsudx sort arrays, any dimension, direction or mode aptsorm sort arrays, any dimension, direction or mode (one column) aptsort sort data, any direction or mode (multiple columns) aptsorm sort data, any direction or mode (one column) aptsort sorted or unsorted tables, eliminate duplicate rows aptdeld sorted or unsorted tables, reduce to unique rows aptdeld special characters (see "delimiter") special relationships (also see "concentric", "in", "parallel") special relationships (also see "orthogonal", "overlap") special relationships (also see "perpendicular", "tangent") special relationships (also see "test") special relationships (see "coaxial", "coincident", "colinear") spectrum, Planck or Wien, randomly sample from aptxnup sphere (see also "quadric surface") sphere and a circular cylinder, distance between, proximal pts aptspcy sphere and a plane, distance between, proximal pts aptplsp, aptplis sphere at the origin, equation of aptesis sphere tangent to a plane? aptplsp, aptplis sphere tangent to four specified tangent spheres, radius of aptspht, aptsphk sphere tangent to three specified tangent spheres, center of aptsphl, aptsphk sphere through 2 points, and on a axis, center and radius of aptspha sphere through 4 specified points, center and radius of aptsphp sphere, distance from a plane to a, intersection aptplqu sphere, distance from a plane to a, intersection aptplsp, aptplis sphere, distance from a point to a aptwhis sphere, distance from a, to a circular cylinder aptspcy sphere, intersection with a linear track aptrkis sphere, intersection with a linear track (1-D) aptrkrl sphere, intersection with a major plane (none, pt or circle?) aptplis sphere, intersection with a plane (none, point or circle?) aptplsp, aptplis sphere, intersection with another sphere (none, pt or circle?) aptspsp sphere, project points onto aptpros, aptprsp sphere, proximal point to a circular cylinder, distance aptspcy sphere, proximal point to a plane, distance aptplsp, aptplis sphere, radially project points onto aptpros, aptprsp sphere, randomly sample points in aptlocs, aptbrks sphere, randomly sample points in an annulus of aptlocs, aptbrks sphere, randomly sample points on aptlocs, aptbrks sphere, tile the surface with planar triangles aptil08 sphere, tile the surface with planar triangles aptil12, aptil20 spheres coincident? aptspsp spheres tangent to four specified tangent spheres, radii of aptspht, aptsphk spheres tangent? aptspsp spherical annulus, randomly sample points in aptlocs, aptbrks spherical coordinates, direction between two points in aptdist spherical coordinates, distance between two points in aptdist spherical coordinates, intersection of a linear track in (1-D) aptrkrl spherical coordinates, move a point in aptmove, aptmovs spherical coordinates, transform to or from aptcsys, aptcsyv spherical coordinates, volume bounded by surfaces of aptbrkn, aptbrkp spherical coordinates, volume bounded by surfaces of aptbrks, aptbrkv spherical sector, randomly sample direction vectors in a aptscar spherically project points onto a cylinder aptprcy square brackets, use as field delimiters [] aptfbrk square matrices, product of two aptmprd squeeze a character out of a character string aptchsq stakes (see "grapestakes") standard deviation of an array aptmean standard form of the equation of a quadric surface aptaxis statistics, mean value and standard deviation aptmean Steiner vertex of a triangle, distances, vertex weights aptsver stem and ASCII subscript, combine into a character string aptsuba, aptsubb stem and ASCII subscripts, combine into a character string aptsubb stem and integer subscripts, combine into a character string aptsubi store integer ASCII characters into a character string aptchia store one character string into another aptchmv straight line that cuts a triangle into equal areas, perimeters aptrabc, aptrcab straight line that cuts a triangle into equal areas, perimeters aptrcut, aptrext strange characters, find in a character string and replace aptchip striation, randomly sample with, aliased bins aptalsb striation, randomly sample with, cumulative PDF bins aptcums striation, randomly sample with, equal-probability bins apteqsb, apteqxs striation, randomly sample with, uniform PDF's aptrfpt, aptrint string (see "bit string", "character string") string, character, copy digits from, into individual words aptatob string, character, form from a stem and ASCII subscript aptsuba, aptsubb string, character, form from a stem and ASCII subscripts aptsubb string, character, form from a stem and integer subscripts aptsubi string, character, reduce multiple blanks to a single blank in aptblsq string, longer than eight characters, parse from input line aptffip, aptchap string, remove leading and/or trailing blanks from a character aptrmbl subdivide spherical pentagon into 5 spherical triangles aptpent subscript, append to a stem in a single character string aptsuba, aptsubb subscript, translate a character string into stem and aptsubr, aptsubs subscripted variable, form from a stem and an ASCII subscript aptsuba, aptsubb subscripted variable, form from a stem and ASCII subscripts aptsubb subscripted variable, form from a stem and integer subscripts aptsubi subscripted variable, translate into variable and subscript aptsubr, aptsubs subscripts, append to a stem in a single character string aptsubi subscripts, ASCII, append to a stem in a character string aptsubb subscripts, integer, append to a stem in a character string aptsubi subscripts, translate a character string into stem and multiple aptsubm substitute (see "replace") subtract big integers (arrays of digits in spec number base) aptbsub sudoku puzzle, find duplicate characters in row, column, box aptsudu sudoku puzzle, solve a aptsudo, aptsudx Sudoku puzzle, solve a aptsudo, aptsudx sum of two big integers (any base), find the aptbadd sum of vectors, with specified coefficients aptvadd, aptvsum sum of vectors, with specified coefficients (major plane) aptvadc, aptvsuc surface (see also "plane", "quadrangle", "quadric", "triangle") surface function, quadric, value at a specifed point aptqval surface of cone, randomly sample points on aptscns, aptscnv surface of cylinder, randomly sample points on aptlocy surface of disk, randomly sample points on aptlocd surface of parallelogram, randomly sample points on aptspar, aptbrks surface of sphere, randomly sample points on aptlocs surface of sphere, tile with planar triangles aptil08 surface of sphere, tile with planar triangles aptil12, aptil20 surface of triangle, randomly sample points on apttlod surface of triangle, randomly sample points on (major plane) aptqloc, apttloc surface, quadric, extreme curvatures at a point on aptcris surface, quadric, fit to edges and center of a quadrangle aptqhyp surface, quadric, fit to nine points aptqfit surface, quadric, intersection with a linear track, normal aptrkis, aptwhis surface, quadric, minimum and maximum points on aptqext, aptqexv surface, quadric, side of, distance of a point aptwhis, aptquad surface, quadric, side of, distance of a point (approx) (OBS) aptqprr, aptqprt surfaces of a brick, areas of aptbrkv surfaces of a brick, distance from a point to aptbrkn surfaces of a tetrahedron, distance from a point to aptetrn surfaces of the coordinate system, volume bounded by (brick) aptbrkn, aptbrkp surfaces of the coordinate system, volume bounded by (brick) aptbrks, aptbrkv surfaces, quadric, intersections or proximal points of aptintq surfaces, quadric, plane tangent to family of, at a point aptqnor surfaces, quadric, triple point of intersection of three aptripq surfaces, quadric, vector normal to family of, at a point aptqnor symbol (delimited character string), convert mode of aptchai symbol (delimited character string), convert mode of aptchtp symbol (delimited character string), convert mode of aptsubs, aptsubm symbol (delimited character string), find in another string aptchfs symbol (delimited character string), find in another string aptquot, aptfbrk symbols, parse a character string into aptffip, aptchap symbols, read and parse an input line into aptffip, aptchap symbols, translate into integer and floating point aptchtp symbols, translate into integer and floating point aptffip, aptchap symbols, translate into integer and floating point aptsubs, aptsubm symmetric around a specified axis, cylinder aptclis, aptcyis symmetric around major axis, cone aptcois, aptcnis symmetric around major axis, cylinder aptcyis symmetric around major axis, ellipsoid aptelis symmetric around major axis, hyperboloid of two sheets aptelis symmetric around the z axis, cone Through two specified points aptcniz symmetry axes of a quadric surface, and center, type aptaxis symmetry axes, intercepts of a quadric surface on aptqupt symmetry center of a quadric surface, and type, axes aptaxis symmetry center of a quadric surface, and type, extrema aptqext symmetry operations (see "reflection", "rotation", "inversion") symmetry operations on a quadric surface aptrois syzygy (see "intersection", "tangent") T table look-up, by binary search, ascending or descending order aptlook table, eliminate lines containing duplicate values aptdeld table, randomize order of aptrand table, reduce to lines containing unique values aptdeld tabulated PDF, generate bins to randomly sample, aliased aptalin tabulated PDF, generate bins to randomly sample, cumulative aptcump tabulated PDF, generate bins to randomly sample, equal-prob apteqin, apteqxn tabulated PDF, randomly sample bins, aliased aptalsb, aptalst tabulated PDF, randomly sample bins, cumulative aptcums, aptcumt tabulated PDF, randomly sample from aliased bins aptalsh, aptalsl tabulated PDF, randomly sample from cumulative PDF bins aptalsh, aptalsl tabulated PDF, randomly sample from equal-probability bins apteqsb, apteqxs tangent circle, to two specified tangent circles, center of aptcirl, aptcirk tangent circles (major plane)? aptcinc tangent circles, to three specified tangent circles, radii of aptcirt, aptcirk tangent circular cylinders? aptcycy tangent plane, to a family of quadric surfaces, at a point aptqnor tangent sphere, to three specified tangent spheres, center of aptsphl, aptsphk tangent spheres, to four specified tangent spheres, radii of aptspht, aptsphk tangent spheres? aptspsp tangent to a quadric surface, a cylindrical quadric surface aptqupr tangent to four specified tangent spheres, radii of spheres aptspht, aptsphk tangent to three specified tangent circles, radii of circles aptcirt, aptcirk tangent vectors, at extreme curvatures on a quadric surface aptcris tangent, line and circle (major plane)? aptlnic tangent, plane and circular cylinder? aptplcy, aptplis tangent, plane and sphere? aptplsp, aptplis tangent, sphere and circular cylinder? aptspcy tangent, two quadric surfaces? aptintq terminate after error detection, and write messages before aptnerr terms of geometric series, find ratio given first, sum, number aptgrat test for a bit string in a machine word aptbitf test for a character string in another character string aptchfp test for a delimited character string in another string aptchfs test for a delimited character string in another string aptquot, aptfbrk test for circle being tangent to a line (major plane) aptlnic test for circles being coincident (major plane) aptcinc test for circles being tangent (major plane) aptcinc test for data values between two specified limits (any mode) aptword test for line being in a plane aptlnpl, aptlntr test for line being tangent to a circle (major plane) aptlnic test for lines being coincident or parallel aptlnln test for lines being coincident or parallel (major plane) aptlnlc test for parabolic trajectory tangent to a line aptparl test for parabolic trajectory tangent to a plane aptparp, aptparq test for parabolic trajectory tangent to a quadric surface aptparq test for plane being tangent to a sphere aptplsp, aptplis test for planes being coincident or parallel aptplpl, aptplis test for point being inside a brick, distances from surfaces aptbrkn test for point being inside a quadrangle (major plane) aptqinc test for point being inside a tetrahedron aptetrn test for point being inside a triangle (major plane) apttinc test for point being on a line aptptln test for point being on a line (major plane) aptptlc test for points being coincident aptdist, aptvdis test for points being coincident (major axis) aptvdil test for points being coincident (major plane) aptvdic test for points being colinear aptpfit test for points being colinear apttrid, aptvpln test for points being colinear (major plane) apttric test for quadrangle convex, boomerang or bowtie (major plane) aptqvac test for sphere being tangent to a plane aptplsp, aptplis test for spheres being coincident aptspsp test for spheres being tangent aptspsp test for uniformly accelerated trajectory tangent to a line aptparl test for uniformly accelerated trajectory tangent to a plane aptparp, aptparq test for uniformly accelerated trajectory tangent to a quadric aptparq test for vectors being parallel aptvaxb, aptvxun test for vectors being parallel (major plane) aptvaxc, aptvxuc test for vectors being perpendicular (major plane) aptvdoc test for vectors being perpendicular or orthogonal aptvdos, aptvdot tetrahedron centroid and volume apttetd tetrahedron edge at specified angles from two other edges aptetrv tetrahedron volume and centroid apttetd tetrahedron, distances of a point from faces of aptetrn tetrahedron, distances of faces from a point aptetrn tetrahedron, generate a regular apttetr tetrahedron, local coordinates of a point in aptetrl tetrahedron, point in, local coordinates of aptetrl tetrahedron, point inside or outside? aptetrn tetrahedron, point with given local coordinates aptetrw tetrahedron, randomly sample points in aptetrp third-order polonomial, real and complex roots of aptcubs three points in major plane, circle through aptcirc three points, circle through, center and orientation of aptcirp three points, equation of plane through aptpfit three points, plane through, vector normal to aptvpln, aptpfit three points, vector normal to plane through aptvpln, aptpfit three quadric surfaces, triple point of intersection of aptripq tile the surface of a sphere with planar triangles aptil08 tile the surface of a sphere with planar triangles aptil12, aptil20 time-dependent accelerated trajectory, position and velocity on aptacct times and positions on a parabolic trajectory aptparb, aptpart times and positions on a time-dependent accelerated trajectory aptacct totient function for arithmetic modulo an integer, find Euler's aptpfac track (see also "line", "direction vector") track, linear, intersection with a cylinder (1-D) aptrkcl track, linear, intersection with a line (major plane) aptrklc track, linear, intersection with a major plane aptrksl track, linear, intersection with a plane aptrkpl track, linear, intersection with a quadric surface, normal aptrkis track, linear, intersection with a sphere (1-D) aptrkrl track, linear, intersection with an axisymmetric surface aptrkcy, aptrkis track, parabolic, intersection with a line aptparl track, parabolic, intersection with a plane aptparp, aptparq track, parabolic, intersection with a point aptparh, aptpars track, parabolic, intersection with a quadric surface aptparq track, parabolic, proximal point to a line aptparl track, parabolic, proximal point to a plane aptparp track, parabolic, proximal point to a point aptpars track, parabolic, proximal point to a quadric surface aptparx trailing blanks, remove leading and/or from a character string aptrmbl trajectory of parabolic motion vs time aptparb, aptpart trajectory of time-dependent accelerated motion vs time aptacct trajectory, parabolic, intersection with a line aptparl trajectory, parabolic, intersection with a plane aptparp, aptparq trajectory, parabolic, intersection with a point aptparh, aptpars trajectory, parabolic, intersection with a quadric surface aptparq trajectory, parabolic, points proximal to a quadric surface aptparq trajectory, parabolic, position and velocity on, vs time aptparb trajectory, parabolic, proximal point to a line aptparl trajectory, parabolic, proximal point to a plane aptparp trajectory, parabolic, proximal point to a point aptpars transform coordinate systems aptcsys, aptcsyv translate a character string into a stem and subscript aptsubr, aptsubs translate a character string into a stem and subscripts, more aptsubm translate a character string into floating point mode aptchtp translate a character string into floating point mode aptffip, aptchap translate a character string into floating point mode aptsubs, aptsubm translate a character string into integer mode aptchtp translate a character string into integer mode aptffip, aptchap translate a character string into integer mode aptsubs, aptsubm translate a character string into integer mode (oct, dec, hex) aptchai translate a decimal number into a hexadecimal string apthexa translate a hexadecimal string into a decimal number apthexd translate a point any distance in any direction, any coords aptmove translate a point any distance in any direction, cylindrical aptmove, aptmvcy translate a point any distance in any direction, spherical aptmove, aptmovs translate a point through space apttran translate a point through space (major plane) apttrac translate a quadric surface through space apttris translate an integer into ASCII characters (oct, dec, hex) aptchia transverse axis of a hyperbola or hyperbolic cylinder aptplqu triangle (see also "plane", "quadrangle", "surface") triangle altitude vectors, lengths, intersections apttrig triangle area (major plane) apttric triangle area and normal vector aptvpln, apttrig triangle area, centroid and edge lengths apttrid, apttrig triangle center of circumscribed circle, perp bisectors apttrig triangle center of inscribed circle, angle bisectors apttrig triangle centroid (major plane) apttric triangle centroid, area and edge lengths apttrid, apttrig triangle edge lengths (major plane) apttric triangle edge lengths, area and centroid apttrid, apttrig triangle edges, distances from a point to (major plane) aptqdic, aptqinc triangle edges, distances from a point to (major plane) apttinc triangle medians, intersections, centroid of triangle apttrig triangle othocenter (intersection of altitudes) apttrig triangle vertex angle, cosine of aptvang triangle vertex angle, sine and cosine of (major plane) aptvanc triangle, bisector of a vertex angle aptbang, apttrig triangle, bisector of a vertex angle (major plane) aptbanc triangle, cosine and sine of a vertex angle (major plane) aptvanc triangle, cosine of a vertex angle aptvang triangle, distance of point from plane of aptript, apttrip triangle, distances from a point to edges of aptripl, apttrip triangle, distances from a point to edges of (major plane) aptqdic, aptqinc triangle, distances from a point to edges of (major plane) apttinc triangle, equilateral, projection onto specified triangle aptripp, aptripr triangle, extend two edges of to double area aptrext triangle, intersection with a linear track aptrkpl triangle, local coordinates of a point in aptripl, apttrip triangle, local coordinates of a point in (major plane) aptripc triangle, normal vector and area aptvpln, apttrig triangle, perpendicular bisector of an edge of aptpbln, apttrig triangle, perpendicular bisector of an edge of (major plane) aptpblc triangle, planar point nearest nonplanar point apttrip triangle, plane bisecting an edge of aptpbln triangle, point in, local coordinates of aptripl, apttrip triangle, point in, local coordinates of (major plane) aptripc triangle, point inside or outside (major plane)? apttinc triangle, point inside or outside? apttrip triangle, point with given local coordinates (major plane) aptrloc triangle, point with given local coordinates (vertex weights) aptrlop triangle, project an equilateral triangle onto a specified aptripp, aptripr triangle, randomly sample points in apttlod triangle, randomly sample points in (major plane) aptqloc, apttloc triangle, real, find if three edge lengths can form aptrich triangle, sine and cosine of a vertex angle (major plane) aptvanc triangle, spherical, 5 subdivided from spherical pentagon aptpent triangle, Steiner vertex of a, distances, vertex weights aptsver triangle, straight line that cuts into equal areas, perimeters aptrabc, aptrcab triangle, straight line that cuts into equal areas, perimeters aptrcut, aptrext triangle, vector bisecting a vertex angle aptbang, apttrig triangle, vector bisecting a vertex angle (major plane) aptbanc triangle, vector normal to plane of aptvpln, aptpfit triangles, planar, tile the surface of a sphere with aptil08 triangles, planar, tile the surface of a sphere with aptil12, aptil20 triangles, tile the surface of a sphere with aptil08 triangles, tile the surface of a sphere with aptil12, aptil20 triangular face, distance from a point to a aptript, apttrip triangular face, point nearest external point apttrip triangular face, point on which side? aptript, apttrip triangular faces of tetrahedron, distances from a point to the aptetrn triple point of intersection of three quadric surfaces aptripq triple product of three vectors, scalar aptrips triple product of three vectors, vector aptripv twist points around an axis depending on axial position aptwist twist points around an axis depending on radial position aptwirl two lines coincident or parallel (major plane) aptlnlc two lines coincident or parallel? aptlnln two lines, distance between aptlnln two lines, distance between (major plane) aptlnlc two lines, intersection between (major plane) aptlnlc two planes coincident or parallel? aptplpl, aptplis two planes, rotation operator to make parallel aptrotp, aptrott two points coincident (major plane)? aptvdic two points coincident? aptdist, aptvdis two points, cone on a major axis through aptcnis two points, cone on the z axis through aptcniz two points, distance and vector between aptdist, aptvdis two points, distance and vector between (major plane) aptvdic two points, sphere through, on a specified axis aptspha two spheres, intersection between (none, point or circle?) aptspsp two vectors parallel (major plane)? aptvaxc, aptvxuc two vectors parallel? aptvaxb, aptvxun two vectors perpendicular (major plane)? aptvdoc two vectors perpendicular? aptvdos, aptvdot two vectors, rotation operator to make parallel aptrott, aptrotv two-dimensional subroutines (see "major plane") two-point interpolation or extrapolation, any linear/log axes aptintp type character (oct, dec, hex), convert integer into aptchia type character, convert to integer (oct, dec, hex) aptchai type character, convert to integer and floating point aptchtp type character, convert to integer and floating point aptffip, aptchap type character, convert to integer and floating point aptsubs, aptsubm type convert delimited character strings aptchai type convert delimited character strings aptchtp type convert delimited character strings aptsubs, aptsubm type convert delimited character strings, read and aptffip, aptchap type convert integers into ASCII characters (oct, dec, hex) aptchia type integer, convert delimited character strings to aptchai type integer, convert delimited character strings to aptchtp type integer, convert delimited character strings to aptffip, aptchap type integer, convert delimited character strings to aptsubs, aptsubm type integer, convert into ASCII characters (oct, dec, hex) aptchia type of quadric surface, symmetry center and axes aptaxis type of quadric surface, symmetry center and extrema aptqext type real, convert delimited character strings to aptchtp type real, convert delimited character strings to aptffip, aptchap type real, convert delimited character strings to aptsubs, aptsubm U u-v plane (see "major plane") un, Boolean operation "union" on two machine words aptbitb unary Boolean operation "comp", do on a machine word aptbitb unexponentiated factors of an integer, find aptafac uniform acceleration, parabolic trajectory produced by aptparb uniform PDF of a direction vector in space (scalar) aptsciz uniform PDF of direction vectors in space aptscap, aptscat uniform PDF of direction vectors in space aptscav uniform PDF of direction vectors in space, in spherical segment aptscar uniform PDFs, randomly sample in specified bins aptalsh uniformly accelerated motion, parabolic trajectory of aptparb, aptparh uniformly accelerated motion, parabolic trajectory of aptparl, aptparp uniformly accelerated motion, parabolic trajectory of aptparq, aptpars uniformly accelerated motion, parabolic trajectory of aptparx uniformly accelerated trajectory, given points on vs time aptpart uniformly accelerated trajectory, intersection with line aptparl uniformly accelerated trajectory, intersection with plane aptparp, aptparq uniformly accelerated trajectory, intersection with point aptparh, aptpars uniformly accelerated trajectory, intersection with quadric aptparq uniformly accelerated trajectory, path length vs time aptparb uniformly accelerated trajectory, points on vs time aptparb uniformly accelerated trajectory, proximal point to line aptparl uniformly accelerated trajectory, proximal point to plane aptparp uniformly accelerated trajectory, proximal point to point aptpars uniformly accelerated trajectory, proximal point to surface aptparx uniformly accelerated trajectory, velocities on vs time aptparb uniformly randomly sample a direction vector (scalar) aptsciz, aptscmz uniformly randomly sample direction vectors aptscan, aptscap uniformly randomly sample direction vectors aptscat, aptscav uniformly randomly sample direction vectors aptscma uniformly randomly sample direction vectors in a plane aptscad uniformly randomly sample in specified bins aptalsh uniformly randomly sample integers from a sequence aptrins, aptrint uniformly randomly sample points in a circle aptlocd, aptbrks uniformly randomly sample points in a circular annulus aptlocd, aptbrks uniformly randomly sample points in a cone aptscnv, aptbrks uniformly randomly sample points in a conical annulus aptscnv, aptbrks uniformly randomly sample points in a conical frustrum aptscnv, aptbrks uniformly randomly sample points in a cylinder aptlocy, aptbrks uniformly randomly sample points in a cylindrical annulus aptlocy, aptbrks uniformly randomly sample points in a parallelogram aptspar, aptbrks uniformly randomly sample points in a parallelopiped aptspar, aptbrks uniformly randomly sample points in a quadrangle (major plane) aptqloc uniformly randomly sample points in a sphere aptlocs, aptbrks uniformly randomly sample points in a spherical annulus aptlocs, aptbrks uniformly randomly sample points in a tetrahedron aptetrp uniformly randomly sample points in a triangle apttlod uniformly randomly sample points in a triangle (major plane) apttloc uniformly randomly sample points on a circle aptlocd, aptbrks uniformly randomly sample points on a cone aptscns, aptscnv uniformly randomly sample points on a cylinder aptlocy, aptbrks uniformly randomly sample points on a sphere aptlocs, aptbrks uniformly randomly sample values from an interval aptrfps, aptrfpt uniformly randomly sample without replacement aptrans uniformly scale points or vectors, and operator to aptsclu union of two machine words (Boolean operation .or. or .un.) aptbitb unique directions between pairs of points in one or two groups aptdirt, aptprim unique values or lines, reduce an array or table to aptdeld unit direction vector (see "direction vector") unit vector perpendicular to two other vectors aptvxun unit vector perpendicular to two other vectors (major plane) aptvxuc unit vector, normalize aptvuna, aptvunb unit vector, normalize (major plane) aptvuac, aptvubc unit vector, normalize (scalar) aptvunz, aptvusz unit vectors, cosine of angle between aptvdos, aptvdot unit vectors, cosine of angle between (major plane) aptvdoc unit vectors, sine of angle between two aptvaxb, aptvxun unit vectors, sine of angle between two (major plane) aptvaxc, aptvxuc units cm-g-sh-keV, 1999 values of the fundamental constants in aptconl units, SI, 1999 values of the fundamental constants in aptconm unpack bits from machine words into a type integer*1 array aptbite unsorted or sorted tables, eliminate duplicate rows aptdeld unsorted or sorted tables, reduce to unique rows aptdeld upper case characters, randomly sample from aptchrn V v-w plane (see "major plane") value of a big integer in a specified number base, find the aptbasb value of a decimal number in a specified number base, find the aptbase value of continued fraction, find, given coefficients aptcofv values (positive) and indices of the min and max of an array aptmnlg values and indices of the minimum and maximum of an array aptmnlg, aptmnmx values between two specified limits (any mode)? aptword values of the fundamental constants, 1999 aptconl, aptconm values, limit range of aptfdad, aptfdav values, limit range of, vector components aptvtol values, limit range of, vector components (major plane) aptvtoc values, lower limits on vector components aptvlim values, lower limits on vector components (major plane) aptvlic values, randomly sample (see also "bins") values, randomly sample, from a linear PDF aptslid, aptsliv values, randomly sample, from a list of bins aptalsh, aptalsl values, randomly sample, from a Normal PDF aptnorm values, randomly sample, from a Planck or Wien spectrum aptxnup values, randomly sample, from a power-law PDF aptspod values, randomly sample, from a range of values (floating-pt) aptrfps, aptrfpt values, randomly sample, from a range of values (integer) aptrins, aptrint values, randomly sample, from a relativistic Maxwellian PDF aptmaxw values, randomly sample, from an equal-probability PDF apteqxs values, randomly sample, from an exponential PDF aptexpl, aptexps variable, subscripted, form from a stem and ASCII subscript aptsuba, aptsubb variable, subscripted, form from a stem and ASCII subscripts aptsubb variable, subscripted, form from a stem and integer subscripts aptsubi variable, subscripted, translate into variable and subscript aptsubr, aptsubs variable, subscripted, translate into variable and subscripts aptsubm vary angle of rotation with position along rotation axis aptwist vary angle of rotation with radius from rotation axis aptwirl vector (see also "direction vector", "line", "track") vector and plane, rotate onto another vector and plane aptrott vector at specified angles from two other vectors aptetru, aptetrv vector between two points aptdist, aptvdis vector between two points (major plane) aptvdic vector bisecting a vertex angle of a quadrangle aptbang vector bisecting a vertex angle of a quadrangle (major plane) aptbanc vector bisecting a vertex angle of a triangle aptbang, apttrig vector bisecting a vertex angle of a triangle (major plane) aptbanc vector components in a plane aptvplp vector components parallel and perpendicular to another vector aptvpap vector components parallel to another vector (major plane) aptvcvc vector components, limit range of values of aptvtol vector components, limit range of values of (major plane) aptvtoc vector components, lower limits on values of aptvlim vector components, lower limits on values of (major plane) aptvlic vector cross product aptvaxb, aptvxun vector cross product (major plane) aptvaxc, aptvxuc vector differences aptvdis vector differences (major plane) aptvdic vector direction and distance between two points aptdist, aptvdis vector direction and distance between two points (major plane) aptvdic vector dot product aptvdos, aptvdot vector dot product (major plane) aptvdoc vector length aptvunb, aptvusz vector magnitude aptvunb, aptvusz vector normal to a family of quadric surfaces, at a point aptqnor vector parallel to a line between two points aptvdis vector perpendicular to a line (major plane) aptvplc vector perpendicular to a plane aptvpln, aptpfit vector perpendicular to a triangle edge, through a vertex apttrig vector product of three vectors aptripv vector product of two vectors aptvaxb, aptvxun vector product of two vectors (major plane) aptvaxc, aptvxuc vector scalar product aptvdos, aptvdot vector scalar product (major plane) aptvdoc vector sum of two vectors aptvadd, aptvsum vector sum of two vectors (major plane) aptvadc, aptvsuc vector triple product aptripv vector, cylindrically scale, relative to a specified axis aptsclr vector, direction (see "direction vector") vector, invert (major plane) aptinvc vector, invert, and operator to aptinvp, aptsclu vector, isotropically scale, and operator to aptsclu vector, linearly scale (major plane) aptsclc vector, linearly scale, relative to a specified plane aptscll vector, matrix operation on aptmopv vector, normal, of a quadric surface function at a point aptwhis vector, normalize aptvuna, aptvunb vector, normalize (major plane) aptvuac, aptvubc vector, normalize (scalar) aptvunz, aptvusz vector, reflect (major plane) aptrefc vector, reflect a aptrefl, aptrefs vector, reflect a, analyze the operator to aptrefq vector, rotate a aptmopv, aptrot* vector, rotate by a specified angle (major plane) aptrotc vector, rotation operator to make parallel to another vector aptrott, aptrotv vector, uniformly scale, and operator to aptsclu vectors between pairs of points in a group, unique unit aptdirt vectors of edges of a triangle apttrig vectors parallel (major plane)? aptvaxc, aptvxuc vectors parallel? aptvaxb, aptvxun vectors perpendicular (major plane)? aptvdoc vectors perpendicular? aptvdos, aptvdot vectors, cross product of two aptvaxb, aptvxun vectors, cross product of two (major plane) aptvaxc, aptvxuc vectors, cylindrically scale, relative to a specified axis aptsclr vectors, dot product of two aptvdos, aptvdot vectors, linearly combine aptvadd, aptvsum vectors, linearly combine (major plane) aptvadc, aptvsuc vectors, linearly scale (major plane) aptsclc vectors, linearly scale, relative to a specified plane aptscll vectors, randomly sample at a fixed scatter angle aptscan, aptscaz vectors, randomly sample at a fixed scatter angle aptscma, aptscmz vectors, randomly sample in a specified plane aptscad vectors, randomly sample, cosine-power PDF aptscap, aptscar vectors, randomly sample, cosine-power PDF aptscav vectors, randomly sample, isotropic aptscap, aptscat vectors, randomly sample, isotropic aptscav vectors, randomly sample, isotropic, limits aptscar vectors, randomly sample, uniform aptscap, aptscat vectors, randomly sample, uniform aptscav vectors, randomly sample, uniform, limits aptscar vectors, rotate (see "operator", "rotate", "rotation") vectors, scalar product of three aptrips vectors, transform the coordinate system of aptcsyv vectors, vector product of three aptripv velocity on a parabolic trajectory vs time aptparb, aptpart velocity on a parabolic trajectory, initial aptparh, aptpart velocity on a parabolic trajectory, initial, to intersect point aptparh velocity on a time-dependent accelerated trajectory vs time aptacct velocity on a uniformly accelerated trajectory vs time aptparb velocity, randomly sample from a relativistic Maxwellian PDF aptmaxw vertex angle and coordinate, equation of cone with given aptcois vertex angle of a quadrangle, bisecting vector aptbang vertex angle of a quadrangle, bisecting vector (major plane) aptbanc vertex angle of a quadrangle, cosine of aptvang vertex angle of a quadrangle, sine and cosine of (major plane) aptvanc vertex angle of a triangle, bisecting vector aptbang, apttrig vertex angle of a triangle, bisecting vector (major plane) aptbanc vertex angle of a triangle, cosine of aptvang vertex angle of a triangle, sine and cosine of (major plane) aptvanc vertex angles of a triangle apttrig vertex areas and shape type of a quadrangle (major plane) aptqvac vertex coordinate and angle, equation of cone with given aptcois vertex of a tetrahedron, given two edges, angles of third edge aptetrv vertex weights (see "local coordinates") vertex weights of Steiner vertex of a triangle aptsver vertices of a cube aptcube vertices of a regular dodecahedron aptdode vertices of a regular icosahedron apticos vertices of a regular octahedron aptocta vertices of a regular polygon aptpoly vertices of a regular polygon (major plane) aptpolc vertices of a regular tetrahedron apttetr vertices, edges of a quadrangle, quadric surface fit to aptqhyp vertices, project equilateral triangle onto triangle with given aptripp view factor projection of points onto a plane aptradv view plane, perspective, project points onto aptpers view point, perspective projection with a specified aptpers view point, perspective projection with specified (major plane) aptperc volume bounded by coordinate surfaces, randomly sample pts in aptbrks volume of a brick (bounded by coordinate surfaces) aptbrkv volume of a conical annulus, randomly sample points in aptscnv volume of a conical frustrum, randomly sample points in aptscnv volume of a cylindrical annulus, randomly sample points in aptlocy volume of a parallelopiped, given three edge vectors aptrips volume of a parallelopiped, randomly sample points in aptspar, aptbrks volume of a spherical annulus, randomly sample points in aptlocs volume of a tetrahedron apttetd volume of a tetrahedron, randomly sample points in aptetrp W w-x plane (see "major plane") wall, Steiner, of a triangle aptsver weighted sum of two vectors aptvadd, aptvsum weighted sum of two vectors (major plane) aptvadc, aptvsuc weights (see "local coordinates") Wien or Planck spectrum, randomly sample values from aptxnup word, machine, contract bits from a type integer*1 array into aptbitp word, machine, expand bits from into a type integer*1 array aptbite word, machine, find bit string in a aptbitf word, machine, find character string in a aptchfp word, machine, find complement of a aptbitb word, machine, find delimited character string in aptchfs word, machine, find delimited character string in aptquot, aptfbrk word, machine, move a bit string from one to another aptbitm word, machine, put random bits into a bit string in a aptbitr word, machine, shift right or left, with wrap-around or pull aptbits words, machine, find Boolean "nor" and "not" of aptbitb words, machine, find equivalence, complement of aptbitb words, machine, find exclusive or, intersection, union of aptbitb wrap-around shift right or left of a machine word aptbits write error message to specified IOCs after error detection aptnerr X x-ray diffraction pattern, crystal directions that determine it aptdirt, aptprim x-y plane (see "major plane") xnor, Boolean operation "equivalence" on two machine words aptbitb xor, Boolean operation "exclusive or" on two machine words aptbitb xyz coordinates (see "rectangular", "cartesian") Y y-z plane (see "major plane") Z z axis, equation of a cone on, through two specified points aptcniz z-r plane (see "major plane") z-r plane, equation of a cone through two points in aptcnis z-r plane, intersection of a 3-D linear track with a line in aptrkcy z-r plane, move a point in, with a specified direction aptmove, aptmvcy z-x plane (see "major plane") =============================================================================== UCRL-WEB-209832