GEOMETRY
Geometric Calculations

GEOMETRY is a C++ library which performs certain geometric calculations in 2, 3 and N space.

These calculations include angles, areas, containment, distances, intersections, lengths, and volumes.

Some geometric objects can be described in a variety of ways. For instance, a line has implicit, explicit and parametric representations. The names of routines often will specify the representation used, and there are routines to convert from one representation to another.

Another useful task is the delineation of a standard geometric object. For instance, there is a routine that will return the location of the vertices of an octahedron, and others to produce a series of "equally spaced" points on a circle, ellipse, sphere, or within the interior of a triangle.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

GEOMETRY is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Programs:

GEOMPACK, a C++ library which computes the Delaunay triangulation and Voronoi diagram of 2D data.

POLYGON_MOMENTS, a C++ library which computes arbitrary moments of a polygon.

TABLE_DELAUNAY, a C++ program which reads a file of 2d point coordinates and computes the Delaunay triangulation.

TET_MESH, a C++ library which defines and analyzes tetrahedral meshes.

TETRAHEDRON_PROPERTIES, a C++ program which computes properties of a tetrahedron whose vertex coordinates are read from a file.

TETRAHEDRONS, a dataset directory which contains examples of tetrahedrons;

TRIANGLES, a dataset directory which contains examples of triangles;

TRIANGULATE, a C program which triangulates a (possibly nonconvex) polygon.

TRIANGULATION, a C++ library which defines and analyzes triangulations.

TRIANGULATION_DISPLAY_OPENGL, a C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_TRIANGLE_NEIGHBORS, a C++ program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.

Reference:

Gerard Bashein, Paul Detmer,
Centroid of a Polygon,
in Graphics Gems IV,
edited by Paul Heckbert,
AP Professional, 1994,
ISBN: 0123361559,
LC: T385.G6974.
SF Bockman,
Generalizing the Formula for Areas of Polygons to Moments,
American Mathematical Society Monthly,
Volume 96, Number 2, February 1989, pages 131-132.
Adrian Bowyer, John Woodwark,
A Programmer's Geometry,
Butterworths, 1983,
ISBN: 0408012420.
Paulo Cezar Pinto Carvalho, Paulo Roma Cavalcanti,
Point in Polyhedron Testing Using Spherical Polygons,
in Graphics Gems V,
edited by Alan Paeth,
Academic Press, 1995,
ISBN: 0125434553,
LC: T385.G6975.
Daniel Cohen,
Voxel Traversal along a 3D Line,
in Graphics Gems IV,
edited by Paul Heckbert,
AP Professional, 1994,
ISBN: 0123361559,
LC: T385.G6974.
Thomas Cormen, Charles Leiserson, Ronald Rivest,
Introduction to Algorithms,
MIT Press, 2001,
ISBN: 0262032937,
LC: QA76.C662.
Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
Computational Geometry,
Springer, 2000,
ISBN: 3-540-65620-0,
LC: QA448.D38.C65.
Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
LINPACK User's Guide,
SIAM, 1979,
ISBN13: 978-0-898711-72-1,
LC: QA214.L56.
James Foley, Andries vanDam, Steven Feiner, John Hughes,
Computer Graphics, Principles and Practice,
Second Edition,
Addison Wesley, 1995,
ISBN: 0201848406,
LC: T385.C5735.
Martin Gardner,
The Mathematical Carnival,
Knopf, 1975,
ISBN: 0394494067,
LC: QA95.G286.
Priamos Georgiades,
Signed Distance From Point To Plane,
in Graphics Gems III,
edited by David Kirk,
Academic Press, 1992,
ISBN: 0124096735,
LC: T385.G6973.
Branko Gruenbaum, Geoffrey Shephard,
Pick's Theorem,
The American Mathematical Monthly,
Volume 100, Number 2, February 1993, pages 150-161.
John Harris, Horst Stocker,
Handbook of Mathematics and Computational Science,
Springer, 1998,
ISBN: 0-387-94746-9,
LC: QA40.S76.
Barry Joe,
GEOMPACK - a software package for the generation of meshes using geometric algorithms,
Advances in Engineering Software,
Volume 13, 1991, pages 325-331.
Anwei Liu, Barry Joe,
Quality Local Refinement of Tetrahedral Meshes Based on 8-Subtetrahedron Subdivision,
Mathematics of Computation,
Volume 65, Number 215, July 1996, pages 1183-1200.
Jack Kuipers,
Quaternions and Rotation Sequences,
Princeton, 1998,
ISBN: 0691102988,
LC: QA196.K85.
Robert Miller,
Computing the Area of a Spherical Polygon,
in Graphics Gems IV,
edited by Paul Heckbert,
Academic Press, 1994,
ISBN: 0123361559,
LC: T385.G6974.
Albert Nijenhuis, Herbert Wilf,
Combinatorial Algorithms for Computers and Calculators,
Second Edition,
Academic Press, 1978,
ISBN: 0-12-519260-6,
LC: QA164.N54.
Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,
Spatial Tesselations: Concepts and Applications of Voronoi Diagrams,
Second Edition,
Wiley, 2000,
, ISBN: 0-471-98635-6,
LC: QA278.2.O36.
Joseph ORourke,
Computational Geometry,
Second Edition,
Cambridge, 1998,
ISBN: 0521649765,
LC: QA448.D38.
Edward Saff, Arno Kuijlaars,
Distributing Many Points on a Sphere,
The Mathematical Intelligencer,
Volume 19, Number 1, 1997, pages 5-11.
Philip Schneider, David Eberly,
Geometric Tools for Computer Graphics,
Elsevier, 2002,
ISBN: 1558605940,
LC: T385.S334.
Peter Schorn, Frederick Fisher,
Testing the Convexity of a Polygon,
in Graphics Gems IV,
edited by Paul Heckbert,
AP Professional, 1994,
ISBN: 0123361559,
LC: T385.G6974.
Moshe Shimrat,
Algorithm 112: Position of Point Relative to Polygon,
Communications of the ACM,
Volume 5, Number 8, August 1962, page 434.
Kenneth Stephenson,
Introduction to Circle Packing, The Theory of Discrete Analytic Functions,
Cambridge, 2005,
ISBN: 0521823560,
LC: QA640.7S74.
Allen VanGelder,
Efficient Computation of Polygon Area and Polyhedron Volume,
in Graphics Gems V,
edited by Alan Paeth,
AP Professional, 1995,
ISBN: 0125434553,
LC: T385.G6975.
Daniel Zwillinger, Steven Kokoska,
Standard Probability and Statistical Tables,
CRC Press, 2000,
ISBN: 1-58488-059-7,
LC: QA273.3.Z95.

Source Code:

geometry.cpp, the source code.
geometry.hpp, the include file.

Examples and Tests:

geometry_test.cpp, a sample calling program.
geometry_test.txt, the output file.

List of Routines:

ANGLE_BOX_2D "boxes" an angle defined by three points in 2D.
ANGLE_CONTAINS_RAY_2D determines if an angle contains a ray, in 2D.
ANGLE_DEG_2D returns the angle in degrees swept out between two rays in 2D.
ANGLE_HALF_2D finds half an angle in 2D.
ANGLE_RAD_2D returns the angle in radians swept out between two rays in 2D.
ANGLE_RAD_3D returns the angle between two vectors in 3D.
ANGLE_RAD_ND returns the angle between two vectors in ND.
ANGLE_TURN_2D computes a turning angle in 2D.
ANGLEI_DEG_2D returns the interior angle in degrees between two rays in 2D.
ANGLEI_RAD_2D returns the interior angle in radians between two rays in 2D.
ANNULUS_AREA_2D computes the area of a circular annulus in 2D.
ANNULUS_SECTOR_AREA_2D computes the area of an annular sector in 2D.
ANNULUS_SECTOR_CENTROID_2D computes the centroid of an annular sector in 2D.
r8_atan computes the inverse tangent of the ratio Y / X.
BALL_UNIT_SAMPLE_2D picks a random point in the unit ball in 2D.
BALL_UNIT_SAMPLE_3D picks a random point in the unit ball in 3D.
BALL_UNIT_SAMPLE_ND picks a random point in the unit ball in ND.
BASIS_MAP_3D computes the matrix which maps one basis to another.
BOX_01_CONTAINS_POINT_2D reports if a point is contained in the unit box in 2D.
BOX_01_CONTAINS_POINT_ND reports if a point is contained in the unit box in ND.
BOX_CONTAINS_POINT_2D reports if a point is contained in a box in 2D.
BOX_CONTAINS_POINT_ND reports if a point is contained in a box in ND.
BOX_RAY_INT_2D: intersection ( box, ray ) in 2D.
BOX_SEGMENT_CLIP_2D uses a box to clip a line segment in 2D.
CIRCLE_ARC_POINT_NEAR_2D : nearest point on a circular arc.
CIRCLE_AREA_2D computes the area of a circle in 2D.
CIRCLE_DIA2IMP_2D converts a diameter to an implicit circle in 2D.
CIRCLE_EXP_CONTAINS_POINT_2D determines if an explicit circle contains a point in 2D.
CIRCLE_EXP2IMP_2D converts a circle from explicit to implicit form in 2D.
CIRCLE_IMP_CONTAINS_POINT_2D determines if an implicit circle contains a point in 2D.
CIRCLE_IMP_LINE_PAR_INT_2D: ( implicit circle, parametric line ) intersection in 2D.
CIRCLE_IMP_POINT_DIST_2D: distance ( implicit circle, point ) in 2D.
CIRCLE_IMP_POINT_DIST_SIGNED_2D: signed distance ( implicit circle, point ) in 2D.
CIRCLE_IMP_POINT_NEAR_2D: nearest ( implicit circle, point ) in 2D.
CIRCLE_IMP_POINTS_2D returns N equally spaced points on an implicit circle in 2D.
CIRCLE_IMP_POINTS_3D returns points on an implicit circle in 3D.
CIRCLE_IMP_POINTS_ARC_2D returns N points on an arc of an implicit circle in 2D.
CIRCLE_IMP_PRINT_2D prints an implicit circle in 2D.
CIRCLE_IMP_PRINT_2D prints an implicit circle in 3D.
CIRCLE_IMP2EXP_2D converts a circle from implicit to explicit form in 2D.
CIRCLE_LLR2IMP_2D converts a circle from LLR to implicit form in 2D.
CIRCLE_LUNE_AREA_2D returns the area of a circular lune in 2D.
CIRCLE_LUNE_CENTROID_2D returns the centroid of a circular lune in 2D.
CIRCLE_PPR2IMP_3D converts a circle from PPR to implicit form in 3D.
CIRCLE_PPR2IMP_2D converts a circle from PPR to implicit form in 2D.
CIRCLE_SECTOR_AREA_2D computes the area of a circular sector in 2D.
CIRCLE_SECTOR_CENTROID_2D returns the centroid of a circular sector in 2D.
CIRCLE_SECTOR_CONTAINS_POINT_2D : is a point inside a circular sector?
CIRCLE_SECTOR_PRINT_2D prints a circular sector in 2D.
CIRCLE_TRIANGLE_AREA_2D returns the area of a circle triangle in 2D.
CIRCLE_TRIPLE_ANGLE_2D returns an angle formed by three circles in 2D.
CIRCLES_IMP_INT_2D: finds the intersection of two implicit circles in 2D.
CONE_AREA_3D computes the surface area of a right circular cone in 3D.
CONE_CENTROID_3D returns the centroid of a cone in 3D.
CONE_VOLUME_3D computes the volume of a right circular cone in 3D.
CONV3D converts 3D data to a 2D projection.
COS_DEG returns the cosine of an angle given in degrees.
COT_DEG returns the cotangent of an angle given in degrees.
COT_RAD returns the cotangent of an angle.
CSC_DEG returns the cosecant of an angle given in degrees.
CUBE_SHAPE_3D describes a cube in 3D.
CUBE_SIZE_3D gives "sizes" for a cube in 3D.
CYLINDER_POINT_DIST_3D determines the distance from a cylinder to a point in 3D.
CYLINDER_POINT_DIST_SIGNED_3D: signed distance from cylinder to point in 3D.
CYLINDER_POINT_INSIDE_3D determines if a cylinder contains a point in 3D.
CYLINDER_POINT_NEAR_3D: nearest point on a cylinder to a point in 3D.
CYLINDER_SAMPLE_3D samples a cylinder in 3D.
CYLINDER_VOLUME_3D determines the volume of a cylinder in 3D.
DEGREES_TO_RADIANS converts an angle from degrees to radians.
DGE_DET computes the determinant of a matrix factored by SGE_FA.
DGE_FA factors a general matrix.
DGE_SL solves a system factored by SGE_FA.
DIRECTION_PERT_3D randomly perturbs a direction vector in 3D.
DIRECTION_UNIFORM_2D picks a random direction vector in 2D.
DIRECTION_UNIFORM_3D picks a random direction vector in 3D.
DIRECTION_UNIFORM_ND generates a random direction vector in ND.
DISK_POINT_DIST_3D determines the distance from a disk to a point in 3D.
DMS_TO_RADIANS converts an angle from degrees/minutes/seconds to radians.
DODEC_SHAPE_3D describes a dodecahedron in 3D.
DODEC_SIZE_3D gives "sizes" for a dodecahedron in 3D.
DUAL_SHAPE_3D constructs the dual of a shape in 3D.
DUAL_SIZE_3D determines sizes for a dual of a shape in 3D.
ELLIPSE_AREA_2D returns the area of an ellipse in 2D.
ELLIPSE_POINT_DIST_2D finds the distance from a point to an ellipse in 2D.
ELLIPSE_POINT_NEAR_2D finds the nearest point on an ellipse in 2D.
ELLIPSE_POINTS_2D returns N points on an tilted ellipse in 2D.
ELLIPSE_POINTS_ARC_2D returns N points on a tilted elliptical arc in 2D.
ENORM0_ND computes the Euclidean norm of a (X-Y) in N space.
GET_SEED returns a random seed for the random number generator.
GLOB2LOC_3D converts from a global to a local coordinate system in 3D.
HALFPLANE_CONTAINS_POINT_2D reports if a half-plane contains a point in 2d.
HALFSPACE_IMP_TRIANGLE_INT_3D: intersection ( implicit halfspace, triangle ) in 3D.
HALFSPACE_NORM_TRIANGLE_INT_3D: intersection ( normal halfspace, triangle ) in 3D.
HALFSPACE_TRIANGLE_INT_3D: intersection ( halfspace, triangle ) in 3D.
HAVERSINE computes the haversine of an angle.
HELIX_SHAPE_3D computes points on a helix in 3D.
HEXAGON_AREA_2D returns the area of a regular hexagon in 2D.
HEXAGON_CONTAINS_POINT_2D finds if a point is inside a hexagon in 2D.
HEXAGON_SHAPE_2D returns points on the unit regular hexagon in 2D.
HEXAGON_UNIT_AREA_2D returns the area of a unit regular hexagon in 2D.
HEXAGON_VERTICES_2D returns the vertices of the unit hexagon in 2D.
I4_DEDEKIND_FACTOR computes a function needed for a Dedekind sum.
I4_DEDEKIND_SUM computes the Dedekind sum of two I4's.
I4_FACTORIAL2 computes the double factorial function N!!
I4_GCD finds the greatest common divisor of two I4's.
I4_LCM computes the least common multiple of two I4's.
I4_MAX returns the maximum of two I4's.
I4_MIN returns the smaller of two I4's.
I4_MODP returns the nonnegative remainder of I4 division.
I4_SIGN returns the sign of an I4.
I4_SWAP switches two I4's.
I4_UNIFORM returns a scaled pseudorandom I4.
I4_WRAP forces an I4 to lie between given limits by wrapping.
I4COL_COMPARE compares columns I and J of an I4COL
I4COL_FIND_ITEM searches an I4COL for a given value.
I4COL_FIND_PAIR_WRAP wrap searches an I4COL for a pair of items.
I4COL_SORT_A ascending sorts the columns of an integer array.
I4COL_SORTED_UNIQUE_COUNT counts unique elements in an ICOL array.
I4COL_SWAP swaps two columns of an integer array.
I4MAT_PRINT prints an I4MAT, with an optional title.
I4MAT_PRINT_SOME prints some of an I4MAT.
I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
I4ROW_COMPARE compares two rows of an I4ROW.
I4ROW_SORT_A ascending sorts the rows of an I4ROW.
I4ROW_SWAP swaps two rows of an I4ROW.
I4VEC_COPY copies an I4VEC.
I4VEC_HEAP_D reorders an I4VEC into a descending heap.
I4VEC_INDICATOR_NEW sets an I4VEC to the indicator vector.
I4VEC_LCM returns the least common multiple of an I4VEC.
I4VEC_PRINT prints an I4VEC.
I4VEC_PRODUCT multiplies the entries of an I4VEC.
I4VEC_REVERSE reverses the elements of an I4VEC.
I4VEC_SORT_HEAP_A ascending sorts an I4VEC using heap sort.
I4VEC_SORTED_UNIQUE finds unique elements in a sorted I4VEC.
I4VEC_UNIFORM_NEW returns a scaled pseudorandom I4VEC.
I4VEC_ZERO zeroes an I4VEC.
I4VEC2_COMPARE compares pairs of integers stored in two vectors.
I4VEC2_SORT_A ascending sorts a vector of pairs of integers.
I4VEC2_SORTED_UNIQUE finds unique elements in a sorted I4VEC2.
ICOS_SHAPE describes a icosahedron.
ICOS_SIZE gives "sizes" for an icosahedron.
LINE_EXP_IS_DEGENERATE_ND finds if an explicit line is degenerate in ND.
LINE_EXP_NORMAL_2D computes the unit normal vector to a line in 2D.
LINE_EXP_PERP_2D computes a line perpendicular to a line and through a point.
LINE_EXP_POINT_DIST_2D: distance ( explicit line, point ) in 2D.
LINE_EXP_POINT_DIST_3D: distance ( explicit line, point ) in 3D.
LINE_EXP_POINT_DIST_SIGNED_2D: signed distance ( explicit line, point ) in 2D.
LINE_EXP_POINT_NEAR_2D computes the point on an explicit line nearest a point in 2D.
LINE_EXP_POINT_NEAR_3D: nearest point on explicit line to point in 3D.
LINE_EXP2IMP_2D converts an explicit line to implicit form in 2D.
LINE_EXP2PAR_2D converts a line from explicit to parametric form in 2D.
LINE_EXP2PAR_3D converts an explicit line into parametric form in 3D.
LINE_IMP_IS_DEGENERATE_2D finds if an implicit point is degenerate in 2D.
LINE_IMP_POINT_DIST_2D: distance ( implicit line, point ) in 2D.
LINE_IMP_POINT_DIST_SIGNED_2D: signed distance ( implicit line, point ) in 2D.
LINE_IMP2EXP_2D converts an implicit line to explicit form in 2D.
LINE_IMP2PAR_2D converts an implicit line to parametric form in 2D.
LINE_PAR_POINT_DIST_2D: distance ( parametric line, point ) in 2D.
LINE_PAR_POINT_DIST_3D: distance ( parametric line, point ) in 3D.
LINE_PAR_POINT_NEAR_2D: nearest point on parametric line to point in 2D.
LINE_PAR_POINT_DIST_3D: distance ( parametric line, point ) in 3D.
LINE_PAR2EXP_2D converts a parametric line to explicit form in 2D.
LINE_PAR2EXP_2D converts a parametric line to explicit form in 3D.
LINE_PAR2IMP_2D converts a parametric line to implicit form in 2D.
LINES_EXP_ANGLE_3D finds the angle between two explicit lines in 3D.
LINES_EXP_ANGLE_ND returns the angle between two explicit lines in ND.
LINES_EXP_DIST_3D computes the distance between two explicit lines in 3D.
LINES_EXP_DIST_3D_2 computes the distance between two explicit lines in 3D.
LINES_EXP_EQUAL_2D determines if two explicit lines are equal in 2D.
LINES_EXP_INT_2D determines where two explicit lines intersect in 2D.
LINES_EXP_NEAR_3D computes nearest points on two explicit lines in 3D.
LINES_EXP_PARALLEL_2D determines if two lines are parallel in 2D.
LINES_EXP_PARALLEL_3D determines if two lines are parallel in 3D.
LINES_IMP_ANGLE_2D finds the angle between two implicit lines in 2D.
LINES_IMP_DIST_2D determines the distance between two implicit lines in 2D.
LINES_IMP_INT_2D determines where two implicit lines intersect in 2D.
LINES_PAR_ANGLE_2D finds the angle between two parametric lines in 2D.
LINES_PAR_ANGLE_3D finds the angle between two parametric lines in 3D.
LINES_PAR_DIST_3D finds the distance between two parametric lines in 3D.
LINES_PAR_INT_2D determines where two parametric lines intersect in 2D.
LOC2GLOB_3D converts from a local to global coordinate system in 3D.
LVEC_PRINT prints a logical vector.
MINABS finds a local minimum of F(X) = A * abs ( X ) + B.
MINQUAD finds a local minimum of F(X) = A * X^2 + B * X + C.
OCTAHEDRON_SHAPE_3D describes an octahedron in 3D.
OCTAHEDRON_SIZE_3D returns size information for an octahedron in 3D.
PARABOLA_EX finds the extremal point of a parabola determined by three points.
PARABOLA_EX2 finds the extremal point of a parabola determined by three points.
PARALLELOGRAM_AREA_2D computes the area of a parallelogram in 2D.
PARALLELOGRAM_AREA_3D computes the area of a parallelogram in 3D.
PARALLELOGRAM_CONTAINS_POINT_2D determines if a point is inside a parallelogram in 2D.
PARALLELOGRAM_CONTAINS_POINT_3D determines if a point is inside a parallelogram in 3D.
PARALLELOGRAM_POINT_DIST_3D: distance ( parallelogram, point ) in 3D.
PARALLELEPIPED_CONTAINS_POINT_3D determines if a point is inside a parallelepiped in 3D.
PARALLELEPIPED_POINT_DIST_3D: distance ( parallelepiped, point ) in 3D.
PERM_CHECK checks that a vector represents a permutation.
PERM_INV inverts a permutation "in place".
PLANE_EXP_GRID_3D computes points and lines making up a planar grid in 3D.
PLANE_EXP_POINT_DIST_3D: distance ( explicit plane, point ) in 3D.
PLANE_EXP_NORMAL_3D finds the normal to an explicit plane in 3D.
PLANE_EXP_PRO2 produces 2D coordinates of points that lie in a plane, in 3D.
PLANE_EXP_PRO3 projects points orthographically onto a plane, in 3D.
PLANE_EXP_PROJECT_3D projects points through a point onto a plane in 3D.
PLANE_EXP2IMP_3D converts an explicit plane to implicit form in 3D.
PLANE_EXP2NORMAL_3D converts an explicit plane to normal form in 3D.
PLANE_IMP_IS_DEGENERATE_3D is TRUE if an implicit plane is degenerate.
PLANE_IMP_LINE_PAR_INT_3D: intersection ( implicit plane, parametric line ) in 3D.
PLANE_IMP_POINT_DIST_3D: distance ( point, implicit plane ) in 3D.
PLANE_IMP_POINT_DIST_SIGNED_3D: signed distance ( implicit plane, point) in 3
PLANE_IMP_POINT_NEAR_3D: nearest point on a implicit plane to a point in 3D.
PLANE_IMP_SEGMENT_NEAR_3D: nearest ( implicit plane, line segment ) in 3D
PLANE_IMP_TRIANGLE_INT_3D: intersection ( implicit plane, triangle ) in 3D.
PLANE_IMP_TRIANGLE_INT_ADD_3D is a utility for PLANE_IMP_TRIANGLE_INT_3D.
PLANE_IMP_TRIANGLE_NEAR_3D: nearest ( implicit plane, triangle ) in 3D.
PLANE_IMP2EXP_3D converts an implicit plane to explicit form in 3D.
PLANE_IMP2NORMAL_3D converts an implicit plane to normal form in 3D.
PLANE_NORMAL_BASIS_3D finds two perpendicular vectors in a plane in 3D.
PLANE_NORMAL_LINE_EXP_INT_3D: intersection of plane and line in 3D.
PLANE_NORMAL_QR_TO_XYZ: QR_TO_XYZ coordinates for a normal form plane.
PLANE_NORMAL_TETRAHEDRON_INTERSECT intersects a plane and a tetrahedron.
PLANE_NORMAL_TRIANGLE_INT_3D: intersection ( normal plane, triangle ) in 3D.
PLANE_NORMAL_UNIFORM_3D generates a random normal plane in 3D.
PLANE_NORMAL_UNIFORM_ND generates a random normal plane in ND.
PLANE_NORMAL_XYZ_TO_QR: XYZ to QR coordinates for a normal form plane.
PLANE_NORMAL2EXP_3D converts a normal plane to explicit form in 3D.
PLANE_NORMAL2IMP_3D converts a normal form plane to implicit form in 3D.
PLANES_IMP_ANGLE_3D: dihedral angle between implicit planes in 3D.
POINTS_AVOID_POINT_NAIVE_2D finds if a point is "far enough" from a set of points in 2D.
POINTS_BISECT_LINE_IMP_2D finds the implicit line bisecting the line between two points in 2D.
POINTS_BISECT_LINE_PAR_2D finds the parametric line bisecting the line between two points in 2D.
POINTS_CENTROID_2D computes the discrete centroid of a point set in 2D.
POINTS_COLIN_2D estimates the colinearity of 3 points in 2D.
POINTS_COLIN_3D estimates the colinearity of 3 points in 3D.
POINTS_DIST_2D finds the distance between two points in 2D.
POINTS_DIST_3D finds the distance between two points in 3D.
POINTS_DIST_ND finds the distance between two points in ND.
POINTS_HULL_2D computes the convex hull of a set of nodes in 2D.
POINTS_PLOT plots a pointset.
POINTS_POINT_NEAR_NAIVE_2D finds the nearest point to a given point in 2D.
POINTS_POINT_NEAR_NAIVE_3D finds the nearest point to a given point in 3D.
POINTS_POINT_NEAR_NAIVE_ND finds the nearest point to a given point in ND.
POINTS_POINTS_NEAR_NAIVE_2D finds the nearest point to given points in 2D.
POINTS_POINTS_NEAR_NAIVE_3D finds the nearest point to given points in 3D.
POLAR_TO_XY converts polar coordinates to XY coordinates.
POLYGON_1_2D integrates the function 1 over a polygon in 2D.
POLYGON_ANGLES_2D computes the interior angles of a polygon in 2D.
POLYGON_AREA_2D computes the area of a polygon in 2D.
POLYGON_AREA_2D_2 computes the area of a polygon in 2D.
POLYGON_AREA_3D computes the area of a polygon in 3D.
POLYGON_AREA_3D_2 computes the area of a polygon in 3D.
POLYGON_CENTROID_2D computes the centroid of a polygon in 2D.
POLYGON_CENTROID_2D_2 computes the centroid of a polygon in 2D.
POLYGON_CENTROID_3D computes the centroid of a polygon in 3D.
POLYGON_CONTAINS_POINT_2D finds if a point is inside a simple polygon in 2D.
POLYGON_CONTAINS_POINT_2D_2 finds if a point is inside a convex polygon in 2D.
POLYGON_DIAMETER_2D computes the diameter of a polygon in 2D.
POLYGON_EXPAND_2D expands a polygon in 2D.
POLYGON_INRAD_DATA_2D determines polygonal data from its inner radius in 2D.
POLYGON_IS_CONVEX determines whether a polygon is convex in 2D.
POLYGON_LATTICE_AREA_2D computes the area of a lattice polygon in 2D.
POLYGON_NORMAL_3D computes the normal vector to a polygon in 3D.
POLYGON_OUTRAD_DATA_2D determines polygonal data from its outer radius in 2D.
POLYGON_SIDE_DATA_2D determines polygonal data from its side length in 2D.
POLYGON_SOLID_ANGLE_3D calculates the projected solid angle of a 3D plane polygon.
POLYGON_X_2D integrates the function X over a polygon in 2D.
POLYGON_Y_2D integrates the function Y over a polygon in 2D.
POLYGON_XX_2D integrates the function X*X over a polygon in 2D.
POLYGON_XY_2D integrates the function X*Y over a polygon in 2D.
POLYGON_YY_2D integrates the function Y*Y over a polygon in 2D.
POLYHEDRON_AREA_3D computes the surface area of a polyhedron in 3D.
POLYHEDRON_CENTROID_3D computes the centroid of a polyhedron in 3D.
POLYHEDRON_CONTAINS_POINT_3D determines if a point is inside a polyhedron.
POLYHEDRON_VOLUME_3D computes the volume of a polyhedron in 3D.
POLYHEDRON_VOLUME_3D_2 computes the volume of a polyhedron in 3D.
POLYLINE_ARCLENGTH_ND computes the arclength of points on a polyline in ND.
POLYLINE_INDEX_POINT_ND evaluates a polyline at a given arclength in ND.
POLYLINE_LENGTH_ND computes the length of a polyline in ND.
POLYLINE_POINTS_ND computes equally spaced points on a polyline in ND.
POLYLOOP_ARCLENGTH_ND computes the arclength of points on a polyloop in ND.
POLYLOOP_POINTS_ND computes equally spaced points on a polyloop in ND.
PROVEC projects a vector from M space into N space.
PYRAMID_VOLUME_3D computes the volume of a pyramid with square base in 3D.
QUAD_AREA_2D computes the area of a quadrilateral in 2D.
QUAD_AREA2_2D computes the area of a quadrilateral in 2D.
QUAD_AREA_3D computes the area of a quadrilateral in 3D.
QUAD_CONTAINS_POINT_2D finds if a point is inside a convex quadrilateral in 2D.
QUAD_CONVEX_RANDOM returns a random convex quadrilateral.
QUAD_POINT_DIST_2D finds the distance from a point to a quadrilateral in 2D.
QUAD_POINT_DIST_SIGNED_2D: signed distanct ( quadrilateral, point ) in 2D.
QUAD_POINT_NEAR_2D computes the nearest point on a quadrilateral in 2D.
QUAT_CONJ conjugates a quaternion.
QUAT_INV inverts a quaternion.
QUAT_MUL multiplies two quaternions.
QUAT_NORM computes the norm of a quaternion.
R4_ABS returns the absolute value of an R4.
R4_NINT returns the nearest integer to an R4.
R8_ABS returns the absolute value of an R8.
R8_ACOS computes the arc cosine function, with argument truncation.
R8_ASIN computes the arc sine function, with argument truncation.
R8_EPSILON returns the R8 round off unit.
R8_HUGE returns a "huge" R8.
R8_MAX returns the maximum of two R8s.
R8_MIN returns the minimum of two R8's.
R8_MODP returns the nonnegative remainder of R8 division.
R8_NINT returns the nearest integer to an R8.
R8_NORMAL_01 returns a unit pseudonormal R8.
R8_PI returns the value of pi.
R8_SIGN returns the "sign" of an R8.
R8_SIGN_OPPOSITE_STRICT is TRUE if two R8's are strictly of opposite sign.
R8_SWAP switches two R8s.
R8_UNIFORM returns a scaled pseudorandom R8.
R8_UNIFORM_01 returns a unit pseudorandom R8.
R82VEC_PART_QUICK_A reorders an R82VEC as part of a quick sort.
R82VEC_PERMUTE permutes an R82VEC in place.
R82VEC_PRINT prints an R82VEC.
R82VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R82VEC.
R82VEC_SORT_QUICK_A ascending sorts an R82VEC using quick sort.
R8MAT_COPY copies one R8MAT to another.
R8MAT_DET_2D computes the determinant of a 2 by 2 R8MAT.
R8MAT_DET_3D computes the determinant of a 3 by 3 R8MAT.
R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
R8MAT_DET_5D computes the determinant of a 5 by 5 R8MAT.
R8MAT_INVERSE_2D inverts a 2 by 2 R8MAT using Cramer's rule.
R8MAT_INVERSE_3D inverts a 3 by 3 R8MAT using Cramer's rule.
R8MAT_MV multiplies a matrix times a vector.
R8MAT_PRINT prints an R8MAT, with an optional title.
R8MAT_PRINT_SOME prints some of an R8MAT.
R8MAT_SOLVE uses Gauss-Jordan elimination to solve an N by N linear system.
R8MAT_SOLVE_2D solves a 2 by 2 linear system using Cramer's rule.
R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
R8MAT_UNIFORM_NEW returns a scaled pseudorandom R8MAT.
R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
R8MAT_UNIFORM_01_NEW returns a unit pseudorandom R8MAT.
R8VEC_ANGLE_3D computes the angle between two vectors in 3D.
R8VEC_ANY_NORMAL returns some normal vector to V1.
R8VEC_BRACKET searches a sorted R8VEC for successive brackets of a value.
R8VEC_COPY copies an R8VEC.
R8VEC_CROSS_PRODUCT_2D finds the cross product of a pair of R8VEC's in 2D.
R8VEC_CROSS_PRODUCT_AFFINE_2D finds the affine cross product in 2D.
R8VEC_CROSS_PRODUCT_3D computes the cross product of two R8VEC's in 3D.
R8VEC_CROSS_PRODUCT_AFFINE_3D computes the affine cross product in 3D.
R8VEC_DISTANCE returns the Euclidean distance between two R8VEC's.
R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's.
R8VEC_DOT_PRODUCT_AFFINE computes the affine dot product.
R8VEC_EQ is true two R8VEC's are equal.
R8VEC_GT == ( A1 > A2 ) for R8VEC's.
R8VEC_LT == ( A1 < A2 ) for R8VEC's.
R8VEC_MAX returns the value of the maximum element in an R8VEC.
R8VEC_MEAN returns the mean of an R8VEC.
R8VEC_MIN returns the value of the minimum element in an R8VEC.
R8VEC_NEGATIVE_STRICT: all entries of R8VEC are strictly negative.
R8VEC_NORM returns the L2 norm of an R8VEC.
R8VEC_NORM_AFFINE returns the affine L2 norm of an R8VEC.
R8VEC_NORMAL_01_NEW returns a unit pseudonormal R8VEC.
R8VEC_NORMSQ returns the squared L2 norm of an R8VEC.
R8VEC_NORMSQ_AFFINE returns the squared affine L2 norm of an R8VEC.
R8VEC_POSITIVE_STRICT: all entries of R8VEC are strictly positive.
R8VEC_PRINT prints an R8VEC.
R8VEC_PRINT_2D prints a 2D vector.
R8VEC_PRINT_3D prints a 3D vector.
R8VEC_SCALAR_TRIPLE_PRODUCT finds the scalar triple product in 3D.
R8VEC_SWAP swaps the entries of two R8VEC's.
R8VEC_UNIFORM_NEW returns a scaled pseudorandom R8VEC.
R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.
R8VEC_UNIFORM_UNIT_NEW generates a random direction vector in ND.
R8VEC_VARIANCE returns the variance of an R8VEC.
R8VEC_ZERO zeroes an R8VEC.
RADEC_DISTANCE_3D - angular distance, astronomical units, sphere in 3D.
RADEC_TO_XYZ converts right ascension/declination to (X,Y,Z) coordinates.
RADIANS_TO_DEGREES converts an angle from radians to degrees.
RADIANS_TO_DMS converts an angle from radians to degrees/minutes/seconds.
RANDOM_INITIALIZE initializes the RANDOM random number generator.
ROTATION_AXIS_VECTOR_3D rotates a vector around an axis vector in 3D.
ROTATION_AXIS2MAT_3D converts a rotation from axis to matrix format in 3D.
ROTATION_AXIS2QUAT_3D converts a rotation from axis to quaternion format in 3D.
ROTATION_MAT_VECTOR applies a marix rotation to a vector in 3d.
ROTATION_MAT2AXIS_3D converts a rotation from matrix to axis format in 3D.
ROTATION_MAT2QUAT_3D converts a rotation from matrix to quaternion format in 3D.
ROTATION_QUAT_VECTOR applies a quaternion rotation to a vector in 3d.
ROTATION_QUAT2AXIS_3D converts a rotation from quaternion to axis format in 3D.
ROTATION_QUAT2MAT_3D converts a rotation from quaternion to matrix format in 3D.
RTP_TO_XYZ converts (R,Theta,Phi) to (X,Y,Z) coordinates.
S_LEN_TRIM returns the length of a string to the last nonblank.
SEC_DEG returns the secant of an angle given in degrees.
SEGMENT_CONTAINS_POINT_1D reports if a line segment contains a point in 1D.
SEGMENT_CONTAINS_POINT_2D reports if a line segment contains a point in 2D.
SEGMENT_POINT_COORDS_2D: coordinates of a point on a line segment in 2D.
SEGMENT_POINT_COORDS_3D: coordinates of a point on a line segment in 3D.
SEGMENT_POINT_DIST_2D: distance ( line segment, point ) in 2D.
SEGMENT_POINT_DIST_3D: distance ( line segment, point ) in 3D.
SEGMENT_POINT_NEAR_2D finds the point on a line segment nearest a point in 2D.
SEGMENT_POINT_NEAR_3D finds the point on a line segment nearest a point in 3D.
SEGMENTS_CURVATURE_2D computes the curvature of two line segments in 2D.
SEGMENTS_DIST_2D computes the distance between two line segments in 2D.
SEGMENTS_DIST_3D computes the distance between two line segments in 3D.
SEGMENTS_DIST_3D_OLD computes the distance between two line segments in 3D.
SEGMENTS_INT_1D computes the intersection of two line segments in 1D.
SEGMENTS_INT_2D computes the intersection of two line segments in 2D.
SHAPE_POINT_DIST_2D: distance ( regular shape, point ) in 2D.
SHAPE_POINT_NEAR_2D: nearest point ( regular shape, point ) in 2D.
SHAPE_PRINT_3D prints information about a polyhedron in 3D.
SHAPE_RAY_INT_2D: intersection ( regular shape, ray ) in 2D.
SIMPLEX_LATTICE_LAYER_POINT_NEXT: next simplex lattice layer point.
SIMPLEX_LATTICE_POINT_NEXT returns the next simplex lattice point.
SIMPLEX_UNIT_LATTICE_POINT_ND: count lattice points.
SIMPLEX_UNIT_VOLUME_ND computes the volume of the unit simplex in ND.
SIMPLEX_VOLUME_ND computes the volume of a simplex in ND.
SIN_DEG returns the sine of an angle given in degrees.
SIN_POWER_INT evaluates the sine power integral.
SOCCER_SHAPE_3D describes a truncated icosahedron in 3D.
SOCCER_SIZE_3D gives "sizes" for a truncated icosahedron in 3D.
SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
SPHERE_CAP_AREA_2D computes the surface area of a spherical cap in 2D.
SPHERE_CAP_AREA_3D computes the surface area of a spherical cap in 3D.
SPHERE_CAP_AREA_ND computes the area of a spherical cap in ND.
SPHERE_CAP_VOLUME_2D computes the volume of a spherical cap in 2D.
SPHERE_CAP_VOLUME_3D computes the volume of a spherical cap in 3D.
SPHERE_CAP_VOLUME_ND computes the volume of a spherical cap in ND.
SPHERE_DIA2IMP_3D converts a diameter to an implicit sphere in 3D.
SPHERE_DISTANCE_XYZ computes great circle distances on a sphere.
SPHERE_DISTANCE1 computes great circle distances on a sphere.
SPHERE_DISTANCE2 computes great circle distances on a sphere.
SPHERE_DISTANCE3 computes great circle distances on a sphere.
SPHERE_EXP_CONTAINS_POINT_3D determines if an explicit sphere contains a point in 3D.
SPHERE_EXP_POINT_NEAR_3D finds the nearest point on an explicit sphere to a point in 3D.
SPHERE_EXP2IMP_3D converts a sphere from explicit to implicit form in 3D.
SPHERE_EXP2IMP_ND finds an N-dimensional sphere through N+1 points.
SPHERE_IMP_AREA_3D computes the surface area of an implicit sphere in 3D.
SPHERE_IMP_AREA_ND computes the surface area of an implicit sphere in ND.
SPHERE_IMP_CONTAINS_POINT_3D determines if an implicit sphere contains a point in 3D.
SPHERE_IMP_GRID_ICOS_SIZE sizes an icosahedral grid on a sphere.
SPHERE_IMP_GRIDFACES_3D produces a grid of triangles on an implicit sphere in 3D.
SPHERE_IMP_LINE_PROJECT_3D projects a line onto an implicit sphere in 3D.
SPHERE_IMP_LOCAL2XYZ_3D converts local to XYZ coordinates on an implicit sphere in 3D.
SPHERE_IMP_POINT_NEAR_3D finds the nearest point on an implicit sphere to a point in 3D.
SPHERE_IMP_POINT_PROJECT_3D projects a point onto an implicit sphere, in 3D.
SPHERE_IMP_VOLUME_3D computes the volume of an implicit sphere in 3D.
SPHERE_IMP_VOLUME_ND computes the volume of an implicit sphere in ND.
SPHERE_IMP_ZONE_AREA_3D computes the surface area of a spherical zone in 3D.
SPHERE_IMP_ZONE_VOLUME_3D computes the volume of a spherical zone in 3D.
SPHERE_IMP2EXP_3D converts a sphere from implicit to explicit form in 3D.
SPHERE_K computes a factor useful for spherical computations.
SPHERE_TRIANGLE_ANGLES_TO_AREA computes the area of a spherical triangle.
SPHERE_TRIANGLE_CONTAINS_POINT determines if a spherical triangle contains a point.
SPHERE_TRIANGLE_SIDES_TO_ANGLES computes spherical triangle angles.
SPHERE_TRIANGLE_VERTICES_TO_ANGLES computes the angles of a spherical triangle.
SPHERE_TRIANGLE_VERTICES_TO_AREA computes the area of a spherical triangle.
SPHERE_TRIANGLE_VERTICES_TO_CENTROID gets a spherical triangle centroid.
SPHERE_TRIANGLE_VERTICES_TO_ORIENTATION seeks the orientation of a spherical triangle.
SPHERE_TRIANGLE_VERTICES_TO_SIDES_3D computes spherical triangle sides.
SPHERE_UNIT_AREA_ND computes the surface area of a unit sphere in ND.
SPHERE_UNIT_AREA_VALUES returns some areas of the unit sphere in ND.
SPHERE_UNIT_SAMPLE_2D picks a random point on the unit sphere (circle) in 2D.
SPHERE_UNIT_SAMPLE_3D picks a random point on the unit sphere in 3D.
SPHERE_UNIT_SAMPLE_3D_2 is a BAD method for sampling the unit sphere in 3D.
SPHERE_UNIT_SAMPLE_ND picks a random point on the unit sphere in ND.
SPHERE_UNIT_SAMPLE_ND_2 picks a random point on the unit sphere in ND.
SPHERE_UNIT_SAMPLE_ND_3 picks a random point on the unit sphere in ND.
SPHERE_UNIT_VOLUME_ND computes the volume of a unit sphere in ND.
SPHERE_UNIT_VOLUME_VALUES returns some volumes of the unit sphere in ND.
SPHERE01_DISTANCE_XYZ computes great circle distances on a unit sphere.
SPHERE01_POLYGON_AREA returns the area of a spherical polygon.
SPHERE01_POLYGON_AREA_KARNEY returns the area of a spherical polygon.
SPHERE01_TRIANGLE_ANGLES_TO_AREA: area of a spherical triangle on the unit sphere.
SPHERE01_TRIANGLE_SIDES_TO_ANGLES: angles of spherical triangle on unit sphere.
SPHERE01_TRIANGLE_VERTICES_TO_ANGLES: angles of spherical triangle on unit sphere.
SPHERE01_TRIANGLE_VERTICES_TO_AREA: area of a spherical triangle on unit sphere.
SPHERE01_TRIANGLE_VERTICES_TO_CENTROID: centroid of spherical triangle on unit sphere.
SPHERE01_TRIANGLE_VERTICES_TO_MIDPOINTS gets the midsides of a spherical triangle.
SPHERE01_TRIANGLE_VERTICES_TO_SIDES_3D: sides of spherical triangle on unit sphere.
STRING_2D groups line segments into connected lines in 2D.
SUPER_ELLIPSE_POINTS_2D returns N points on a tilted superellipse in 2D.
TAN_DEG returns the tangent of an angle given in degrees.
TETRAHEDRON_BARYCENTRIC_3D returns the barycentric coordinates of a point in 3D.
TETRAHEDRON_CENTROID_3D computes the centroid of a tetrahedron in 3D.
TETRAHEDRON_CIRCUMSPHERE_3D computes the circumsphere of a tetrahedron in 3D.
TETRAHEDRON_CONTAINS_POINT_3D: a tetrahedron contains a point in 3D.
TETRAHEDRON_DIHEDRAL_ANGLES_3D computes dihedral angles of a tetrahedron.
TETRAHEDRON_EDGE_LENGTH_3D returns edge lengths of a tetrahedron in 3D.
TETRAHEDRON_FACE_ANGLES_3D returns the 12 face angles of a tetrahedron 3D.
TETRAHEDRON_FACE_AREAS_3D returns the 4 face areas of a tetrahedron 3D.
TETRAHEDRON_INSPHERE_3D finds the insphere of a tetrahedron in 3D.
TETRAHEDRON_LATTICE_LAYER_POINT_NEXT: next tetrahedron lattice layer point.
TETRAHEDRON_LATTICE_POINT_NEXT returns the next tetrahedron lattice point.
TETRAHEDRON_QUALITY1_3D: "quality" of a tetrahedron in 3D.
TETRAHEDRON_QUALITY2_3D: "quality" of a tetrahedron in 3D.
TETRAHEDRON_QUALITY3_3D computes the mean ratio of a tetrahedron.
TETRAHEDRON_QUALITY4_3D computes the minimum solid angle of a tetrahedron.
TETRAHEDRON_RHOMBIC_SHAPE_3D describes a rhombic tetrahedron in 3D.
TETRAHEDRON_RHOMBIC_SIZE_3D gives "sizes" for a rhombic tetrahedron in 3D.
TETRAHEDRON_SAMPLE_3D returns random points in a tetrahedron.
TETRAHEDRON_SHAPE_3D describes a tetrahedron in 3D.
TETRAHEDRON_SIZE_3D gives "sizes" for a tetrahedron in 3D.
TETRAHEDRON_SOLID_ANGLES_3D computes solid angles of a tetrahedron.
TETRAHEDRON_UNIT_LATTICE_POINT_NUM_3D: count lattice points.
TETRAHEDRON_VOLUME_3D computes the volume of a tetrahedron in 3D.
TIMESTAMP prints the current YMDHMS date as a time stamp.
TMAT_INIT initializes the geometric transformation matrix.
TMAT_MXM multiplies two geometric transformation matrices.
TMAT_MXP multiplies a geometric transformation matrix times a point.
TMAT_MXP2 multiplies a geometric transformation matrix times N points.
TMAT_MXV multiplies a geometric transformation matrix times a vector.
TMAT_ROT_AXIS applies an axis rotation to the geometric transformation matrix.
TMAT_ROT_VECTOR applies a rotation about a vector to the geometric transformation matrix.
TMAT_SCALE applies a scaling to the geometric transformation matrix.
TMAT_SHEAR applies a shear to the geometric transformation matrix.
TMAT_TRANS applies a translation to the geometric transformation matrix.
TORUS_AREA_3D returns the area of a torus in 3D.
TORUS_VOLUME_3D computes the volume of a torus in 3D.
TP_TO_XYZ converts unit spherical TP coordinates to XYZ coordinates.
TRIANGLE_ANGLES_2D computes the angles of a triangle in 2D.
TRIANGLE_ANGLES_2D_NEW computes the angles of a triangle in 2D.
TRIANGLE_ANGLES_3D computes the angles of a triangle in 3D.
TRIANGLE_ANGLES_3D_NEW computes the angles of a triangle in 3D.
TRIANGLE_AREA_2D computes the area of a triangle in 2D.
TRIANGLE_AREA_3D computes the area of a triangle in 3D.
TRIANGLE_AREA_3D_2 computes the area of a triangle in 3D.
TRIANGLE_AREA_3D_3 computes the area of a triangle in 3D.
TRIANGLE_AREA_HERON computes the area of a triangle using Heron's formula.
TRIANGLE_AREA_VECTOR_3D computes the area vector of a triangle in 3D.
TRIANGLE_BARYCENTRIC_2D finds the barycentric coordinates of a point in 2D.
TRIANGLE_CENTROID_2D computes the centroid of a triangle in 2D.
TRIANGLE_CENTROID_3D computes the centroid of a triangle in 3D.
TRIANGLE_CIRCUMCENTER_2D computes the circumcenter of a triangle in 2D.
TRIANGLE_CIRCUMCENTER_2D_2 computes the circumcenter of a triangle in 2D.
TRIANGLE_CIRCUMCENTER computes the circumcenter of a triangle in ND.
TRIANGLE_CIRCUMCIRCLE_2D computes the circumcircle of a triangle in 2D.
TRIANGLE_CIRCUMCIRCLE_2D_2 computes the circumcircle of a triangle in 2D.
TRIANGLE_CIRCUMRADIUS_2D computes the circumradius of a triangle in 2D.
TRIANGLE_CONTAINS_LINE_EXP_3D finds if a line is inside a triangle in 3D.
TRIANGLE_CONTAINS_LINE_PAR_3D: finds if a line is inside a triangle in 3D.
TRIANGLE_CONTAINS_POINT_2D_1 finds if a point is inside a triangle in 2D.
TRIANGLE_CONTAINS_POINT_2D_2 finds if a point is inside a triangle in 2D.
TRIANGLE_CONTAINS_POINT_2D_3 finds if a point is inside a triangle in 2D.
TRIANGLE_DIAMETER_2D computes the diameter of a triangle in 2D.
TRIANGLE_EDGE_LENGTH_2D returns edge lengths of a triangle in 2D.
TRIANGLE_GRIDPOINTS_2D computes gridpoints within a triangle in 2D.
TRIANGLE_INCENTER_2D computes the incenter of a triangle in 2D.
TRIANGLE_INCIRCLE_2D computes the inscribed circle of a triangle in 2D.
TRIANGLE_INRADIUS_2D computes the inradius of a triangle in 2D.
TRIANGLE_IS_DEGENERATE_ND finds if a triangle is degenerate in ND.
TRIANGLE_LATTICE_LAYER_POINT_NEXT: next triangle lattice layer point.
TRIANGLE_LATTICE_POINT_NEXT returns the next triangle lattice point.
TRIANGLE_LINE_IMP_INT_2D finds where an implicit line intersects a triangle in 2D.
TRIANGLE_ORIENTATION_2D determines the orientation of a triangle in 2D.
TRIANGLE_ORTHOCENTER_2D computes the orthocenter of a triangle in 2D.
TRIANGLE_POINT_DIST_2D: distance ( triangle, point ) in 2D.
TRIANGLE_POINT_DIST_3D: distance ( triangle, point ) in 3D.
TRIANGLE_POINT_DIST_SIGNED_2D: signed distance ( triangle, point ) in 2D.
TRIANGLE_POINT_NEAR_2D computes the nearest triangle point to a point in 2D.
TRIANGLE_QUALITY_2D: "quality" of a triangle in 2D.
TRIANGLE_RIGHT_LATTICE_POINT_NUM_2D: count lattice points.
TRIANGLE_SAMPLE returns random points in a triangle.
TRIANGLE_UNIT_LATTICE_POINT_NUM_2D: count lattice points.
TRIANGLE_XSI_TO_XY_2D converts from barycentric to XY coordinates in 2D.
TRIANGLE_XY_TO_XSI_2D converts from XY to barycentric in 2D.
TRUNCATED_OCTAHEDRON_SHAPE_3D describes a truncated octahedron in 3D.
TRUNCATED_OCTAHEDRON_SIZE_3D gives "sizes" for a truncated octahedron in 3D.
TUBE_2D constructs a "tube" of given width around a path in 2D.
TUPLE_NEXT2 computes the next element of an integer tuple space.
VECTOR_DIRECTIONS_ND returns the direction angles of a vector in ND.
VECTOR_ROTATE_2D rotates a vector around the origin in 2D.
VECTOR_ROTATE_3D rotates a vector around an axis vector in 3D.
VECTOR_ROTATE_BASE_2D rotates a vector around a base point in 2D.
VECTOR_SEPARATION_2D finds the angular separation between vectors in 2D.
VECTOR_SEPARATION_3D finds the angular separation between vectors in 3D.
VECTOR_SEPARATION_ND finds the angular separation between vectors in ND.
VECTOR_UNIT_ND normalizes a vector in ND.
VOXELS_DIST_L1_3D computes the L1 distance between voxels in 3D.
VOXELS_DIST_L1_ND computes the L1 distance between voxels in ND.
VOXELS_LINE_3D computes voxels along a line in 3D.
VOXELS_REGION_3D arranges a set of voxels into contiguous regions in 3D.
VOXELS_STEP_3D computes voxels along a line from a given point in 3D.
XY_TO_POLAR converts XY coordinates to polar coordinates.
XYZ_TO_RADEC converts (X,Y,Z) to right ascension/declination coordinates.
XYZ_TO_RTP converts (X,Y,Z) to (R,Theta,Phi) coordinates.
XYZ_TO_TP converts (X,Y,Z) to (Theta,Phi) coordinates.

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Last revised on 14 April 2013.

GEOMETRY Geometric Calculations