Geometric Calculations

GEOMETRY is a FORTRAN90 library which carries out geometric calculations in 2, 3 and N dimensional 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.


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


GEOMETRY is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

DUTCH, a FORTRAN90 library which carries out tasks in computational geometry.

GEOMPACK, a FORTRAN90 library which can compute Delaunay triangulations Voronoi diagrams and other information, written by Barry Joe.

POLYGON_MOMENTS, a FORTRAN90 library which computes arbitrary moments of a polygon.

SIMPLEX_COORDINATES, a FORTRAN90 library which computes the Cartesian coordinates of the vertices of a regular simplex in M dimensions.

SPHERE_GRID, a FORTRAN90 library which provides a number of ways of generating grids of points, or of points and lines, or of points and lines and faces, over the unit sphere.

TABLE_DELAUNAY, a FORTRAN90 program which reads a file of point coordinates in the TABLE format and writes out the Delaunay triangulation.

TET_MESH, a FORTRAN90 library which carries out various operations on tetrahedral meshes.

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

TRIANGULATION, a FORTRAN90 library which defines and analyzes triangulations.

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

TRIANGULATION_TRIANGLE_NEIGHBORS, a FORTRAN90 program which reads data defining a triangulation, determines the neighboring triangles of each triangle, and writes that information to a file.


  1. Gerard Bashein, Paul Detmer,
    Centroid of a Polygon,
    in Graphics Gems IV,
    edited by Paul Heckbert,
    AP Professional, 1994,
    ISBN: 0123361559,
    LC: T385.G6974.
  2. SF Bockman,
    Generalizing the Formula for Areas of Polygons to Moments,
    American Mathematical Society Monthly,
    Volume 96, Number 2, February 1989, pages 131-132.
  3. Adrian Bowyer, John Woodwark,
    A Programmer's Geometry,
    Butterworths, 1983,
    ISBN: 0408012420.
  4. 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.
  5. Daniel Cohen,
    Voxel Traversal along a 3D Line,
    in Graphics Gems IV,
    edited by Paul Heckbert,
    AP Professional, 1994,
    ISBN: 0123361559,
    LC: T385.G6974.
  6. Thomas Cormen, Charles Leiserson, Ronald Rivest,
    Introduction to Algorithms,
    MIT Press, 2001,
    ISBN: 0262032937,
    LC: QA76.C662.
  7. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
    Computational Geometry,
    Springer, 2000,
    ISBN: 3-540-65620-0,
    LC: QA448.D38.C65.
  8. Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
    LINPACK User's Guide,
    SIAM, 1979,
    ISBN13: 978-0-898711-72-1,
    LC: QA214.L56.
  9. James Foley, Andries vanDam, Steven Feiner, John Hughes,
    Computer Graphics, Principles and Practice,
    Second Edition,
    Addison Wesley, 1995,
    ISBN: 0201848406,
    LC: T385.C5735.
  10. Martin Gardner,
    The Mathematical Carnival,
    Knopf, 1975,
    ISBN: 0394494067,
    LC: QA95.G286.
  11. Priamos Georgiades,
    Signed Distance From Point To Plane,
    in Graphics Gems III,
    edited by David Kirk,
    Academic Press, 1992,
    ISBN: 0124096735,
    LC: T385.G6973.
  12. Branko Gruenbaum, Geoffrey Shephard,
    Pick's Theorem,
    The American Mathematical Monthly,
    Volume 100, Number 2, February 1993, pages 150-161.
  13. John Harris, Horst Stocker,
    Handbook of Mathematics and Computational Science,
    Springer, 1998,
    ISBN: 0-387-94746-9,
    LC: QA40.S76.
  14. Barry Joe,
    GEOMPACK - a software package for the generation of meshes using geometric algorithms,
    Advances in Engineering Software,
    Volume 13, 1991, pages 325-331.
  15. 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.
  16. Jack Kuipers,
    Quaternions and Rotation Sequences,
    Princeton, 1998,
    ISBN: 0691102988,
    LC: QA196.K85.
  17. 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.
  18. Albert Nijenhuis, Herbert Wilf,
    Combinatorial Algorithms for Computers and Calculators,
    Second Edition,
    Academic Press, 1978,
    ISBN: 0-12-519260-6,
    LC: QA164.N54.
  19. 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.
  20. Joseph ORourke,
    Computational Geometry,
    Second Edition,
    Cambridge, 1998,
    ISBN: 0521649765,
    LC: QA448.D38.
  21. Edward Saff, Arno Kuijlaars,
    Distributing Many Points on a Sphere,
    The Mathematical Intelligencer,
    Volume 19, Number 1, 1997, pages 5-11.
  22. Philip Schneider, David Eberly,
    Geometric Tools for Computer Graphics,
    Elsevier, 2002,
    ISBN: 1558605940,
    LC: T385.S334.
  23. 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.
  24. Moshe Shimrat,
    Algorithm 112: Position of Point Relative to Polygon,
    Communications of the ACM,
    Volume 5, Number 8, August 1962, page 434.
  25. Kenneth Stephenson,
    Introduction to Circle Packing, The Theory of Discrete Analytic Functions,
    Cambridge, 2005,
    ISBN: 0521823560,
    LC: QA640.7S74.
  26. 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.
  27. Daniel Zwillinger, Steven Kokoska,
    Standard Probability and Statistical Tables,
    CRC Press, 2000,
    ISBN: 1-58488-059-7,
    LC: QA273.3.Z95.

Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the FORTRAN90 source codes.

Last revised on 12 April 2013.