STRIPACK_DELAUNAY
Delaunay Diagrams on Unit Spheres

STRIPACK_DELAUNAY is a FORTRAN90 program which interactively determines the Delaunay diagram of a set of points on a sphere.

The set of points is read from a file, and the Delaunay triangulation, once computed, is written out to another file, described as a series of triplets of point indexes.

According to Steven Fortune, it is possible to compute the Delaunay triangulation of points on the surface of a sphere by computing their convex hull, regarded as a 3D pointset. If the sphere is the unit sphere at the origin, the facet normals are the Voronoi vertices.

Usage:

stripack_delaunay node_filename

• node_filename is the name of a file containing the (X,Y,Z) coordinates of points on the unit sphere.

Licensing:

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

Languages:

STRIPACK_DELAUNAY is available in a FORTRAN90 version.

Related Data and Programs:

GEOMETRY, a FORTRAN90 library which computes various geometric quantities, including grids on spheres.

SPHERE_CVT, a FORTRAN90 library which creates a mesh of well-separated points on a unit sphere using Centroidal Voronoi Tessellations.

SPHERE_DELAUNAY, a FORTRAN90 program which computes the Delaunay triangulation of points on a sphere.

SPHERE_DESIGN_RULE, a FORTRAN90 library which returns point sets on the surface of the unit sphere, known as "designs", which can be useful for estimating integrals on the surface, among other uses.

SPHERE_GRID, a dataset directory containing files which describe sets of points on the unit sphere.

SPHERE_QUAD, a FORTRAN90 library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

SPHERE_STEREOGRAPH, a FORTRAN90 library which computes the stereographic mapping between points on the unit sphere and points on the plane Z = 1; a generalized mapping is also available.

SPHERE_VORONOI, a FORTRAN90 program which computes and plots the Voronoi diagram of points on the unit sphere.

SPHERE_VORONOI_DISPLAY_OPENGL, a C++ program which displays a sphere and randomly selected generator points, and then gradually colors in points in the sphere that are closest to each generator.

SPHERE_XYZF_DISPLAY, a MATLAB program which reads XYZF information defining points and faces, and displays a unit sphere, the points, and the faces, in the MATLAB 3D graphics window. This can be used, for instance, to display Voronoi diagrams or Delaunay triangulations on the unit sphere.

STRIPACK, a FORTRAN90 library which can compute the Delaunay triangulation or Voronoi diagram of a set of points on the unit sphere.

STRIPACK_VORONOI, a FORTRAN90 program which reads a set of points on the unit sphere, computes the Voronoi diagram, and writes it to a file.

TOMS772, a FORTRAN77 library which is the original text of the STRIPACK program.

XYZF_DISPLAY, a MATLAB program which reads XYZF information defining points and faces in 3D, and displays an image using OpenGL.

Reference:

1. Thomas Ericson, Victor Zinoviev,
   Codes on Euclidean Spheres,
   Elsevier, 2001,
   ISBN: 0444503293,
   LC: QA166.7E75
2. Gerald Folland,
   How to Integrate a Polynomial Over a Sphere,
   American Mathematical Monthly,
   Volume 108, Number 5, May 2001, pages 446-448.
3. Jacob Goodman, Joseph ORourke, editors,
   Handbook of Discrete and Computational Geometry,
   Second Edition,
   CRC/Chapman and Hall, 2004,
   ISBN: 1-58488-301-4,
   LC: QA167.H36.
4. AD McLaren,
   Optimal Numerical Integration on a Sphere,
   Mathematics of Computation,
   Volume 17, Number 84, October 1963, pages 361-383.
5. Robert Renka,
   Algorithm 772:
   STRIPACK: Delaunay Triangulation and Voronoi Diagram on the Surface of a Sphere,
   ACM Transactions on Mathematical Software,
   Volume 23, Number 3, September 1997, pages 416-434.
6. Edward Saff, Arno Kuijlaars,
   Distributing Many Points on a Sphere,
   The Mathematical Intelligencer,
   Volume 19, Number 1, 1997, pages 5-11.

Source Code:

• stripack_delaunay.f90, the source code.

Examples and Tests:

SPHERE_GRID_ICOS1_F1 is a grid of 12 points based on an icosahedron.

• sphere_grid_icos1_f1.xyz, the point coordinates.
• sphere_grid_icos1_f1.xyzf, the triangles forming the Delaunay triangulation.
• sphere_grid_icos1_f1.png, a PNG image after display by XYZF_DISPLAY.
• sphere_grid_icos1_f1.txt, the output from the program.

SPHERE_GRID_ICOS1_F2 is a grid of 42 points based on an icosahedron.

• sphere_grid_icos1_f2.xyz, the point coordinates.
• sphere_grid_icos1_f2.xyzf, the triangles forming the Delaunay triangulation.
• sphere_grid_icos1_f2.png, a PNG image after display by XYZF_DISPLAY.
• sphere_grid_icos1_f2.txt, the output from the program.

SPHERE_GRID_ICOS1_F3 is a grid of 92 points based on an icosahedron.

• sphere_grid_icos1_f3.xyz, the point coordinates.
• sphere_grid_icos1_f3.xyzf, the triangles forming the Delaunay triangulation.
• sphere_grid_icos1_f3.png, a PNG image after display by XYZF_DISPLAY.
• sphere_grid_icos1_f3.txt, the output from the program.

List of Routines:

• MAIN is the main program for STRIPACK_DELAUNAY
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• I4MAT_WRITE writes an I4MAT file.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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