SPHERE_DELAUNAY
Delaunay Triangulation of Points on the Unit Sphere

SPHERE_DELAUNAY is a MATLAB library which computes the Delaunay triangulation of points on the unit sphere.

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

SPHERE_DELAUNAY uses this approach, by calling MATLAB's convhulln function to generate the convex hull. The information defining the convex hull is actually the desired triangulation of the points. Since this computation is so easy, other parts of the program are designed to analyze the resulting Delaunay triangulation and return other information, such as the areas of the triangles and so on.

Licensing:

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

Languages:

SPHERE_DELAUNAY is available in a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

SPHERE_CVT, a MATLAB library which creates a mesh of well-separated points on a unit sphere by applying the Centroidal Voronoi Tessellation (CVT) iteration.

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 MATLAB 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.

SPHERE_VORONOI, a MATLAB program which computes the Voronoi diagram of points on a 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_XYZ_DISPLAY, a MATLAB program which reads XYZ information defining points in 3D, and displays a unit sphere and the points in the MATLAB graphics window.

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 graphics window. This can be used, for instance, to display Voronoi diagrams or Delaunay triangulations on the unit sphere.

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

STRIPACK_INTERACTIVE, a FORTRAN90 program which reads an XYZ file of 3D points on the unit sphere, computes the Delaunay triangulation, and writes it to a file.

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

Reference:

1. 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.
2. 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.

Source Code:

• arc_cosine.m, computes the inverse cosine function;
• arc_sine.m, computes the inverse cosine function;
• atan4.m, computes the inverse tangent;
• i4col_compare.m, compares two columns of an I4COL;
• i4col_sort_a.m, ascending sorts the columns of an I4COL;
• i4col_swap.m, swaps two columns of an I4COL;
• i4mat_transpose_print.m, prints an I4MAT, transposed;
• i4mat_transpose_print_some.m, prints some of an I4MAT, transposed;
• icos_shape.m, returns the shape of an icosahedron.
• icos_size.m, returns size information for an icosahedron.
• r8mat_transpose_print.m, prints the transpose of an R8MAT;
• r8mat_transpose_print_some.m, prints some of the transpose of an R8MAT;
• r8mat_uniform_01.m, returns a unit pseudorandom R8MAT;
• r8vec_normal_01.m, returns unit pseudonormal R8VEC.
• r8vec_polarize.m, decomposes an R8VEC into normal and parallel components;
• r8vec_print.m, prints an R8VEC.
• r8vec_uniform_01.m, returns a unit pseudorandom R8VEC.
• sort_heap_external.m, external sorts a list of values into ascending order;
• sphere_delaunay.m, returns the Delaunay triangulation of points on the unit sphere.
• sphere_grid_icos_size.m, returns size information for an icosahedral grid on the sphere.
• sphere_gridpoints_icos2.m, returnspoints of an icosahedral grid on the sphere.
• stri_angles_to_area.m, computes the area of a spherical triangle;
• stri_sides_to_angles.m, computes the angles of a spherical triangle from its sides;
• stri_vertices_to_area.m, computes the area of a spherical triangle from its vertices;
• stri_vertices_to_sides.m, computes the sides of a spherical triangle from its sides;
• timestamp.m, prints the current YMDHMS date as a timestamp;
• triangulation_neighhbor_triangles.m, determines triangle neighbors in a triangulation;
• uniform_on_sphere01_map.m, returns uniform random points on the unit sphere.

Examples and Tests:

• sphere_delaunay_test.m, a sample calling program.
• sphere_delaunay_test_output.txt, the output file.
• sphere_delaunay_test01.m, test 1;
• sphere_delaunay_test02.m, test 2;
• sphere_delaunay_test03.m, test 3;
• test02_plot1.png, a TRISURF plot of the Delaunay triangulation.
• test02_plot2.png, a PATCH plot of the Delaunay triangulation.
• test03_plot1.png, a TRISURF plot of the Delaunay triangulation.
• test03_plot2.png, a PATCH plot of the Delaunay triangulation.

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