SPHERE_VORONOI
Voronoi Diagram of Points on the Unit Sphere

SPHERE_VORONOI is a MATLAB library which computes the Voronoi diagram 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_VORONOI uses this approach, by calling MATLAB's convhulln function to generate the convex hull. The information defining the convex hull is actually the Delaunay triangulation of the points. From here, it is possible to compute the Voronoi vertices, and to determine how these vertices are joined to form the Voronoi polygons.

The code, as presented here, is quite preliminary. In particular, the process of converting the Delaunay information into information about the Voronoi polygons is inefficient. I suspect, though, that I can compute the centroids almost immediately, without having to go through the tedious process of determining the ordering of the Voronoi vertices that constitute each Voronoi polygon. If I can clear that up, then it should be possible to apply this simple algorithm to systems with hundreds of points.

Licensing:

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

Languages:

SPHERE_VORONOI 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_DELAUNAY, a MATLAB 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 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_QUAD, a MATLAB library which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

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 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 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.

VORONOI_PLOT, a MATLAB program which plots the Voronoi neighborhoods of points using L1, L2, LInfinity or arbitrary LP norms;

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;
• 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;
• i4list_print.m, prints an I4LIST;
• i4mat_transpose_print.m, prints an I4MAT, transposed;
• i4mat_transpose_print_some.m, prints some of an I4MAT, transposed;
• i4vec_print.m, prints an I4VEC.
• r8mat_transpose_print.m, prints the transpose of an R8MAT;
• r8mat_transpose_print_some.m, prints some of the transpose of an R8MAT;
• r8vec_normal_01.m, returns unit pseudonormal R8VEC.
• 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.
• 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_centroid.m, computes the centroid of a spherical triangle from its vertices;
• stri_vertices_to_orientation.m, attempts to define an orientation for 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.
• voronoi_areas.m, determines the area of each Voronoi polygon, given the Delaunay triangulation of points on the unit sphere and the location of the Voronoi vertices and the explicit description of the Voronoi polygons.
• voronoi_areas_direct.m, determines the area of each Voronoi polygon, given the Delaunay triangulation of points on the unit sphere and the location of the Voronoi vertices.
• voronoi_centroids.m, computes the centroids of each polygon in a Voronoi diagram.
• voronoi_data.m, computes Voronoi areas and centroids directly.
• voronoi_order.m, determines the order of each Voronoi polygon, given the Delaunay triangulation of points on the unit sphere.
• voronoi_polygons.m, determines the Voronoi polygons, given the Delaunay triangulation of points on the unit sphere.
• voronoi_vertices.m, determines the location of Voronoi vertices, given the Delaunay triangulation of points on the unit sphere.

Examples and Tests:

• sphere_voronoi_test.m, a sample calling program.
• sphere_voronoi_test01.m, demonstrates some of the Voronoi calculations.
• sphere_voronoi_test02.m, displays the Voronoi diagram on a sphere.
• sphere_voronoi_test03.m, compares naive and direct approaches to computing the area of the Voronoi polygons.
• sphere_voronoi_test_output.txt, the output file.
• figure_01.png, an image of the Voronoi diagram made using PATCH.

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