SPHERE_CVT
Centroidal Voronoi Tessellation
Unit Sphere

SPHERE_CVT, a MATLAB library which iteratively approximates a centroidal Voronoi tessellation (CVT) on the unit sphere.

The CVT approximation algorithm used here is quite simple. We start with XYZ, an arbitrary set of points on the unit sphere. We compute the convex hull, from that the Delaunay triangulation, and from that the Voronoi diagram. Finally, we compute the centroids of the Voronoi polygons, and overwrite XYZ with this data. This iteration may be carried out repeatedly. While in the plane, the Voronoi cells all tend to the same shape and area, on a sphere there are certain constraints. A typical CVT, if it has properly converged, will generally have 12 pentagons, with the other polygons being hexagons. Cases in which polygons of degree 4 or 7 occur indicate that the iteration is not near enough to completion. Because of the occurrence of two polygonal shapes, the areas of the cells will tend to two separate values.

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

Related Data and Programs:

FLORIDA_CVT_GEO, MATLAB programs which explore the creation of a centroidal Voronoi Tessellation (CVT) of the state of Florida, based solely on geometric considerations.

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

sphere_cvt_test

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_LEBEDEV_RULE, a dataset directory which contains sets of points on a sphere which can be used for quadrature rules of a known precision;

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, a MATLAB program which computes the Voronoi diagram of points on a sphere.

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 3D 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.

Reference:

1. Qiang Du, Vance Faber, Max Gunzburger,
   Centroidal Voronoi Tessellations: Applications and Algorithms,
   SIAM Review,
   Volume 41, Number 4, December 1999, pages 637-676.
2. 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.
3. Lili Ju, Qiang Du, Max Gunzburger,
   Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations,
   Parallel Computing,
   Volume 28, 2002, pages 1477-1500.
4. 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_cvt_step.m, carries out one CVT step on the unit sphere; actually it simply returns the centroids of the Voronoi polygons associated with a given set of points.
• sphere_delaunay.m, returns the Delaunay triangulation of points on the unit sphere.
• sphere_voronoi_plot.m, plots the Voronoi diagram 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.
• 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, determines the centroid of each Voronoi polygon, given the Delaunay triangulation of points on the unit sphere.
• 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.