# SPHERE_VORONOI Voronoi Diagram of Points on the Unit Sphere

SPHERE_VORONOI, 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.

### 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_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.

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:

Last revised on 27 March 2019.