# sphere_cvt

sphere_cvt, a MATLAB code 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.

### Languages:

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

### Related Data and Programs:

florida_cvt_geo, MATLAB codes which explore the creation of a centroidal voronoi tessellation (cvt) of the state of florida, based solely on geometric considerations.

geometry, a MATLAB code which computes various geometric quantities, including grids on spheres.

sphere_delaunay, a MATLAB code 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 code 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 code which approximates an integral over the surface of the unit sphere by applying a triangulation to the surface;

sphere_voronoi, a MATLAB code which computes the voronoi diagram of points on a 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:

Last revised on 12 January 2021.