February – Voronoi tesselation on a sphere

Voronoi tessellation consists neighborhood areas of a given set of points, or Voronoi cells: every cell is surrounded by the area that consists of locations that are closer to the cell than any other cell. In addition to the artistic value, Voronoi Diagram, or Voronoi tessellation, have applications in wireless communications (among the million other things).

The next Matlab code produces a Voronoi tessellation on a sphere Poisson points as cells. It is based in Grady Wrights codes openly available in Github.

%%Create Voronoi tesselation around Poisson points on a Sphere.

clear all;
close all;

addpath(fullfile(cd,'rbfsphere'));
density = 100;
X = poissononsphere(density)'; %Poisson points in spherical coordinates.
voronoiSph(X); %This function is downloaded from gradywright's Github. It is placed in the folder 'rbfsphere'.

function refc = poissononsphere(density)
  yMin = -1; yMax = 1;
  xMin = -pi; xMax = pi;
  
  xDelta = xMax - xMin; yDelta = yMax - yMin; %Rectangle dimensions
  numbPoints = poissrnd(density);    %Number of points in the area is a Poisson variable of intensity given as density
  x = xDelta*(rand(numbPoints,1)) + xMin;    %Pick points from uniform distribution
  y = yDelta*(rand(numbPoints,1)) + yMin;    %Map referencepoints to geographical coordinates
  ref = [x y]';

  refs = [x'; asin(y)'];%Map geographical coordinates to Cartesian coordinates on a unit circle
  r = 1;
  refc = [r*sin(refs(2,:)+pi/2).*cos(refs(1,:)+pi);...
          r*sin(refs(2,:)+pi/2).*sin(refs(1,:)+pi);...
          r*cos(refs(2,:)+pi/2)];
end

References:

• gradywright Github