# Distribution of areas of Voronoi cells on a sphere

I have two sets of, respectively, N and M points, which are independently, randomly allocated on a sphere. I consider the Voronoi tessellation of the sphere by the N points, and I want to find how many of the other M points are on average in a Voronoi cell, and the standard deviation (or variance) of this number. As I am drawing point locations from a uniform distribution, I am thinking of looking at the statistics of Voronoi cell areas (first two moments), and then estimate accordingly the mean and variance of the density of points out of the M ones per cell. However, it looks like there is no analytical estimation of the PDF for cell area distribution for the spherical Voronoi tessellation. Any reference/suggestion that comes to mind? Otherwise, any alternative approach for the solution of my problem? I stress that I am looking for analytical solutions.

• How do you build your spherical Voronoi cells ? Using $\mathbb{R}^3$ distance or by arc (= geodesic) distance ? – Jean Marie Apr 29 '20 at 16:45
• @JeanMarie I am sticking to the Euclidean distance in $\mathbb{R}^3$ mostly because, I have that easily implemented in the scipy.spatial.SphericalVoronoi method. – maurizio Apr 29 '20 at 16:50
• Are you aware of the "Lambert cylindrical equal area projection", as described p. 5 of this document ? – Jean Marie Apr 29 '20 at 18:50
• Sorry, I forgot to attach the URL of the document : citeseerx.ist.psu.edu/viewdoc/… – Jean Marie Apr 29 '20 at 19:42
• @JeanMarie Thanks. I did not know this mapping. I am going to see if it can be used. I will follow up. – maurizio Apr 29 '20 at 21:00