Nearest Neighbor and Contact Distance Distribution for Binomial Point Process on Spherical Surfaces

This letter characterizes the statistics of the contact distance and the nearest neighbor (NN) distance for binomial point processes (BPP) spatially-distributed on spherical surfaces. We consider a setup of $n$ concentric spheres, with each sphere $S_k$ has a radius $r_k$ and $N_k$ points that are uniformly distributed on its surface. For that setup, we obtain the cumulative distribution function (CDF) of the distance to the nearest point from two types o observation points: (i) the observation point is not a part of the point process and located on a concentric sphere with a radius $r_e<r_k\forall k$, which corresponds to the contact distance distribution, and (ii) the observation point belongs to the point process, which corresponds to the nearest-neighbor (NN) distance distribution.