Stochastic Properties of Minimal Arc Distance and Cosine Similarity between a Random Point and Prespecified Sites on Sphere

TL;DR

This paper analyzes the statistical properties of minimal arc distance and cosine similarity between a random point and fixed sites on a sphere, deriving distributions, moments, and verifying results through simulations.

Contribution

It introduces new analytical expressions for the distribution and moments of arc distance and cosine similarity on spheres, applicable to wireless communication and related fields.

Findings

Derived the CDF and PDF of arc distance within spherical triangles.

Computed moments of minimal arc distance and cosine similarity.

Validated analytical results with extensive Monte Carlo simulations.

Abstract

In applications such as wireless communication, it is important to study the statistical properties of $L_{2}$, the minimal arc distance between a random point (e.g., a cellphone user) uniformly distributed on a sphere to a set of pre-defined seeds (e.g., wireless towers) on that sphere. In this study, we first derive the distribution (CDF) and density (PDF) functions of the arc distance between a selected vertex of a spherical triangle to a random point uniformly distributed within this triangle. Next, using computational techniques based on spherical Voronoi diagram and triangular partition of Voronoi cells, we derive moments of $L_{2}$ and $\cos L_{2}$. These results are verified by extensive Monte Carlo simulations.