Distributional Properties of Nearest-Site Angular Distances on the Sphere
Abstract
Nearest-site distances arise in many applications involving spherical or directional domains, including global geospatial analysis, wireless communications, spherical clustering, and cosine-similarity-based data analysis. In this paper, we study the distributional and computational properties of L2, the minimal angular great-circle distance from a uniformly distributed random point on a sphere to a set of prespecified sites on the same sphere. We first derive the cumulative distribution function (CDF) and probability density function (PDF) of L0, the angular great-circle distance from a fixed vertex of a spherical triangle to a random point uniformly distributed within that triangle. We then extend these triangle-level results to convex spherical polygons and use spherical Voronoi diagrams, triangulations of Voronoi cells, and numerical integration to obtain computable distributional and moment formulas for L2. In addition, we derive explicit formulas for selected moments of (L2), which are relevant to cosine similarity and spherical data analysis. Extensive Monte Carlo simulations validate the proposed CDF, PDF, and moment formulas and demonstrate computational efficiency of our method relative to generic numerical integration and simulation-based alternatives.