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 $L_2$, 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 $L_0$, 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 $L_2$. In addition, we derive explicit formulas for selected moments of $\cos(L_2)$, 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.

翻译：最近邻点距离在涉及球面或方向域的众多应用中出现，包括全球地理空间分析、无线通信、球面聚类以及基于余弦相似度的数据分析。本文研究了从球面上均匀分布的随机点到同一球面上预设站点集合的最小角大圆距离 \(L_2\) 的分布与计算性质。我们首先推导了从球面三角形固定顶点到该三角形内均匀分布随机点的角大圆距离 \(L_0\) 的累积分布函数（CDF）和概率密度函数（PDF）。随后将这些三角形层面的结果推广到凸球面多边形，并利用球面Voronoi图、Voronoi胞腔的三角剖分以及数值积分，得到了 \(L_2\) 的可计算分布与矩公式。此外，我们还推导了 \(\cos(L_2)\) 选定矩的显式公式，这些矩与余弦相似度及球面数据分析相关。大量蒙特卡洛模拟验证了所提出的CDF、PDF和矩公式，并展示了我们的方法相对于通用数值积分和基于模拟的替代方案的计算效率。