Distance distribution between two random points in arbitrary polygons

Distance distributions are a key building block in many subfields in mathematics, science and engineering. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random points each uniformly randomly distributed in respective arbitrary polygon regions. The proposed framework is based on measure theory and uses polar decomposition for simplifying and calculating the integrals to obtain closed form results. We validate our proposed framework by comparison with simulations and published closed form results in the literature for simple cases. We illustrate the versatility and advantage of the proposed framework by deriving closed form results for a case not yet reported in the literature. Finally, we also develop a Mathematica implementation of the proposed framework which allows a user to define any two arbitrary (concave or convex) polygons, with or without holes, which may be disjoint or overlap or coincide and determine the distance distribution numerically.

Automated summary

This paper presents a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two randomly distributed points. The framework is validated through simulations and comparison with existing literature, and its versatility and advantages are demonstrated by deriving closed form results for a case not yet reported. Additionally, a Mathematica implementation allows users to define and determine distance distributions numerically for arbitrary polygons.