Random Distances Associated with Arbitrary Polygons: An Algorithmic Approach between Two Random Points

TL;DR

This paper introduces an algorithmic method to compute the distribution of distances between two random points in arbitrary polygons, utilizing integral geometry and triangulation, with applications in wireless network modeling.

Contribution

It provides a novel algorithmic framework for deriving point distance distributions in polygons, including triangles and rings, expanding probabilistic modeling tools for wireless networks.

Findings

Derived closed-form and algorithmic PDDs for triangles.

Extended PDDs to arbitrary polygons via triangulation.

Provided PDDs for ring geometries.

Abstract

This report presents a new, algorithmic approach to the distributions of the distance between two points distributed uniformly at random in various polygons, based on the extended Kinematic Measure (KM) from integral geometry. We first obtain such random Point Distance Distributions (PDDs) associated with arbitrary triangles (i.e., triangle-PDDs), including the PDD within a triangle, and that between two triangles sharing either a common side or a common vertex. For each case, we provide an algorithmic procedure showing the mathematical derivation process, based on which either the closed-form expressions or the algorithmic results can be obtained. The obtained triangle-PDDs can be utilized for modeling and analyzing the wireless communication networks associated with triangle geometries, such as sensor networks with triangle-shaped clusters and triangle-shaped cellular systems with highly directional antennas. Furthermore, based on the obtained triangle-PDDs, we then show how to obtain the PDDs associated with arbitrary polygons through the decomposition and recursion approach, since any polygons can be triangulated, and any geometry shapes can be approximated by polygons with a needed precision. Finally, we give the PDDs associated with ring geometries. The results shown in this report can enrich and expand the theory and application of the probabilistic distance models for the analysis of wireless communication networks.