# Distance Distribution Between Two Random Nodes in Arbitrary Polygons

## TL;DR

This paper introduces a new measure-theoretic framework for deriving the exact probability density function of distances between two uniformly distributed random nodes in any arbitrary polygonal regions, enhancing analytical tools in stochastic geometry.

## Contribution

It presents a novel, measure theory-based method for analytically computing the distance distribution between nodes in arbitrary polygons, including convex and concave shapes, with a practical Mathematica implementation.

## Key findings

- Validated framework against simulations and existing results
- Derived new closed-form solutions for complex polygon cases
- Provided a versatile computational tool for arbitrary polygons

## Abstract

Distance distributions are a key building block in stochastic geometry modelling of wireless networks and in many other fields in mathematics and science. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random nodes each uniformly randomly distributed in respective arbitrary (convex or concave) polygon regions (which may be disjoint or overlap or coincide). 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 polygons and conveniently determine the distance distribution numerically.