Distance Distributions in Regular Polygons

TL;DR

This paper derives exact formulas for the distribution of distances between points inside regular polygons, accounting for polygon shape and reference point location, and provides an algorithm for general cases.

Contribution

It introduces a novel method to compute distance distributions in regular polygons, including a closed-form PDF and an algorithm for arbitrary reference points.

Findings

Derived the exact CDF of distance from a node to a reference point inside a polygon.

Obtained the closed-form PDF of distances between nodes in a polygon.

Proposed an algorithm to determine distance distributions for any reference point location.

Abstract

This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular $\el$-sided polygon. Using this result, we obtain the closed-form probability density function (PDF) of the Euclidean distance between any arbitrary reference point and its $n$-th neighbour node, when $N$ nodes are uniformly and independently distributed inside a regular $\ell$-sided polygon. First, we exploit the rotational symmetry of the regular polygons and quantify the effect of polygon sides and vertices on the distance distributions. Then we propose an algorithm to determine the distance distributions given any arbitrary location of the reference point inside the polygon. For the special case when the arbitrary reference point is located at the center of the polygon, our framework reproduces the existing result in the literature.