We propose and implement an algorithm to compute the exact cumulative density function (CDF) of the distance from an arbitrary reference point to a randomly located node within an arbitrarily shaped (convex or concave) simple polygon. Using this result, we also obtain the closed-form probability density function (PDF) of the Euclidean distance between an arbitrary reference point and its i th neighbor node when N nodes are uniformly and independently distributed inside the arbitrarily shaped polygon. The implementation is based on the recursive approach proposed by Ahmadi and Pan [1] in order to obtain the distance distributions associated with arbitrary triangles. The algorithm in [1] is extended for arbitrarily shaped polygons by using a modified form of the shoelace formula. This modification allows tractable computation of the overlap area between a disk of radius r centered at the arbitrary reference point and the arbitrarily shaped polygon, which is a key part of the implementation. The obtained distance distributions can be used in the modeling of wireless networks, especially in the context of emerging ultra-dense small cell deployment scenarios, where network regions can be arbitrarily shaped. They can also be applied in other branches of science, such as forestry, mathematics, operations research, and material sciences.

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.