Computing Exact Closed-Form Distance Distributions in Arbitrarily Shaped Polygons with Arbitrary Reference Point

Abstract

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.

On the distance between random events on a network

“…The problem we study in this paper belongs to a larger class of problems whose main objective is to study the distance between random points on a given metric space. Some of the most well-studied variations are cases developed for compact path-connected subspaces of R 2 or R 3 , with norms L 1 or L 2 acting as the metric functions [27,33]. There are three main aspects that define the different variations found on the literature: the geometric shape of the space, the metric function used to measure the distances in the space, and the distribution of the location of the points [12,38].…”

On the Distribution of Random Geometric Graphs

“…It is clear from (2) that the joint pdf f Rn of inter-node distances is highly important for the graph distribution and its entropy. For n = 2, the sought pdf reduces to the pdf of the distance between two nodes, which has been extensively studied for various shapes of the embedding space K (e.g., see [24]- [27]). Obtaining the joint pdf analytically for n > 2 is very challenging and no such results have been reported previously.…”

“…The joint distribution of all n(n − 1)/2 inter-node distances is greatly relevant for the distribution of the RGG. Finding distance distributions is a very challenging task in probabilistic geometry, as it often leads to intractable definite integrals; existing literature focuses on the distance between two nodes or the distances between a node and its neighbours (e.g., see [24]- [27]). We derive the joint distribution of the inter-node distances in closed-form, for n = 3 nodes confined in a disk in R 2 ; to our knowledge, this is the first time such a result is obtained.…”

Frequency hopping on a 5G millimeter-wave uplink

“…For both E 1 and E 2 , the final expression reduces to the evaluation of E X [·], which is the average with respect to the location X of a single mobile. These two averages can be obtained either by Monte Carlo simulation (by simulating the location of a mobile within A 1 \A sec and A 2 , respectively) or by using numerical integration after the distribution of the CDF of the normalized received power is evaluated using the approach provided in [15].…”