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

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 closedform 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. Index Terms Distance distributions; Kinematic Measure; triangles; polygons; ring geometries; wireless communication networks 2 I. PDD WITHIN A TRIANGLE A B C x ! " # (i) $ % & % ! €  A B C x ! " # €  (ii) ! % & % ' ( # (iii) ' ( # % & % ' A B C x ! " # €   a c b Fig. 1: KM over an arbitrary triangle. △ABC is an arbitrary triangle with side lengths |CB| = a, |AC| = b, and |AB| = c, internal angles ∠A = α, ∠B = β, and ∠C = γ, and area ||△ABC|| = S. Without loss of generality (WLOG), a ≥ b ≥ c, and let side CB be on x-axis, as shown in Fig. 1. For simplicity, we can have a = 1 and other edges normalized correspondingly. The Probability Density Function (PDF) of the PDD, denoted as fD(d), can be scaled to any size of triangles with a = s by fsD = 1 s fD( d s ) , (1) where fsD is the corresponding PDF of the PDD within the triangle with a = s. Such a scaling is applicable to any polygons. When calculating the length of the chord produced by a line intersecting with the triangle with orientation θ with regard to x-axis, there are three cases in terms of the range of θ: (i) 0 ≤ θ ≤ γ, (ii) γ ≤ θ ≤ π− β, (iii) π− β ≤ θ ≤ π. We then design a systematic algorithmic procedure for the numerical integration of the intended PDD, based on which the corresponding closed-form expression is also derived, as shown below in detail. Specifically, let θ increase from 0 to π with a fixed small step of δθ (e.g., δθ = π 180 ). For each θ with a set of lines intersecting with the triangle, G is the line which produces the longest chord of length base. The distance between the two tangents parallel with G, i.e., the support lines G1 and G2 which completely encompass the whole triangle, is pm. The distance between G1 and G and that between G2 and G are p1 and p2, respectively. Obviously, pm = p1+ p2. With p increasing from 0 to pm with a fixed small step δp (e.g., δp = 1 1,000 ), we obtain pm δp chords. For each chord of length l calculated based on trigonometry, we obtain fG(d), based on which the PDF of the PDD can be calculated. The derivation is summarized in Fig. 2, providing the regularity to help obtain the PDF symbolically as δθ and δp go to 0.