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This report presents a new, algorithmic approach to the distributions of thedistance between two points distributed uniformly at random in variouspolygons, based on the extended Kinematic Measure (KM) from integral geometry.We first obtain such random Point Distance Distributions (PDDs) associated witharbitrary 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 mathematicalderivation process, based on which either the closed-form expressions or thealgorithmic results can be obtained. The obtained triangle-PDDs can be utilizedfor modeling and analyzing the wireless communication networks associated withtriangle geometries, such as sensor networks with triangle-shaped clusters andtriangle-shaped cellular systems with highly directional antennas. Furthermore,based on the obtained triangle-PDDs, we then show how to obtain the PDDsassociated with arbitrary polygons through the decomposition and recursionapproach, since any polygons can be triangulated, and any geometry shapes canbe approximated by polygons with a needed precision. Finally, we give the PDDsassociated with ring geometries. The results shown in this report can enrichand expand the theory and application of the probabilistic distance models forthe analysis of wireless communication networks.

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