  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 closed-form 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.
