Random Distances Associated with Arbitrary Triangles: A Systematic Approach between Two Random Points

摘要

It has been known that the distribution of the random distances between twouniformly distributed points within a convex polygon can be obtained based onits chord length distribution (CLD). In this report, we first verify theexisting known CLD for arbitrary triangles, and then derive and verify thedistance distribution between two uniformly distributed points within anarbitrary triangle by simulation. Furthermore, a decomposition and recursionapproach is applied to obtain the random point distance distribution betweentwo arbitrary triangles sharing a side. As a case study, the explicitdistribution functions are derived when two congruent isosceles triangles withthe acute angle equal to $\frac{\pi}{6}$ form a rhombus or a concave 4-gon.