Exact computation of the CDF of the Euclidean distance between a point   and a random variable uniformly distributed in disks, balls, or polyhedrons   and application to PSHA
  We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this set: a union of disjoint disks, a union of disjoint balls, a union of disjoint line segments, and the boundary of a polyhedron. In the first three cases, we provide closed-form expressions of the cumulative distribution function and the density. In the last case, we propose an algorithm with complexity O(n ln n), n being the number of edges of the polyhedron, that computes exactly the cumulative distribution function. An application of these results to probabilistic seismic hazard analysis and extensions are discussed. 
Vincent Guigues
math.PR