Algorithms and a Library for the Exact Computation of the Cumulative Distribution Function of the Euclidean Distance Between a Point and a Random Variable Uniformly Distributed in Disks, Balls, or Polygones and Application to Probabilistic Seismic Hazard Analysis

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 two algorithms with complexity [Formula: see text], [Formula: see text] 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. Finally, we present an open source library, available at https://github.com/vguigues/Areas_Library , that implements the algorithms presented in this paper.