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

TL;DR

This paper derives exact formulas and algorithms for the cumulative distribution function of the Euclidean distance between a point and a uniformly distributed random variable in various 3D sets, with applications to seismic hazard analysis.

Contribution

It provides closed-form expressions for certain sets and an efficient algorithm for polyhedrons, advancing the computation of distance distributions in 3D spaces.

Findings

Closed-form CDF and density for disks, balls, and line segments.

An O(n ln n) algorithm for polyhedrons' CDF.

Application to probabilistic seismic hazard analysis.

Abstract

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.