Chord-Length Distributions and Related Quantities for Spheroids

Abstract

The chord-length distributions are derived that result when spheroids are randomly traversed by straight lines. The first part of the article applies generally to convex domains in three-dimensional or two-dimensional space; the relationships between the chord-length distributions and their moments for different types of randomness are summarized. Subsequently the chord-length distributions, the point-pair distance distributions, and the geometric reduction factors are derived by a suitable transformation from the distributions for the sphere. All integrals can be resolved and the resulting formulae are valid for both prolate and oblate spheroids. The moments of the chord-length distributions are obtained by the same transformation from those for the sphere. The solutions for ellipses are given in the Appendix and contain Legendre integrals.