Estimation of the distribution of ellipsoids with random test lines.

Abstract

The distribution F(lambda) of the length lambda of chords delivered by a random test line of unit length, which intersects a sufficiently large number of ellipsoids of different sizes but of uniform shape randomly oriented and dispersed in a three-dimensional space of unit volume, is give by (formula: see text). In this equation, phi(e) is a coefficient determined by two eccentricities of the ellipsoids and N(r) is the distribution function of their major semi-axes r. In the case of spheres phi(e) is 1, but otherwise it is always smaller than 1. N(r) can be further expressed as the product of the probability distribution p(r) of r and the total number Nvo of the ellipsoids in a unit volume. On account of the above equation, it is possible to extend the application of the histometrical methods of spheres with the use of chord length immediately to a group of ellipsoids of uniform shape, so far as the estimation of the parameters of p(r) is concerned. Even with a group of ellipsoids of different shapes, the same principle is valid, provided that all the subgroups of ellipsoids of different eccentricities have a common p(r). Nvo is most practically estimated from Nvo=Tv(h)/2h, in which Tv(h) is the number of tangential points made by ellipsoids with a test plane of unit surface area during its transit h in the direction perpendicular to itself.