Random paths through rectangles and cubes

Abstract The sizes of particles embedded at random in an opaque material can be estimated from the lengths of paths through the particles made by a straight line probe. The length distribution of these paths is known to be related to length distributions of paths resulting from other random mechanisms. These random mechanisms for generating paths are described. When the particles are rectangles embedded in a plane surface, or cubes embedded in three-dimensional Euclidean space, the resulting path length distributions under the different randomnesses are known, but take complicated forms. Recurrence formulae are derived for calculating the moments of these distributions, and the moments are tabulated. Simple relationships between the moments under the different randomnesses are found.