The random tangential projection of a surface

Abstract

Suppose a smooth surface S is exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency with S, there is a corresponding point registration in an image plane H placed perpendicular to the beam. The registering rays form a cylindrical surface which registers in H as a smooth curve C (except possibly for a number of cusps). Properties of C when the beam is taken isotropic random in space are investigated, and stereological applications are noted. The phenomenon of screening of projecting rays, which occurs for example with opaque surfaces exposed to light beams, is considered. Limiting processes permit extensions of the formulae to the cases in which S contains curved edges and in which any boundary curves of S also register. Finally, various related types of random tangent to a surface are considered.