The random tangential projection of a surface

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Abstract
Suppose a smooth surfaceSis exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency withS, there is a corresponding point registration in an image planeHplaced perpendicular to the beam. The registering rays form a cylindrical surface which registers inHas a smooth curveC(except possibly for a number of cusps). Properties ofCwhen 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 whichScontains curved edges and in which any boundary curves ofSalso register. Finally, various related types of random tangent to a surface are considered.

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