EFFECTS OF SMOOTHING ON MULTIVARIATE STATISTICAL PROPERTIES OF THE NORMAL TO A ROUGH CURVE OR SURFACE

Much of the practical interest attached to curves and surfaces derives from features of roughness, rather than smoothness. For example, cons iderable attention has been paid to fractal models of curves and surfa ces, for which the notions of a normal and curvature are usually not w ell defined Nevertheless. these quantities are sometimes measurable, b ecause the device for recording a rough surface (such as a stylus or ' 'compass'') adds its own intrinsic smoothness. In this paper we addres s the effect of such smoothing operations on the multivariate statisti cal properties of a normal to the surface. Particular attention is pai d to the validity of commonly assumed unimodal approximations to the d istribution of the normal. It is shown that the actual distribution ma y have more than one mode, although in a range of situations the unimo dal approximation is valid. (C) 1996 Academic Press, Inc.