Approximate implicitization using monoid curves and surfaces

This paper presents an approach to finding an approximate implicit equation and an approximate inversion map of a planar rational parametric curve or a rational parametric surface. High accuracy of the approximation is achiev ed with a relatively small number of low-degree curve segments or surface p atches. By using monoid curves and surfaces, the method eliminates the unde sirable singularities and "phantom" branches normally associated with impli cit representation. The monoids are expressed in exact implicit and paramet ric equations simultaneously, and upper bounds are derived for the approxim ate errors of implicitization and inversion equations. (C) 1999 Academic Pr ess.