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 achieved with a relatively small number of low-degree curve segments or surface patches. By using monoid curves and surfaces, the method eliminates the undesirable singularities and “phantom” branches normally associated with implicit representation. The monoids are expressed in exact implicit and parametric equations simultaneously, and upper bounds are derived for the approximate errors of implicitization and inversion equations.