Finite Representations of Real Parametric Curves and Surfaces

Global parameterizations of parametric algebraic curves or surfaces are defined over infinite parameter domains. In this paper we show how to replace these global real parameterizations with a finite number of alternate bounded parameterizations, each defined over a fixed, bounded part of the real parameter domain space. The new bounded parameterizations together generate all real points of the old one and in particular the points corresponding to infinite parameter values in the old domain. We term such an alternate finite set of bounded parameterizations a finite representation of a real parametric curve or surface. Two solutions are presented for real parametric varieties of arbitrary dimension n. In the first method, a real parametric variety of dimension n is finitely represented in a piecewise fashion by 2" bounded parameterizations with individual pieces meeting with Coo continuity; each bounded parameterization is a map from a unit simplex of the real parameter domain space. In the second method, only a single bounded parameterization is used; it is a map from the unit hypersphere centered at the origin of the real parameter domain space. Both methods start with an arbitrary real parameterization of a real parametric variety and apply projective domain transformations of different types to yield the new bounded parameterizations. Both these methods are implementable in a straightforward fashion. Applications of these results include displaying entire real parametric curves and surfaces, computing nonnal parameterizations of curves and surfaces (settling an open problem for quadric surfaces), and exactly representing an entire real parametric curve or surface in piecewise Bernstein-Bezier form.