Resultants, Implicit Parameterizations, and Intersections of Surfaces

Abstract

A fundamental problem in computer graphics and computer aided design is to convert between a parameterization of a surface and an implicit representation of it. Almost as fundamental is to derive a parameterization for the intersection of two surfaces.

In these problems, it seems that resultants, specifically the Dixon resultant, have been underappreciated. Indeed, several well known papers from ten to twenty years ago reported unsuitability of resultant techniques. To the contrary, we show that the Dixon resultant is an extremely effective and efficient method to compute an implicit representation.

To use resultants to compute a parameterization of an intersection, we introduce the concept of an “implicit parameterization.” Unlike the conventional parameterization of a curve where x, y, and z are each explicitly given as functions of, say, t, we have three implicit functions, one each for (x, t), (y, t), and (z, t). This concept has rarely been mentioned before. We show that given a (conventional) parameterization for one surface and either an implicit equation for the second, or a parameterization for it, it is straightforward to compute an implicit parameterization for the intersection. Doing so is very easy for the Dixon resultant, but can be very daunting even for well respected Gröbner bases programs.

Further, we demonstrate that such implicit parameterizations are useful. We use builtin 3D plotting utilities of a computer algebra system to graph the intersection using our implicit parameterization. We do this for examples that are more complex than the quadric examples usually discussed in intersection papers.