Approximate implicitization of triangular Bézier surfaces

Abstract

We discuss how Dokken's methods of approximate implicitization can be applied to triangular Bézier surfaces in both the original and weak forms. The matrices D and M that are fundamental to the respective forms of approximate implicitization are shown to be constructed essentially by repeated multiplication of polynomials and by matrix multiplication. A numerical approach to weak approximate implicitization is also considered and we show that symmetries within this algorithm can be exploited to reduce the computation time of M. Explicit examples are presented to compare the methods and to demonstrate properties of the approximations.