Approximate Implicitization of Triangular B\'ezier Surfaces

TL;DR

This paper explores methods for approximate implicitization of triangular Bézier surfaces, focusing on matrix construction, numerical approaches, and computational efficiency improvements, with explicit examples demonstrating the techniques.

Contribution

It extends Dokken's approximate implicitization methods to triangular Bézier surfaces, analyzing matrix construction, symmetry exploitation, and providing practical examples.

Findings

Matrices are constructed via polynomial multiplication and matrix multiplication.

Symmetries can be exploited to reduce computation time.

Explicit examples demonstrate the effectiveness of the methods.

Abstract

We discuss how Dokken's methods of approximate implicitization can be applied to triangular B\'ezier surfaces in both the original and weak forms. The matrices $\mathbf{D}$ and $\mathbf{M}$ 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 $\mathbf{M}.$ Explicit examples are presented to compare the methods and to demonstrate properties of the approximations.