Approximate Implicitization of Triangular Bézier Surfaces
For researchers in computer-aided geometric design, this provides an incremental extension of existing implicitization techniques to a specific surface type.
This paper extends Dokken's approximate implicitization methods to triangular Bézier surfaces, showing that the fundamental matrices can be constructed via polynomial multiplication and matrix multiplication, and exploits symmetries to reduce computation time. Examples compare methods and demonstrate approximation properties.
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 $\mathbf{D}$ 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.