Constrained multi-degree reduction of triangular Bézier surfaces using dual Bernstein polynomials

Abstract

We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is $\mathcal{O}\left(n^2{m}^2\right)$, $n$ and $m$ being the degrees of the input and output Bézier surfaces, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global $C^r$ continuity with a prescribed order $r$. Some illustrative examples are given.