Evaluation and Approximate Evaluation of the Multivariate Bernstein-Bézier form on a Regularly Partitioned Simplex

Abstract

Polynomials of the total degree d in m variables have a geometrically intuitive representation in the Bernstein-Be´zier form defined over an m-dimensional simplex. The two algorithms given in this article evaluate the Bernstein-Be´zier form on a large number of points corresponding to a regular partition of the simplicial domain. The first algorithm is an adaptation of isoparametric evaluation. The second is a subdivision algorithm. In contrast to de Casteljau's algorithm, both algorithms have a cost of evaluation per point that is linear in the degree regardless of the number of variables. To demonstrate practicality, implementations of both algorithms on a triangular domain are compared with generic implementations of six algorithms in the literature. © 1994, ACM. All rights reserved.