IJPAM: Volume 103, No. 4 (2015)
MULTIVARIATE BERNSTEIN POLYNOMIALS
Faculty of Technology
Narvik University College
P.O. Box 385, 8505 Narvik, NORWAY
Abstract. Ordinary univariate Bernstein polynomials can be represented in matrix form using factor matrices. In this paper we present the definition and basic properties of such factor matrices extended from the univariate case to the general case of arbitrary number of variables by using barycentric coordinates in the hyper-simplices of respective dimension. The main results in the paper are related to the design of an iterative algorithm for fast convex computation of multivariate Bernstein polynomials based on sparse-matrix factorization. In the process of derivation of this algorithm, we investigate some properties of the factorization, including symmetry, commutativity and differentiability of the factor matrices, and address the relevance of this factorization to the de Casteljau algorithm for evaluating curves and surfaces on Bézier form. A set of representative examples is provided, including a geometric interpretation of the de Casteljau algorithm, and representation by factor matrices of multivariate surfaces and their derivatives in Bézier form. Another new result is the observation that inverting the order of steps of a part of the new factorization algorithm provides a new, matrix-based, algebraic representation of a multivariate generalization of a special case of the de Boor-Cox computational algorithm.
Received: June 25, 2015
AMS Subject Classification: 15A23, 15A27, 65D07, 65D17, 65D10, 65F30, 65F50
Key Words and Phrases: Bernstein polynomials, matrix factorization, de Casteljau algorithm
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DOI: 10.12732/ijpam.v103i4.12 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Pages: 749 - 780