B\'ezier form of dual bivariate Bernstein polynomials

TL;DR

This paper derives recurrence relations and an efficient algorithm for evaluating Bzier coefficients of dual bivariate Bernstein polynomials, facilitating improved polynomial approximation in computer-aided geometric design.

Contribution

It introduces a novel set of recurrence relations and an efficient computational algorithm for dual bivariate Bernstein polynomial coefficients.

Findings

Derived recurrence relations for Bzier coefficients.

Proposed an efficient algorithm for coefficient evaluation.

Discussed applications in geometric approximation.

Abstract

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining B\'{e}zier form of the $L^2$-solution of the problem of best polynomial approximation of B\'{e}zier curve or surface. In this connection, the B\'{e}zier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the B\'{e}zier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.