A RATIONAL PARAMETRIZATION OF BÉZIER LIKE CURVES

Abstract

In this paper, we construct a family of Bernstein functions using a class of rational parametrizations. The new family of rational Bernstein basis on an index α ∈ ( − ∞ , 0 ) U ( 1 , + ∞ ) , and for a given degree k ∈ N ∗ , these basis functions are rationals with a numerator and a denominator each of polynomials of degree k. All of the classical properties as positivity, partition of unity hold for these rational Bernstein bases. They constitute approximation basis functions for spaces of continuous functions. The Bézier curves obtained satisfy the classical properties. We have the classical computational algorithms like the de Casteljau algorithm and the algorithm of subdivision with the similar accuracy. Given a degree k and a control polygon points, all of these algorithms converge to the same Bézier curve as the classical case. That means the Bézier curve is independent of the index α. The classical polynomial Bernstein basis seems to be an asymptotic case of our new class of rational Bernstein basis.