A rational parametrization of B\'ezier like curves

Abstract

In this paper, we construct a family of Bernstein functions using a class of rational parametrization. The new family of rational Bernstein basis on an index $\alpha \in {\left(-\infty \, , \, 0 \right)}\cup {\left(1 \, , \, +\infty\right)}$, and for a given degree $k\in \mathbb{N}^*$, these basis functions are rational with a numerator and a denominator are polynomials of degree k. All of the classical properties as positivity, partition of unity are hold for these rational Bernstein basis and they constitute approximation basis functions for continuous functions spaces. The B\'ezier curves obtained verify the classical properties and we have the classical computational algorithms like the deCasteljau 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 B\'ezier curve as the classical case. That means the B\'ezier curve is independent of the index $\alpha$. The classical polynomial Bernstein basis seems a asymptotic case of our new class of rational Bernstein basis.