A rational parametrization of B\'ezier like curves

TL;DR

This paper introduces a new family of rational Bernstein basis functions parameterized by an index, which retain classical properties and computational algorithms, providing a rational generalization of Bezier curves that converges to classical curves.

Contribution

The paper presents a novel rational Bernstein basis parametrized by an index, extending classical Bezier curves with preserved properties and algorithms.

Findings

The new basis functions are rational with polynomial numerator and denominator.

Classical properties like positivity and partition of unity are maintained.

Algorithms like deCasteljau and subdivision converge to the same Bezier curve.

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.