  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.
