Vandermonde factorizations of a regular Hankel matrix and their application on the computation of Bézier curves

Abstract

In this paper, a new method to compute a Bézier curve of degree n = 2m-1 is introduced, here formulated as a set of points whose coordinates are calculated from two Hankel forms in $\C^m$. From Vandermonde factorizations of the two associated Hankel matrices $H_x$ and $H_y$, the Hankel forms can be easily calculated, thus yielding points on the Bézier curve. Here, a new proof of the existence of a Vandermonde factorization of regular Hankel matrix is given from Pascal matrices techniques. But, even when the Hankel matrix associated to the form is singular, the method can still be used by shifting its skew-diagonal and counteracting it after, which is pratically done without costs.. By comparing this new method with a Pascal matrix method and Casteljau's, we see that the results suggest that this new method is very effective with regard to accuracy and time of computation for various values of n.