Vandermonde factorizations of a regular Hankel matrix and their application on the computation of B\'ezier curves

TL;DR

This paper introduces a novel Vandermonde factorization-based method for efficiently computing Bézier curves of degree 2m-1 using Hankel matrices, offering improved accuracy and computational speed.

Contribution

The paper presents a new proof of Vandermonde factorization for regular Hankel matrices and extends the method to singular cases, enhancing Bézier curve computation techniques.

Findings

Method is accurate and fast for various degrees n.

Effective even when Hankel matrices are singular.

Outperforms Pascal matrix and Casteljau's methods in tests.

Abstract

In this paper, a new method to compute a B\'ezier 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 B\'ezier 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.