VANDERMONDE FACTORIZATIONS OF A REGULAR HANKEL MATRIX AND THEIR APPLICATION TO THE COMPUTATION OF BEZIER CURVES

In this paper each coordinate of a Bezier curve B(s) of degree (n - 1), n - 2m - 1, is expressed as a Hankel form applied to the vector (B-m(c)(s))(T)e(m), where B-m(c)(s) is the m x m Bernstein matrix and c(m) is the mth canonical vector of C-m. These expressions can be easily calculated once we have a Vandermonde factorization of the Hankel matrices associated to the forms. To that end, we begin presenting another proof of the existence of a Vandermonde factorization of a regular Hankel matrix by using Pascal matrices techniques. In the cases where one or both Hankel matrices associated to the forms are ill-conditioned with respect to inversion, we propose shifting their skew-diagonals and counteracting them after, which is done practically without computational costs. By comparing this way of computing a Bezier curve with other current methods, we see that the results suggest that this approach is very promising with regard to accuracy and time of computation, even for large values of n. Examples of the behavior of this kind of method under degree elevation and degree reduction are also presented here.