On computing Bézier curves by Pascal matrix methods

Abstract

The main goal of the paper is to introduce methods which compute Bézier curves faster than Casteljau's method does. These methods are based on the spectral factorization of a $n\times n$ Bernstein matrix, $B^e_n(s)= P_nG_n(s)P_n^{-1}$, where $P_n$ is the $n\times n$ lower triangular Pascal matrix. So we first calculate the exact optimum positive value $t$ in order to transform $P_n$ in a scaled Toeplitz matrix, which is a problem that was partially solved by X. Wang and J. Zhou (2006). Then fast Pascal matrix-vector multiplications and strategies of polynomial evaluation are put together to compute Bézier curves. Nevertheless, when $n$ increases, more precise Pascal matrix-vector multiplications allied to affine transformations of the vectors of coordinates of the control points of the curve are then necessary to stabilize all the computation.