On computing Bézier curves by Pascal matrix methods

The main goal of the paper is to introduce methods that compute Bézier curves faster than Casteljau’s method does. These methods are based on the spectral factorization of an n × n   Bernstein matrix, Bne(s)=PnGn(s)Pn-1, where Pn is the n × n lower triangular Pascal matrix. To that end, we first calculate the exact optimum positive value t in order to transform Pn into a scaled Toeplitz matrix (how to do so is a problem that was partially solved by Wang and Zhou (2006) [6]). Then, fast Pascal matrix–vector multiplications are combined with polynomial evaluations to compute Bézier curves. Nevertheless, when n increases, we need more precise Pascal matrix–vector multiplications to achieve stability in the numerical results. We see here that a Pascal matrix–vector product, combined with a polynomial evaluation and some affine transforms of the vectors of coordinates of the control points, will yield a method that can be used to efficiently compute a Bézier curve of degree n, n ⩽ 60.