On computing Bézier curves by Pascal matrix methods

2010/06/22 by Licio H. Bezerra, Bezerra, Licio H., Leonardo Sacht +1 · 1 citation
Computer Science · Engineering · Mathematics · #15A18 #65F15 #68U07 #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #FOS: Mathematics #Numerical Analysis (math.NA) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1006.4327

openalex publication_date 2010/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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× n Bernstein matrix, B^e_n(s)= P_nG_n(s)P_n^-1, where P_n is the n× 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.

Cited by

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

Related

• Fast and Stable Pascal Matrix Algorithms
• A de Casteljau Algorithm for Bernstein type Polynomials based on (p,q)-integers
• Linear-time algorithm for computing the Bernstein-Bézier coefficients of B-spline basis functions
• Computing the Bézier Control Points of the Lagrangian Interpolant in Arbitrary Dimension
• Efficient modified Jacobi-Bernstein basis transformations