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arXiv · 1006.4327

On computing B\'ezier curves by Pascal matrix methods

Licio H. Bezerra·Leonardo K. Sacht
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Abstract

The main goal of the paper is to introduce methods which compute B\'ezier 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 B\'ezier 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.

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Licio H. Bezerra, Leonardo K. Sacht. 2010-06-22. On computing B\'ezier curves by Pascal matrix methods. https://arxiv.org/abs/1006.4327

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