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Judging or setting weight steady-state of rational Bézier curves and surfaces. (English) Zbl 1324.65023

Summary: Many works have investigated the problem of reparameterizing rational Bézier curves or surfaces via Möbius transformation to adjust their parametric distributions as well as weights, such that the maximal ratio of weights becomes smaller, that some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after Möbius transformation. What’s more the users of computer aided design (CAD) softwares may require some guidelines for designing rational Bézier curves or surfaces with the smallest ratio of weights. In this paper, we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational Bézier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal parametric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational Bézier surfaces with compact derivative bounds.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
68U07 Computer science aspects of computer-aided design
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References:

[1] M S Floater. Derivatives of rational Bézier curves, Comput Aided Geom Design, 1992, 9: 161–174. · Zbl 0760.65013 · doi:10.1016/0167-8396(92)90014-G
[2] G J Wang, T W Sederberg, T Saito. Partial derivatives of rational Bézier surfaces, Comput Aided Geom Design, 1997, 14: 377–381. · Zbl 0906.68164 · doi:10.1016/S0167-8396(96)00033-7
[3] G J Wang, C L Tai. On the convergence of hybrid polynomial approximation to higher derivatives of rational curves, J Comput Appl Math, 2008, 214: 163–174. · Zbl 1138.65017 · doi:10.1016/j.cam.2007.02.018
[4] D Filip, R Magedson, R Markot. Surface algorithms using bounds on derivatives, Comput Aided Geom Design, 1986, 3: 295–311. · Zbl 0632.65013 · doi:10.1016/0167-8396(86)90005-1
[5] A Rockwood. A generalized scanning technique for display of parametrically defined surfaces, IEEE Comput Graph Appl, 1987, 7: 15–26. · doi:10.1109/MCG.1987.276916
[6] J M Zheng. Minimizing the maximal ratio of weights of a rational Bézier curve, Comput Aided Geom Design, 2005, 22: 275–280. · Zbl 1090.65021 · doi:10.1016/j.cagd.2004.12.003
[7] H J Cai, G J Wang. Minimizing the maximal ratio of weights of rational Bézier curves and surfaces, Comput Aided Geom Design, 2010, 27: 746–755. · Zbl 1210.65035 · doi:10.1016/j.cagd.2010.08.001
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