Judging or setting weight steady-state of rational Bézier curves and surfaces

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• 作者：Hong-jie Cai (1) (2)
  Guo-jin Wang (1)
  
• 关键词：65D17 ; 65D18 ; 68U05 ; 68U07 ; rational Bézier curve/surface ; M?bius transformation ; reparameterization ; stable state
• 刊名：Applied Mathematics - A Journal of Chinese Universities
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：29
• 期：4
• 页码：391-398
• 全文大小：301 KB
• 参考文献：1. M S Floater. / Derivatives of rational Bézier curves, Comput Aided Geom Design, 1992, 9: 161-74. CrossRef
  2. G J Wang, T W Sederberg, T Saito. / Partial derivatives of rational Bézier surfaces, Comput Aided Geom Design, 1997, 14: 377-81. CrossRef
  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: 163-74. CrossRef
  4. D Filip, R Magedson, R Markot. / Surface algorithms using bounds on derivatives, Comput Aided Geom Design, 1986, 3: 295-11. CrossRef
  5. A Rockwood. / A generalized scanning technique for display of parametrically defined surfaces, IEEE Comput Graph Appl, 1987, 7: 15-6. CrossRef
  6. J M Zheng. / Minimizing the maximal ratio of weights of a rational Bézier curve, Comput Aided Geom Design, 2005, 22: 275-80. CrossRef
  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: 746-55. CrossRef
  
• 作者单位：Hong-jie Cai (1) (2)
  Guo-jin Wang (1)
  
  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China
  2. Mathematics and Applied Mathematics, Foshan University, Foshan, 528000, China
  
• ISSN：1993-0445

文摘

Many works have investigated the problem of reparameterizing rational Bézier curves or surfaces via M?bius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat 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 M?bius transformation. What’s more the users of computer aided design 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.