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

Applied Mathematics(A Journal of Chinese Universities) 2014年第4期29卷 391-398页

作者： CAI Hong-jie WANG Guo-jin Department of Mathematics Zhejiang University Mathematics and Applied Mathematics Foshan University

Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius 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 MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier 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 B6zier 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 para- metric 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 B6zier surfaces with compact derivative bounds.

关键词： rational Bezier curve/surface Mobius transformation reparameterization stable state.

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