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2次有理Bézier曲线的最优参数化

ISSN号：1000-1239
期刊名称：《计算机研究与发展》
时间：0
分类：P391.41[天文地球—地球物理学]
作者机构：[1]浙江大学计算机图象图形研究所,杭州310027, [2]浙江大学CAD&CG;国家重点实验室,杭州310027
相关基金：国家“九七三”重点基础研究发展规划基金项目（2004CB719400）;国家自然科学基金项目（60673031,60333010） The parametric representation occupies an important place in computer aided geometric design, and arc-length parameterization has a wide applied foreground in the areas dealing with geometric problems. So, it is necessary to compute the arc-length parameterization or the approximate arc-length parameterization. A technique of optimal parameterization of the Bezier curves is successfully extended to the case of degree 2 rational Bezier curves, and the function of parametric speed after re-parameterization is of C1 continuity. Our research is supported by the National Basic Research Program of China （No. 2004CB719400） and the National Natural Science Foundation of China （No. 60673031 and No. 60333010）.

关键词：计算机辅助几何设计, 有理Bezier曲线, 最优参数化, 弧长参数, Mobius变换, computer aided geometric design, rational Bezier curves, optimal parameterization, arclength parameterization, Mobius transformation

中文摘要：

把Bezier曲线的最优参数化技术成功地推广到外形设计系统中更为常用的2次有理Bezier曲线场合．新方法能够事先对曲线进行重新参数化，而不需要在计算过程中对非均匀的参数速率采用动态的补偿算法．其关键是巧妙地化简需要求解的高次有理函数积分公式，使得Mobius参数变换公式并不是基于数值解法来得到近似解，而是简单明了地具有解析形式的精确解．Mobius变换能够保持有理Bezier曲线的控制顶点和形状不变，仅仅改变曲线的参数分布情况．优化后的参数速率保持C1连续．新参数速率关于单位速率的偏离量在L2范数下达到最小，即实现了最优参数化，所得到的参数最为接近弧长参数．新方法简单直接，数值实例验证了算法的正确与有效．

英文摘要：

A technique of optimal parameterization of the Bezier curves is successfully extended to the case of degree 2 rational Bezier curves which are frequently used to shape design. Optimal parameterization brings a prior explicit parameterization instead of ＂on-the-fly＂ compensation for nonuniformity of the parametric speed. After making the formulae much simpler, a tractable closed-form solution rather than a numerical solution is obtained, and an appropriate Mobius transformation for degree 2 rational Bezier curves is found by computing the integrals directly. The re-parameterization by Mebius transformation maintains both the same shape and the same control points of rational Bezier curve, only changes the distribution of the parameter. The parametric speed after re-parameterization is C1 continuous. The deviation of parametric speed from unit-speed reaches the minimum with respect to L2 norm, which means the rational optimal parameterization is ＂closest＂ to the arc-length parameterization. The method is simple, convenient and efficacious. A numerical example is given to illustrate the correctness and validity of the algorithm.

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