2次有理Bézier曲线的最优参数化

Optimal Parameterizations of the Degree 2 Rational Bézier Curves

把Bzier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Bzier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Mbius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.Mbius变换能够保持有理Bzier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.

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.