平面三次PH曲线偶C1Hermite插值解的构造和形状分析

一般情况下,三次PH曲线偶的C^1 Hemite插值问题有四个不同的解。在这四个解中,只有一条曲线能很好地满足几何设计的要求。已有的插值算法都是依赖于构造出所有四个解,利用绝对旋转指标或弹性弯曲能量来找出这条“好”的插值曲线。本文提出一种新的方法以区分这些解,即用由三次PH曲线偶和惟一经典三次插值曲线的速端曲线形成的闭环的弯曲数来区分。对于“合理”的Hemite数据,本文还给出了不需计算和比较所有的四个解便可直接构造“好”的三次PH曲线偶的方法。

CONSTRUCTION AND SHAPE ANALYSIS OF THE COUPLE OF PLANAR PH CUBIC C1 HERMITE INTERPOLATION

In general,the problem of interpolating given first-order Hermite data(end points and derivatives) by the couple of planar Pythagorean-hodograph(PH) cubic curves has four distinct formal solutions.Ordinarily,only one of these interpolants is of acceptable shape.Previous interpolation algorithms have relied on explicitly constructing all four solutions,and invoking a suitable measure of shape,the absolute rotation index or elastic bending energy-to select the"good"interpolant.A new means to differentiate among the solutions is introduced here,namely,the winding number of the closed loop formed by a union of the hodographs of the couple of PH cubics and of the unique "ordinary" cubic interpolant.It also shows that,for"reasonable"Hermite data,the "good" couple of PH cubics can be directly constructed,obviating the need to compute and compare all four solutions.