七次PH曲线G^2[C^1]Hermite插值方法

G^2[C^1] Hermite interpolation using septic PH curves

本文讨论七次PH (Pythagorean hodograph)曲线的G^2[C^1] Hermite插值问题. PH曲线是具有有理形式的等距线的一类多项式参数曲线,其弧长可精确计算,因此在CAD (computer aided design)中有着广泛的应用.本文采用平面参数曲线的复数表示形式,根据导矢曲线因式分解得到的多项式因子次数不同,将七次PH曲线分为3类.第1类七次PH曲线都是正则曲线,且其G^2[C^1]Hermite插值方法已经被研究.本文研究另外两类七次PH曲线的构造方法,并指出对于给定的初始条件,存在不超过6条的偶数条第2类七次PH曲线;而第3类七次PH曲线的存在性不仅对初始条件有约束,而且可以通过用户指定一个具有几何意义的实参进行交互构造.本文最后通过实例构造了这两类曲线对六分之一圆弧曲线的逼近.

In this paper,we examine the problem of G^2[C^1]Hermite interpolation using septic Pythagorean hodograph(PH)curves.PH curves are a special class of polynomial parametric curves,which have a polynomial arc length function and rational offsets,and are thus widely used in computer-aided design.According to different factorizations of their first derivative in complex form,septic PH curves are classified into three classes.The curves in the first class are all regular,and their construction under any G^2[C^1]condition has already been studied.Therefore,in this paper we focus on the remaining two classes.The number of septic PH curves in the second class is even and no more than six.The existence of septic PH curves in the third class is dependent on the initial Hermite data,and users may specify a real parameter to determine the resultant curve.In addition,we provide the approximation of arcs with septic PH curves as examples demonstrating the application of our results.