一类五次OR曲线的构造方法

Construction of a class of quintic curves with rational offsets

本文讨论非PH(Pythagorean hodograph)曲线的一类五次OR曲线(curves with rational offsets)的构造方法.OR曲线是具有有理形式的等距线的一类参数曲线,在CAD中有着广泛的应用.本文采用参数曲线的复数表示形式,根据导矢曲线的因式分解的不同,将非PH曲线的五次OR曲线分为两种类型,并分别给出这两类曲线的构造方法.在给定C1连续的初始条件下,通过指定一个自由实参数来确定构造曲线.本文进一步阐述了这个自由实参数的几何意义.由于五次OR曲线是非正则的曲线,对于第一类曲线,奇异点可以在构造过程中显式地被指定,因此可以有效地避免其在特定曲线段上的出现;而对于第二类曲线,奇异点在曲线中的位置则不易被直接控制.

In this paper, a method for constructing a class of quintic curves with rational offsets is presented.Although this class of planar curves does not include PH curves, such curves are widely applied in CAD because they have rational offsets. A complex variate model is employed to deduce the proposed method. The quintic OR curves are first classified into two classes according to the different factorization of their hodographs, and are then discussed. With the given C1 Hermite data, the curves are determined by specifying a real parameter. This real parameter can be used to adjust the shape of the constructed curves, and affects the parameter values of the cusps.