平面三次参数曲线的分类和拐点奇点分布定理

THE KIND OF PLANAR CUBIC PARAMETRIC CURVES AND THE DISTRIBUTIVE THEORY FOR POINT OF INFLECTION AND POINT OF SINGULARITY

讨论平面三次参数曲线的分类。对四类曲线的拐点或奇点的位置作了全面详尽的分析研究,提出用几个λμ判别函数值正负号的同异来判别拐点奇点位置的新方法。根据各种情况的λμ值范围和有关的边界点或其附近点计算列出了四类曲线的拐点奇点位置分布表。利用分布表很容易设计出所需要的三次参数曲线,能保证曲线是指定的类型并具有指定的拐点或奇点位置。为便于应用提出了设计步骤和计算实例。

This article presents the kind of planar culic parametric cvsves andanalyses the distrifution for point of inflection and the point of singularity attentive-ly. The author found a new method to ascertain the location of point of inflectionand the point of singularity by the sign of value of some discriminativefunction. The λ_μ- plane divides into twenty- seven districts and fourteen?egments dy means of this method. It represents forty-one distrifutive states of thespecial point of four kinds of cusves. The author found four tables. Eyerv tablepresents distributive state of the special point of one hind of curve. It is easyto design a planar cubic parametric curve that satisfies the designative kind ofcurve and designative location of the special point, if we use these tables. Thereare two exsamples in the end of this article.