一类有理曲线曲面的修改方法及计算机实现

本文首先进一步探讨三次Ｈ－样条曲线的一些性质，推导了曲线权函数性质，得到了位相似定理，证明了曲线保凸的充要条件，给出了曲线二阶几何连续的条件。然后，对三次Ｈ－样条形式的有理曲线曲面，给出了权因子的几何解释，提出了一系列修改曲线曲面形状的算法，并将其应用于各种实例，在计算机得以实现。结果表明，采用有理Ｈ－样条方法设计曲线曲面，便于局部修改，形状容易控制，能使造型达到满意效果。

The Modifying Methods and Implementation with Computer of a Class of Rational Curves and Surfaces

Some characteristics of cubic H-spline curves are further explored in this article. Some properties of its weight functions are derived. A theorem of position similarity is obtained. A necessary and sufficient condition for keeping its convexity is proved. A conditions meeting with the second order geometric continuity are given. These characteristics have important meanings in computer-aided geometric design. Thus geometric explanation of weighing factors is given for a class of rational curves and surfaces in cubic H-spline forms, and a set of algorithms for modifying the shapes of the curves and surfaces are presented with application to some practical examples. The class of curves and surfaces are implemented with computers. The results show that the rational H-spline method has the advantages of convenience in local modification and easiness in shape control. The method can makes the result satisfactory.