Abstract: A kind of special curves can be constructed in the surface, which is formed with four sequence and non-coplanar points in space. On this basis, the four rational basis functions with control handle are presented, studying on the functional relations between these two parameters and defining a new function set, it can define a kind of spline curve. The spline curve can approximate the control polygon with arbitrary precision through changing the control handle. Additionally, the curve segments of this kind which meet prescribed tangent direction and bounded curvature at any endpoint, are then easy to connect into a approximating spline curve. Finally, this spline is used to construct the surfaces for spatial mesh, which also has many freedoms.

Key words: basis functions, rational spline, curve approximation

摘要： 在空间四个有序数据点所确定的一个二次曲面上，可以构造一类特殊的曲线。给出了四个形状控制因子的有理基函数，以及通过研究其参数间的函数关系定义函数集，构造一类样条曲线，使得通过改变控制因子能任意精确地逼近控制多边形。这类样条曲线端点处满足一定切线方向和有界曲率，容易将它们拼接成一条逼近样条曲线。利用这些样条构造出逼近样条曲面，具有更多的自由度。

关键词: 基函数, 有理样条, 曲线逼近