Abstract: This paper aims at realizing the conversion from approximation to interpolation using one model. A blending function with one parameter is constructed in the polynomial space. A curve on four-point piecewise scheme is defined based on the blending function. The new curve can be considered as a linear combination of an approximation curve and an interpolation curve which are defined by the same set of control points. The approximation curve is the cubic B-spline curve. The interpolation curve passes through the control points except the first and the last one. The curve is C2 continuous with uniform parameter segmentation and can reach C3 continuity when taking special parameter. In the process of parameter change, the position of the starting point and end point of each curve segment change, while the first and second derivative at these points always remain the same as the cubic B-spline curve. The shape of a curve is closely related with the endpoint conditions, and B-spline curve has good shape preserving property. These factors make the curve can always keep well the characteristic of the control polygon in the process of shape change. By adopting the method of tensor product,the curve is extended to surface. The curve and surface legend shows the effectiveness of the method in modeling design.

Key words: curve and surface design, B-spline method, approximation and interpolation, piecewise combination, shape parameter

摘要： 为了用一种模型实现从逼近到插值的转换，在多项式空间上构造了含一个参数的调配函数，由之定义了基于4点分段的曲线，该曲线可以理解为由相同的一组控制顶点定义的逼近曲线和插值曲线的线性组合，其中的逼近曲线为3次均匀B样条曲线，插值曲线经过除首末点以外的所有控制点。在均匀参数分割下，曲线具有C2连续性，取特殊参数时可达C3连续。在参数变化过程中，曲线各段起点、终点的位置发生改变，但这些点处的一阶、二阶导矢始终保持不变，即始终与3次B样条曲线相同。曲线形状与端点条件密切相关，而B样条曲线具有良好的保形性，这些综合因素使得曲线在形状变化的过程中始终可以较好地保持控制多边形的特征。采用张量积方法将曲线推广至曲面，曲线曲面图例显示了该方法在造型设计中的有效性。

关键词: 曲线曲面设计, B样条方法, 逼近与插值, 分段组合, 形状参数