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

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