集逼近插值于一体的分段3次多项式曲线曲面

为了用一种模型实现逼近与插值的统一，在多项式函数空间上构造了含两组参数的混合函数，并由之定义了基于四点分段的多项式曲线和相应的张量积曲面。当参数取特殊值时，新曲线曲面成为3次均匀B样条曲线曲面。除了继承B样条方法的局部性，自动光滑性等优点之外，新曲线曲面还具有局部形状可调性。限制混合函数中参数的取值范围，可以使新曲线曲面位于控制顶点的凸包内。让混合函数中的一组参数取特定值，可以使新曲线曲面自动插值除边界点以外的控制顶点，且插值曲线曲面的形状依然局部可调。给出了一些曲线曲面图例。

Piecewise cubic polynomial curve and surface combining approximation and interpolation into one

In order to use one model to realize the unification of the approximation and interpolation, a kind of blending function, which contains two groups of parameters, is constructed in the polynomial function space. Based on the blending function, a kind of polynomial curve on four-point piecewise scheme and the corresponding tensor product surface are defined. When taking special parameters, the new curve and surface become the cubic uniform B-spline curve and surface. In addition to the locality and automatic smoothness, the new curve and surface also have local shape adjustability. Limiting the value range of the parameters of the blending function, the new curve and surface can be located within the convex hull of the control points. Letting one set of the parameters take a specific value, the new curve and surface can pass through the control points except the boundary points. The shape of the interpolation curve and surface is still local adjustable. Some curves and surfaces illustrations are given.