中文摘要：

针对三次Cardinal样条与Catmull-Rom样条的不足,提出带形状因子的C2连续五次Cardinal样条与Catmull-Rom样条.首先构造一组带2个形状因子的五次Cardinal样条基函数;然后基于该组基函数定义带形状因子的五次Cardinal样条曲线与曲面,并讨论五次Cardinal样条函数的保单调插值;最后研究对应的一元与二元五次Catmull-Rom样条插值函数,并给出最优一元与二元五次Catmull-Rom样条插值函数的确定方法.实例结果表明,五次Cardinal样条与Catmull-Rom样条无需任何条件即可达到C2连续,且其形状还可通过自带的形状因子进行灵活地调整,利用最优五次Catmull-Rom样条插值函数可获得满意的插值效果.

英文摘要：

In view of the deficiency of the cubic Cardinal spline and Catmull-Rom spline, the C2 continuousquintic Cardinal spline and Catmull-Rom spline with shape factors are presented in this paper. First, a classof quitic Cardinal spline basis functions with two shape factors is constructed. Then, the quintic Cardinalspline curves and surfaces with shape factors are defined on base of the proposed basis functions, and themonotonicity-preserving interpolation with the quintic Cardinal spline function is discussed. Finally, thecorresponding one dimensional and two dimensional quintic Catmull-Rom spline interpolation functions arestudied, and the method of determining the optimal one dimensional and two dimensional quintic Catmull-Rom spline interpolation functions are given. Example results show that, the quintic Cardinal splineand Catmull-Rom spline can not only be C2 continuous without any conditions, but also can be flexibly adjustedby the shape factors. Satisfactory interpolation results can be obtained by using the optimal quinticCatmull-Rom spline interpolation functions.