C² Continuous Quintic Cardinal Spline and Catmull-Rom Spline with Shape Factors

In view of the deficiency of the cubic Cardinal spline and Catmull-Rom spline, the C² continuous quintic Cardinal spline and Catmull-Rom spline with shape factors are presented in this paper. First, a class of quitic Cardinal spline basis functions with two shape factors is constructed. Then, the quintic Cardinal spline curves and surfaces with shape factors are defined on base of the proposed basis functions, and the monotonicity-preserving interpolation with the quintic Cardinal spline function is discussed. Finally, the corresponding one dimensional and two dimensional quintic Catmull-Rom spline interpolation functions are studied, 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 spline and Catmull-Rom spline can not only be C² continuous without any conditions, but also can be flexibly adjusted by the shape factors. Satisfactory interpolation results can be obtained by using the optimal quintic Catmull-Rom spline interpolation functions.