The Quintic Catmull-Rom Spline with Local Parameters

Aiming at the shortcomings in shape adjustment of the quintic Catmull-Rom spline curve, a quintic Catmull-Rom spline curve with local parameters is proposed. Firstly, the quintic Catmull-Rom spline basis functions with local parameters are constructed, and the corresponding quintic Catmull-Rom spline curve with local parameters is defined, which is called the improved quintic Catmull-Rom spline curve. Then, the method for constructing fair improved quintic Catmull-Rom spline curve using strain energy minimization is presented. Finally, the improved quintic Catmull-Rom spline interpolation function is discussed, the interpolation error of the interpolation function is analyzed, and the method for constructing the optimal interpolation function is given. The results of curve design and function interpolation examples show that the proposed curve retains the main characteristics of the previous quintic Catmull-Rom spline curve, such as interpolation, C² continuity, and shape adjustability, and can use the free parameters to achieve local shape adjustment while keeping the control points unchanged. Compared with the optimized previous quintic Catmull-Rom spline curve, the optimized improved quintic Catmull-Rom spline curve has better fairness. The overall error of the optimal improved quintic Catmull-Rom spline interpolation function is smaller than that of the optimal previous quintic Catmull-Rom spline interpolation function.