The C³ Catmull-Rom parametric spline with local shape adjustability

To endow the four-order Catmull-Rom splines with both high-order continuity and local shape adjustabil-ity, a septic four-order Catmull-Rom parametric spline curve and its corresponding surface are proposed. Firstly, a set of septic Catmull-Rom spline basis functions with adjustable degrees of freedom is construct-ed, and their properties are analyzed. Then, based on the basis functions, the associated septic four-order Catmull-Rom parametric spline curves are defined, their geometric properties are discussed, and a compar-ative analysis with existing four-order Catmull-Rom spline curves is conducted. Finally, the curve  formu-lation is extended to surface case, where septic four-order Catmull-Rom parametric spline surfaces are de-fined and their properties are examined. Curves and surfaces design examples show that the proposed sep-tic four-order Catmull-Rom parametric spline not only retain the interpolation properties of its counter-parts, but also achieves C3 continuity and enables flexible local or global shape adjustment by utilizing the incorporated degrees of freedom, thereby providing an effective approach for constructing interpolation curves and surfaces that combine high-order continuity and shape adjustability.