In this paper, we investigate $C^2$ super-smoothness of the full $C^1$ cubic spline space on a Powell-Sabin refined triangulation, for which a B-spline basis can be constructed. Blossoming is used to identify the $C^2$ smoothness conditions between the functionals of the dual basis. Some of these conditions can be enforced without difficulty on general triangulations. Others are more involved but greatly simplify if the triangulation and its corresponding Powell-Sabin refinement possess certain symmetries. Furthermore, it is shown how the $C^2$ smoothness constraints can be integrated into the spline representation by reducing the set of basis functions. As an application of the super-smooth basis functions, a reduced spline space is introduced that maintains the cubic precision of the full $C^1$ spline space.

翻译：本文研究了在Powell-Sabin细化三角剖分上，完整$C^1$三次样条空间的$C^2$超光滑性质，该空间可构造B样条基。我们采用开花方法识别对偶基泛函间的$C^2$光滑性条件。其中部分条件在一般三角剖分上可直接实施，其余条件则较为复杂，但当三角剖分及其对应的Powell-Sabin细化具有特定对称性时将大幅简化。进一步地，本文展示了如何通过缩减基函数集合，将$C^2$光滑约束整合至样条表示中。作为超光滑基函数的应用，我们引入了一个保持完整$C^1$样条空间三次精度的简约样条空间。