In this paper, we address the problem of constructing $C^2$ cubic spline functions on a given arbitrary triangulation $\mathcal{T}$. To this end, we endow every triangle of $\mathcal{T}$ with a Wang-Shi macro-structure. The $C^2$ cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in such space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of $C^2$ cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for $C^2$ joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang-Shi macro-structure is transparent to the user. Stable global bases for the full space of $C^2$ cubics on the Wang-Shi refined triangulation $\mathcal{T}$ are deduced from the local simplex spline basis by extending the concept of minimal determining sets.

翻译：本文研究在给定任意三角剖分$\mathcal{T}$上构造$C^2$三次样条函数的问题。为此，我们在$\mathcal{T}$的每个三角形上赋予Wang-Shi宏结构。在此类细化三角剖分上，$C^2$三次空间具有稳定的维数和最优逼近能力。此外，该空间中的任意样条函数均可通过Hermite插值在每个宏三角形上独立局部构造。我们为单个宏三角形上定义的$C^2$三次空间提供了一个单纯样条基，该基在三角形上表现为Bernstein/B样条基。该基函数继承了单纯样条构造中的递推关系和微分公式，构成非负单位分解，可导出相邻三角形边界的$C^2$拼接条件，并满足类似Marsden恒等式的性质。同时，存在单一控制网以方便对宏三角形上样条函数的控制和早期可视化。凭借这些性质，Wang-Shi宏结构的复杂几何对用户透明。通过扩展最小确定集的概念，我们由局部单纯样条基推导出Wang-Shi细化三角剖分$\mathcal{T}$上$C^2$三次全空间的稳定全局基。