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 this 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.

在本文中，我们解决了在给定的任意三角剖分\({\mathcal {T}}\)上构造\(C^2\)三次样条函数的问题。为此，我们赋予\({\mathcal {T}}\)的每个三角形一个 Wang-Shi 宏观结构。这样一个细化的三角剖分上的\(C^2\)三次空间具有稳定的维度和最优的逼近能力。此外，该空间中的任何样条函数都可以通过 Hermite 插值独立地在每个宏三角形上局部构建。我们为\(C^2\)的空间提供了一个单纯形样条基在单个宏三角形上定义的三次方，其行为类似于三角形上的 Bernstein/B 样条基础。基函数继承了单纯形样条构造的递归关系和微分公式，它们形成了一个非负的统一分区，它们承认\(C^2\)连接相邻三角形边缘的简单条件，并且它们具有 Marsden-like身份。此外，还有一个单一的控制网络，以促进宏三角形上样条函数的控制和早期可视化。由于这些特性，Wang-Shi 宏观结构的复杂几何形状对用户来说是透明的。Wang-Shi 细化三角剖分\({\mathcal {T}}\)上\(C^2\)三次方全空间的稳定全局基通过扩展最小确定集的概念，从局部单纯形样条基推导出来。