Starting from a general B-spline representation for $C^1$ cubic Powell–Sabin splines on arbitrary triangulations, we focus on the construction of a B-spline representation for a particular subspace defined on three-directional triangulations with $C^2$ super-smoothness over each of the macro-triangles. We analyze the properties of the basis and point out the relation with simplex splines. Furthermore, we provide explicit expressions for the B-spline coefficients of any element of the spline space, and derive subdivision rules under dyadic refinement. Finally, we show simple conditions ensuring global $C^2$ smoothness on the domain.

中文翻译：

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三方向三角剖分上的超光滑三次Powell-Sabin样条：B样条表示和细分

从一般的B样条表示开始 $C^{\mathrm{1个}}$ 关于任意三角剖分的三次Powell–Sabin样条曲线，我们专注于针对在三向三角剖分中定义的特定子空间的B样条表示的构造 $C^2$每个宏三角形的超平滑度。我们分析了基础的属性，并指出了与单纯形样条曲线的关系。此外，我们为样条空间中任何元素的B样条系数提供了显式表达式，并在二进式细化条件下得出细分规则。最后，我们展示了简单的条件来确保全球$C^2$ 域上的平滑度。