Building on a result of U. Reif on removable singularities, we construct $C^1$ bi-3 splines that may include irregular points where less or more than four tensor-product patches meet. The resulting space complements PHT splines, is refinable and the refined spaces are nested, preserving for example surfaces constructed from the splines. As in the regular case, each quadrilateral has four degrees of freedom, each associated with one spline and the splines are linearly independent. Examples of use for surface construction and isogeometric analysis are provided.

中文翻译：

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可优化的C1样条曲线元素，用于不规则四边形布局。

基于U. Reif关于可移动奇点的结果，我们构造 $C^{\mathrm{1个}}$bi-3样条曲线，可能包含不规则点，其中少于或多于四个张量积斑块相遇。生成的空间补充了PHT样条曲线，是可精炼的，并且精炼的空间是嵌套的，例如保留由样条曲线构造的曲面。与常规情况一样，每个四边形都具有四个自由度，每个自由度与一个样条线相关联，并且这些样条线是线性独立的。提供了用于表面构造和等几何分析的示例。