Geometrically smooth spline surfaces, generalized to include $n$-sided facets or configurations of $n\ne 4$ quads, can exhibit a curious lack of additional degrees of freedom for modeling or engineering analysis when refined.

This paper establishes a minimal polynomial degree for smooth constructions of multi-sided surfaces that guarantees more flexibility in all directions under refinement. Degree bi-4 is both necessary and sufficient for flexibility-increasing $G^1$-refinability within a bi-quadratic $C^1$ spline complex. Sufficiency is proven by two alternative flexibly $G^1$-refinable constructions exhibiting good highlight line distributions.

中文翻译：

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适合包含在 C 1 bi-2 样条中的最小 G 1 级可精炼多边表面。

几何光滑的样条曲面，概括为包括 $n$的侧面或构型 $n\ne 4$ 四边形，在细化时可能会表现出奇怪的缺乏用于建模或工程分析的额外自由度。

本文为多边曲面的平滑构造建立了最小多项式次数，以保证在细化下在所有方向上都具有更大的灵活性。度 bi-4 是增加灵活性的必要和充分条件 $G^1$- 二次二次方程中的可精炼性 $C^1$样条复合体。充分性由两个备选方案灵活证明$G^1$- 可精炼的结构表现出良好的高光线分布。