Given a finite set of points $P$ sampling an unknown smooth surface $\mathcal{M} \subseteq \mathbb{R}^3$, our goal is to triangulate $\mathcal{M}$ based solely on $P$. Assuming $\mathcal{M}$ is a smooth orientable submanifold of codimension 1 in $\mathbb{R}^d$, we introduce a simple algorithm, Naive Squash, which simplifies the $\alpha$-complex of $P$ by repeatedly applying a new type of collapse called vertical relative to $\mathcal{M}$. Naive Squash also has a practical version that does not require knowledge of $\mathcal{M}$. We establish conditions under which both the naive and practical Squash algorithms output a triangulation of $\mathcal{M}$. We provide a bound on the angle formed by triangles in the $\alpha$-complex with $\mathcal{M}$, yielding sampling conditions on $P$ that are competitive with existing literature for smooth surfaces embedded in $\mathbb{R}^3$, while offering a more compartmentalized proof. As a by-product, we obtain that the restricted Delaunay complex of $P$ triangulates $\mathcal{M}$ when $\mathcal{M}$ is a smooth surface in $\mathbb{R}^3$ under weaker conditions than existing ones.

翻译：给定采样未知光滑曲面$\mathcal{M} \subseteq \mathbb{R}^3$的有限点集$P$，我们的目标仅基于$P$对$\mathcal{M}$进行三角剖分。假设$\mathcal{M}$是$\mathbb{R}^d$中余维1的光滑可定向子流形，我们提出一种简单算法Naive Squash，通过反复应用一种相对于$\mathcal{M}$的新型坍缩操作（称为垂直坍缩）来简化$P$的$\alpha$-复形。Naive Squash还具有一个无需已知$\mathcal{M}$的实用版本。我们建立了保证朴素版与实用版Squash算法均能输出$\mathcal{M}$三角剖分的条件。通过给出$\alpha$-复形中三角形与$\mathcal{M}$所成夹角的界，我们推导出$P$的采样条件，这些条件与现有关于$\mathbb{R}^3$中光滑曲面嵌入的文献结果具有竞争力，同时提供了更具模块化的证明。作为副产品，我们得到当$\mathcal{M}$是$\mathbb{R}^3$中的光滑曲面时，$P$的限制Delaunay复形可在比现有条件更弱的条件下对$\mathcal{M}$进行三角剖分。