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——朴素压缩算法，该算法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。朴素压缩算法还有一个无需已知 $\mathcal{M}$ 的实用版本。我们建立了朴素与实用压缩算法均能输出 $\mathcal{M}$ 三角剖分的条件。通过给出 $\alpha$-复形中三角形与 $\mathcal{M}$ 所成角度的界，我们推导出 $P$ 的采样条件，这些条件与现有文献中关于 $\mathbb{R}^3$ 中光滑曲面嵌入的结果具有竞争力，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，在比现有条件更弱的假设下，$P$ 的限制Delaunay复形可对 $\mathcal{M}$ 进行三角剖分。