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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2411.10388 |
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| _version_ | 1866910898041389056 |
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| author | Attali, Dominique Clémot, Mattéo Dornelas, Bianca B. Lieutier, André |
| author_facet | Attali, Dominique Clémot, Mattéo Dornelas, Bianca B. Lieutier, André |
| contents | 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 $α$-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 $α$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | When alpha-complexes collapse onto codimension-1 submanifolds Attali, Dominique Clémot, Mattéo Dornelas, Bianca B. Lieutier, André Computational Geometry 65D18, 57Q05 F.2.2; I.3.5 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 $α$-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 $α$-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. |
| title | When alpha-complexes collapse onto codimension-1 submanifolds |
| topic | Computational Geometry 65D18, 57Q05 F.2.2; I.3.5 |
| url | https://arxiv.org/abs/2411.10388 |