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DOI: 10.4230/LIPIcs.SoCG.2025.11

When Alpha-Complexes Collapse onto Codimension-1 Submanifolds

Authors: Dominique Attali, Mattéo Clémot, Bianca B. Dornelas, and André Lieutier

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

Given a finite set of points P sampling an unknown smooth surface ℳ ⊆ ℝ³, our goal is to triangulate ℳ based solely on P. Assuming ℳ is a smooth orientable submanifold of codimension 1 in 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 ℳ. Naive Squash also has a practical version that does not require knowledge of ℳ. We establish conditions under which both the naive and practical Squash algorithms output a triangulation of ℳ. We provide a bound on the angle formed by triangles in the α-complex with ℳ, yielding sampling conditions on P that are competitive with existing literature for smooth surfaces embedded in ℝ³, while offering a more compartmentalized proof. As a by-product, we obtain that the restricted Delaunay complex of P triangulates ℳ when ℳ is a smooth surface in ℝ³ under weaker conditions than existing ones.

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Dominique Attali, Mattéo Clémot, Bianca B. Dornelas, and André Lieutier. When Alpha-Complexes Collapse onto Codimension-1 Submanifolds. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{attali_et_al:LIPIcs.SoCG.2025.11,
  author =	{Attali, Dominique and Cl\'{e}mot, Matt\'{e}o and Dornelas, Bianca B. and Lieutier, Andr\'{e}},
  title =	{{When Alpha-Complexes Collapse onto Codimension-1 Submanifolds}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.11},
  URN =		{urn:nbn:de:0030-drops-231630},
  doi =		{10.4230/LIPIcs.SoCG.2025.11},
  annote =	{Keywords: Submanifold reconstruction, triangulation, abstract simplicial complexes, collapses, convexity}
}

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Documents authored by Dornelas, Bianca B.

When Alpha-Complexes Collapse onto Codimension-1 Submanifolds

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