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When Alpha-Complexes Collapse onto Codimension-1 Submanifolds

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

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  • Theory of computation → Computational geometry
Keywords
  • Submanifold reconstruction
  • triangulation
  • abstract simplicial complexes
  • collapses
  • convexity

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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) https://doi.org/10.4230/LIPIcs.SoCG.2025.11

Author Details

Dominique Attali
  • Université Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
Mattéo Clémot
  • Universite Claude Bernard Lyon 1, CNRS, INSA Lyon, LIRIS, Villeurbanne, France
Bianca B. Dornelas
  • Institute of Geometry, TU Graz, Austria
  • Institute for Medical Informatics, Statistics and Documentation, MedUni Graz, Austria
André Lieutier
  • No affiliation, Aix-en-Provence, France

Funding

  • Dornelas, Bianca B.: Funded by the Austrian Science Fund (FWF), grant W1230.

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