3 Search Results for "Dyer, Ramsay"

Document
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.9
Local Criteria for Triangulation of Manifolds

Authors: Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, and Mathijs Wintraecken

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.
Cite as

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, and Mathijs Wintraecken. Local Criteria for Triangulation of Manifolds. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{boissonnat_et_al:LIPIcs.SoCG.2018.9,
  author =	{Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Wintraecken, Mathijs},
  title =	{{Local Criteria for Triangulation of Manifolds}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.9},
  URN =		{urn:nbn:de:0030-drops-87224},
  doi =		{10.4230/LIPIcs.SoCG.2018.9},
  annote =	{Keywords: manifold, simplicial complex, homeomorphism, triangulation}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.255
Riemannian Simplices and Triangulations

Authors: Ramsay Dyer, Gert Vegter, and Mathijs Wintraecken

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of the standard simplex under this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.
Cite as

Ramsay Dyer, Gert Vegter, and Mathijs Wintraecken. Riemannian Simplices and Triangulations. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 255-269, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dyer_et_al:LIPIcs.SOCG.2015.255,
  author =	{Dyer, Ramsay and Vegter, Gert and Wintraecken, Mathijs},
  title =	{{Riemannian Simplices and Triangulations}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{255--269},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.255},
  URN =		{urn:nbn:de:0030-drops-51361},
  doi =		{10.4230/LIPIcs.SOCG.2015.255},
  annote =	{Keywords: Karcher means, barycentric coordinates, triangulation, Riemannian manifold, Riemannian simplices}
}
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