3 Search Results for "Vegter, Gert"

Document
DOI: 10.4230/LIPIcs.SoCG.2021.31
Minimal Delaunay Triangulations of Hyperbolic Surfaces

Authors: Matthijs Ebbens, Hugo Parlier, and Gert Vegter

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we show that every hyperbolic surface of genus g has a simplicial Delaunay triangulation with O(g) vertices, where edges are given by distance paths. Then, we construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we show that the Ω(√g) lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus g is tight for hyperbolic surfaces as well.
Cite as

Matthijs Ebbens, Hugo Parlier, and Gert Vegter. Minimal Delaunay Triangulations of Hyperbolic Surfaces. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ebbens_et_al:LIPIcs.SoCG.2021.31,
  author =	{Ebbens, Matthijs and Parlier, Hugo and Vegter, Gert},
  title =	{{Minimal Delaunay Triangulations of Hyperbolic Surfaces}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{31:1--31:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.31},
  URN =		{urn:nbn:de:0030-drops-138305},
  doi =		{10.4230/LIPIcs.SoCG.2021.31},
  annote =	{Keywords: Delaunay triangulations, hyperbolic surfaces, metric graph embeddings, moduli spaces}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.20
Delaunay Triangulations on Orientable Surfaces of Low Genus

Authors: Mikhail Bogdanov, Monique Teillaud, and Gert Vegter

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
Earlier work on Delaunay triangulation of point sets on the 2D flat torus, which is locally isometric to the Euclidean plane, was based on lifting the point set to a locally isometric 9-sheeted covering space of the torus. Under mild conditions the Delaunay triangulation of the lifted point set, consisting of 9 copies of the input set, projects to the Delaunay triangulation of the input set. We improve and generalize this work. First we present a new construction based on an 8-sheeted covering space, which shows that eight copies suffice for the standard flat torus. Then we generalize this construction to the context of compact orientable surfaces of higher genus, which are locally isometric to the hyperbolic plane. We investigate more thoroughly the Bolza surface, homeomorphic to a sphere with two handles, both because it is the hyperbolic surface with lowest genus, and because triangulations on the Bolza surface have applications in various fields such as neuromathematics and cosmological models. While the general properties (existence results of appropriate covering spaces) show similarities with the results for the flat case, explicit constructions and their proofs are much more complex, even in the case of the apparently simple Bolza surface. One of the main reasons is the fact that two hyperbolic translations do not commute in general. To the best of our knowledge, the results in this paper are the first ones of this kind. The interest of our contribution lies not only in the results, but most of all in the construction of covering spaces itself and the study of their properties.
Cite as

Mikhail Bogdanov, Monique Teillaud, and Gert Vegter. Delaunay Triangulations on Orientable Surfaces of Low Genus. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bogdanov_et_al:LIPIcs.SoCG.2016.20,
  author =	{Bogdanov, Mikhail and Teillaud, Monique and Vegter, Gert},
  title =	{{Delaunay Triangulations on Orientable Surfaces of Low Genus}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.20},
  URN =		{urn:nbn:de:0030-drops-59129},
  doi =		{10.4230/LIPIcs.SoCG.2016.20},
  annote =	{Keywords: covering spaces, hyperbolic surfaces, finitely presented groups, Fuchsian groups, systole}
}
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-dev.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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