License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.20
URN: urn:nbn:de:0030-drops-121787
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12178/
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Boissonnat, Jean-Daniel ; Wintraecken, Mathijs

The Topological Correctness of PL-Approximations of Isomanifolds

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LIPIcs-SoCG-2020-20.pdf (1.0 MB)


Abstract

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d β†’ ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the FrΓ©chet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.

BibTeX - Entry

@InProceedings{boissonnat_et_al:LIPIcs:2020:12178,
  author =	{Jean-Daniel Boissonnat and Mathijs Wintraecken},
  title =	{{The Topological Correctness of PL-Approximations of Isomanifolds}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12178},
  URN =		{urn:nbn:de:0030-drops-121787},
  doi =		{10.4230/LIPIcs.SoCG.2020.20},
  annote =	{Keywords: PL-approximations, isomanifolds, solution manifolds, topological correctness}
}

Keywords: PL-approximations, isomanifolds, solution manifolds, topological correctness
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020


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