The Topological Correctness of PL Approximations of Isomanifolds
Abstract Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$. A natural (and efficient) way to approximate an isomanifo...
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| Autor*in: |
Boissonnat, Jean-Daniel [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2021 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s) 2021 |
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| Übergeordnetes Werk: |
Enthalten in: Foundations of computational mathematics - Springer US, 2001, 22(2021), 4 vom: 13. Juli, Seite 967-1012 |
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| Übergeordnetes Werk: |
volume:22 ; year:2021 ; number:4 ; day:13 ; month:07 ; pages:967-1012 |
| Links: |
|---|
| DOI / URN: |
10.1007/s10208-021-09520-0 |
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OLC2079279181 |
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| 520 | |a Abstract Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$. A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ of the ambient space $${\mathbb {R}}^d$$. 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 and thick triangulation $$\mathcal {T}$$. 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 Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary. | ||
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10.1007/s10208-021-09520-0 doi (DE-627)OLC2079279181 (DE-He213)s10208-021-09520-0-p DE-627 ger DE-627 rakwb eng 510 004 VZ 510 VZ 17,1 ssgn 31.76$jNumerische Mathematik bkl 54.10$jTheoretische Informatik bkl Boissonnat, Jean-Daniel verfasserin aut The Topological Correctness of PL Approximations of Isomanifolds 2021 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2021 Abstract Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$. A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ of the ambient space $${\mathbb {R}}^d$$. 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 and thick triangulation $$\mathcal {T}$$. 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 Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary. Isomanifold Solution manifolds Piecewise-linear approximation Isotopy Fréchet distance Wintraecken, Mathijs aut Enthalten in Foundations of computational mathematics Springer US, 2001 22(2021), 4 vom: 13. Juli, Seite 967-1012 (DE-627)330598139 (DE-600)2050531-0 (DE-576)094479291 1615-3375 nnns volume:22 year:2021 number:4 day:13 month:07 pages:967-1012 https://doi.org/10.1007/s10208-021-09520-0 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT 31.76$jNumerische Mathematik VZ 106408194 (DE-625)106408194 54.10$jTheoretische Informatik VZ 106418815 (DE-625)106418815 AR 22 2021 4 13 07 967-1012 |
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The Topological Correctness of PL Approximations of Isomanifolds |
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The Topological Correctness of PL Approximations of Isomanifolds |
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Boissonnat, Jean-Daniel |
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Foundations of computational mathematics |
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the topological correctness of pl approximations of isomanifolds |
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The Topological Correctness of PL Approximations of Isomanifolds |
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Abstract Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$. A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ of the ambient space $${\mathbb {R}}^d$$. 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 and thick triangulation $$\mathcal {T}$$. 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 Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary. © The Author(s) 2021 |
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Abstract Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$. A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ of the ambient space $${\mathbb {R}}^d$$. 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 and thick triangulation $$\mathcal {T}$$. 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 Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary. © The Author(s) 2021 |
| abstract_unstemmed |
Abstract Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$. A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ of the ambient space $${\mathbb {R}}^d$$. 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 and thick triangulation $$\mathcal {T}$$. 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 Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary. © The Author(s) 2021 |
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The Topological Correctness of PL Approximations of Isomanifolds |
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