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...
Ausführliche Beschreibung
Autor*in: |
Boissonnat, Jean-Daniel [verfasserIn] |
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E-Artikel |
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Sprache: |
Englisch |
Erschienen: |
2021 |
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Anmerkung: |
© The Author(s) 2021 |
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Übergeordnetes Werk: |
Enthalten in: Foundations of Computational Mathematics - Springer-Verlag, 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 |
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DOI / URN: |
10.1007/s10208-021-09520-0 |
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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)SPR047755075 (SPR)s10208-021-09520-0-e DE-627 ger DE-627 rakwb eng Boissonnat, Jean-Daniel verfasserin aut The Topological Correctness of PL Approximations of Isomanifolds 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Solution manifolds (dpeaa)DE-He213 Piecewise-linear approximation (dpeaa)DE-He213 Isotopy (dpeaa)DE-He213 Fréchet distance (dpeaa)DE-He213 Wintraecken, Mathijs aut Enthalten in Foundations of Computational Mathematics Springer-Verlag, 2001 22(2021), 4 vom: 13. Juli, Seite 967-1012 (DE-627)SPR009133062 nnns volume:22 year:2021 number:4 day:13 month:07 pages:967-1012 https://dx.doi.org/10.1007/s10208-021-09520-0 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER AR 22 2021 4 13 07 967-1012 |
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10.1007/s10208-021-09520-0 doi (DE-627)SPR047755075 (SPR)s10208-021-09520-0-e DE-627 ger DE-627 rakwb eng Boissonnat, Jean-Daniel verfasserin aut The Topological Correctness of PL Approximations of Isomanifolds 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Solution manifolds (dpeaa)DE-He213 Piecewise-linear approximation (dpeaa)DE-He213 Isotopy (dpeaa)DE-He213 Fréchet distance (dpeaa)DE-He213 Wintraecken, Mathijs aut Enthalten in Foundations of Computational Mathematics Springer-Verlag, 2001 22(2021), 4 vom: 13. Juli, Seite 967-1012 (DE-627)SPR009133062 nnns volume:22 year:2021 number:4 day:13 month:07 pages:967-1012 https://dx.doi.org/10.1007/s10208-021-09520-0 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER AR 22 2021 4 13 07 967-1012 |
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10.1007/s10208-021-09520-0 doi (DE-627)SPR047755075 (SPR)s10208-021-09520-0-e DE-627 ger DE-627 rakwb eng Boissonnat, Jean-Daniel verfasserin aut The Topological Correctness of PL Approximations of Isomanifolds 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Solution manifolds (dpeaa)DE-He213 Piecewise-linear approximation (dpeaa)DE-He213 Isotopy (dpeaa)DE-He213 Fréchet distance (dpeaa)DE-He213 Wintraecken, Mathijs aut Enthalten in Foundations of Computational Mathematics Springer-Verlag, 2001 22(2021), 4 vom: 13. Juli, Seite 967-1012 (DE-627)SPR009133062 nnns volume:22 year:2021 number:4 day:13 month:07 pages:967-1012 https://dx.doi.org/10.1007/s10208-021-09520-0 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER AR 22 2021 4 13 07 967-1012 |
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10.1007/s10208-021-09520-0 doi (DE-627)SPR047755075 (SPR)s10208-021-09520-0-e DE-627 ger DE-627 rakwb eng Boissonnat, Jean-Daniel verfasserin aut The Topological Correctness of PL Approximations of Isomanifolds 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Solution manifolds (dpeaa)DE-He213 Piecewise-linear approximation (dpeaa)DE-He213 Isotopy (dpeaa)DE-He213 Fréchet distance (dpeaa)DE-He213 Wintraecken, Mathijs aut Enthalten in Foundations of Computational Mathematics Springer-Verlag, 2001 22(2021), 4 vom: 13. Juli, Seite 967-1012 (DE-627)SPR009133062 nnns volume:22 year:2021 number:4 day:13 month:07 pages:967-1012 https://dx.doi.org/10.1007/s10208-021-09520-0 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER AR 22 2021 4 13 07 967-1012 |
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10.1007/s10208-021-09520-0 doi (DE-627)SPR047755075 (SPR)s10208-021-09520-0-e DE-627 ger DE-627 rakwb eng Boissonnat, Jean-Daniel verfasserin aut The Topological Correctness of PL Approximations of Isomanifolds 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr 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 (dpeaa)DE-He213 Solution manifolds (dpeaa)DE-He213 Piecewise-linear approximation (dpeaa)DE-He213 Isotopy (dpeaa)DE-He213 Fréchet distance (dpeaa)DE-He213 Wintraecken, Mathijs aut Enthalten in Foundations of Computational Mathematics Springer-Verlag, 2001 22(2021), 4 vom: 13. Juli, Seite 967-1012 (DE-627)SPR009133062 nnns volume:22 year:2021 number:4 day:13 month:07 pages:967-1012 https://dx.doi.org/10.1007/s10208-021-09520-0 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER AR 22 2021 4 13 07 967-1012 |
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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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doi_str |
10.1007/s10208-021-09520-0 |
up_date |
2024-07-03T14:46:55.079Z |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR047755075</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230509104603.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">220804s2021 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10208-021-09520-0</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR047755075</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s10208-021-09520-0-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Boissonnat, Jean-Daniel</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="4"><subfield code="a">The Topological Correctness of PL Approximations of Isomanifolds</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2021</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s) 2021</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="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. 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