---
_id: '7952'
abstract:
- lang: eng
text: "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 → ℝ^(d-n). A natural (and efficient) way to approximate
an isomanifold is to consider its Piecewise-Linear (PL) approximation based on
a triangulation \U0001D4AF 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 \U0001D4AF. 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. "
alternative_title:
- LIPIcs
article_number: 20:1-20:18
article_processing_charge: No
author:
- first_name: Jean-Daniel
full_name: Boissonnat, Jean-Daniel
last_name: Boissonnat
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Boissonnat J-D, Wintraecken M. The topological correctness of PL-approximations
of isomanifolds. In: 36th International Symposium on Computational Geometry.
Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:10.4230/LIPIcs.SoCG.2020.20'
apa: 'Boissonnat, J.-D., & Wintraecken, M. (2020). The topological correctness
of PL-approximations of isomanifolds. In 36th International Symposium on Computational
Geometry (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2020.20'
chicago: Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness
of PL-Approximations of Isomanifolds.” In 36th International Symposium on Computational
Geometry, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
https://doi.org/10.4230/LIPIcs.SoCG.2020.20.
ieee: J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL-approximations
of isomanifolds,” in 36th International Symposium on Computational Geometry,
Zürich, Switzerland, 2020, vol. 164.
ista: 'Boissonnat J-D, Wintraecken M. 2020. The topological correctness of PL-approximations
of isomanifolds. 36th International Symposium on Computational Geometry. SoCG:
Symposium on Computational Geometry, LIPIcs, vol. 164, 20:1-20:18.'
mla: Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness
of PL-Approximations of Isomanifolds.” 36th International Symposium on Computational
Geometry, vol. 164, 20:1-20:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2020, doi:10.4230/LIPIcs.SoCG.2020.20.
short: J.-D. Boissonnat, M. Wintraecken, in:, 36th International Symposium on Computational
Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.
conference:
end_date: 2020-06-26
location: Zürich, Switzerland
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2020-06-22
date_created: 2020-06-09T07:24:11Z
date_published: 2020-06-01T00:00:00Z
date_updated: 2023-08-02T06:49:16Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2020.20
ec_funded: 1
file:
- access_level: open_access
checksum: 38cbfa4f5d484d267a35d44d210df044
content_type: application/pdf
creator: dernst
date_created: 2020-06-17T10:13:34Z
date_updated: 2020-07-14T12:48:06Z
file_id: '7969'
file_name: 2020_LIPIcsSoCG_Boissonnat.pdf
file_size: 1009739
relation: main_file
file_date_updated: 2020-07-14T12:48:06Z
has_accepted_license: '1'
intvolume: ' 164'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: 36th International Symposium on Computational Geometry
publication_identifier:
isbn:
- 978-3-95977-143-6
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
related_material:
record:
- id: '9649'
relation: later_version
status: public
scopus_import: '1'
status: public
title: The topological correctness of PL-approximations of isomanifolds
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 164
year: '2020'
...