The topological correctness of PL approximations of isomanifolds

Boissonnat J-D, Wintraecken M. 2021. The topological correctness of PL approximations of isomanifolds. Foundations of Computational Mathematics .

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Author
Boissonnat, Jean-Daniel; Wintraecken, MathijsISTA
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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 vector-valued smooth function f : Rd → Rd−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 Rd. 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 Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.
Publishing Year
Date Published
2021-07-13
Journal Title
Foundations of Computational Mathematics
Acknowledgement
First and foremost, we acknowledge Siargey Kachanovich for discussions. We thank Herbert Edelsbrunner and all members of his group, all former and current members of the Datashape team (formerly known as Geometrica), and André Lieutier for encouragement. We further thank the reviewers of Foundations of Computational Mathematics and the reviewers and program committee of the Symposium on Computational Geometry for their feedback, which improved the exposition. This work was funded by the European Research Council under the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric Understanding in Higher Dimensions). This work was also supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002. Mathijs Wintraecken also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 754411.
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Boissonnat J-D, Wintraecken M. The topological correctness of PL approximations of isomanifolds. Foundations of Computational Mathematics . 2021. doi:10.1007/s10208-021-09520-0
Boissonnat, J.-D., & Wintraecken, M. (2021). The topological correctness of PL approximations of isomanifolds. Foundations of Computational Mathematics . Springer Nature. https://doi.org/10.1007/s10208-021-09520-0
Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL Approximations of Isomanifolds.” Foundations of Computational Mathematics . Springer Nature, 2021. https://doi.org/10.1007/s10208-021-09520-0.
J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL approximations of isomanifolds,” Foundations of Computational Mathematics . Springer Nature, 2021.
Boissonnat J-D, Wintraecken M. 2021. The topological correctness of PL approximations of isomanifolds. Foundations of Computational Mathematics .
Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL Approximations of Isomanifolds.” Foundations of Computational Mathematics , Springer Nature, 2021, doi:10.1007/s10208-021-09520-0.
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