# The Topological Correctness of PL-Approximations of Isomanifolds

1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multi-variate multivalued 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 $T$ of the ambient space $R d$. In this paper, we give conditions under which the PL-approximation of an isomani-fold 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 $T$. 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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Cited literature [55 references]

https://hal.inria.fr/hal-02874720
Contributor : Jean-Daniel Boissonnat <>
Submitted on : Monday, June 22, 2020 - 11:43:36 AM
Last modification on : Tuesday, June 23, 2020 - 3:36:19 AM

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• HAL Id : hal-02874720, version 2

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Jean-Daniel Boissonnat, M Wintraecken. The Topological Correctness of PL-Approximations of Isomanifolds. 2020. ⟨hal-02874720v2⟩

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