conference paper published yes Jean-Daniel Boissonnat author Mathijs Wintraecken author 307CFBC8-F248-11E8-B48F-1D18A9856A870000-0002-7472-2220 HeEd department SoCG: Symposium on Computational Geometry ISTplus - Postdoctoral Fellowships project 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 𝒯 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 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. https://research-explorer.ista.ac.at/download/7952/7969/2020_LIPIcsSoCG_Boissonnat.pdf application/pdfno Schloss Dagstuhl - Leibniz-Zentrum für Informatik2020Zürich, Switzerland eng 1868-8969 978-3-95977-143-610.4230/LIPIcs.SoCG.2020.20 164 https://research-explorer.ista.ac.at/record/9649 Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” <i>36th International Symposium on Computational Geometry</i>, vol. 164, 20:1-20:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.20">10.4230/LIPIcs.SoCG.2020.20</a>. Boissonnat, Jean-Daniel, and Mathijs Wintraecken. “The Topological Correctness of PL-Approximations of Isomanifolds.” In <i>36th International Symposium on Computational Geometry</i>, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.20">https://doi.org/10.4230/LIPIcs.SoCG.2020.20</a>. 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. Boissonnat, J.-D., &#38; Wintraecken, M. (2020). The topological correctness of PL-approximations of isomanifolds. In <i>36th International Symposium on Computational Geometry</i> (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.20">https://doi.org/10.4230/LIPIcs.SoCG.2020.20</a> J.-D. Boissonnat and M. Wintraecken, “The topological correctness of PL-approximations of isomanifolds,” in <i>36th International Symposium on Computational Geometry</i>, Zürich, Switzerland, 2020, vol. 164. J.-D. Boissonnat, M. Wintraecken, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Boissonnat J-D, Wintraecken M. The topological correctness of PL-approximations of isomanifolds. In: <i>36th International Symposium on Computational Geometry</i>. Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2020.20">10.4230/LIPIcs.SoCG.2020.20</a> 79522020-06-09T07:24:11Z2025-04-22T13:45:17Z