1. The Topological Correctness of PL Approximations of Isomanifolds
- Author
-
Mathijs Wintraecken, Jean-Daniel Boissonnat, Université Côte d'Azur (UCA), Understanding the Shape of Data (DATASHAPE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria), Institute of Science and Technology [Klosterneuburg, Austria] (IST Austria), ANR-19-P3IA-0002,3IA@cote d'azur,3IA Côte d'Azur(2019), European Project: 339025,EC:FP7:ERC,ERC-2013-ADG,GUDHI(2014), Institute of Science and Technology [Austria] (IST Austria), and ANR-19-P3IA-0002,3IA Côte d'Azur,Nice - Interdisciplinary Institute for Artificial Intelligence(2019)
- Subjects
0209 industrial biotechnology ,Generalization ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,Boundary (topology) ,Isotopy ,02 engineering and technology ,0102 computer and information sciences ,Topology ,[INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG] ,01 natural sciences ,Fréchet distance ,020901 industrial engineering & automation ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,Piecewise-linear approximation ,0202 electrical engineering, electronic engineering, information engineering ,Isomanifolds ,0101 mathematics ,PL-approximations ,Mathematics ,Triangulation (topology) ,and phrases PL-approximations ,Zero set ,Multivalued function ,Applied Mathematics ,010102 general mathematics ,Codimension ,Isomanifold ,Ambient space ,Computational Mathematics ,Solution manifolds ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,[MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT] ,Topological correctness ,020201 artificial intelligence & image processing ,Topological conjugacy ,Analysis - 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: 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., LIPIcs, Vol. 164, 36th International Symposium on Computational Geometry (SoCG 2020), pages 20:1-20:18
- Published
- 2021