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An Elementary and Quantified Version of Whitney's Method

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An Elementary and Quantified Version of Whitney's Method Triangulating submanifolds: An elementary and quantified version of Whitney’s method Jean-Daniel Boissonnat, Siargey Kachanovich, Mathijs Wintraecken To cite this version: Jean-Daniel Boissonnat, Siargey Kachanovich, Mathijs Wintraecken. Triangulating submanifolds: An elementary and quantified version of Whitney’s method. 2018. hal-01950149 HAL Id: hal-01950149 https://hal.inria.fr/hal-01950149 Preprint submitted on 10 Dec 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Triangulating submanifolds: An elementary and quantified version of Whitney’s method Jean-Daniel Boissonnat Université Côte d’Azur, INRIA [Sophia-Antipolis, France] [email protected] Siargey Kachanovich Université Côte d’Azur, INRIA [Sophia-Antipolis, France] [email protected] Mathijs Wintraecken Université Côte d’Azur, INRIA [Sophia-Antipolis, France] [email protected] 1 Abstract 2 2 We quantize Whitney’s construction to prove the existence of a triangulation for any C manifold, 3 so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is 4 completely geometric. 2012 ACM Subject Classification Theory of computation → Computational geometry Keywords and phrases Triangulations, Manifolds, Coxeter triangulations Related Version A full version of this paper is available at https://hal.inria.fr/hal-????? ?? Funding This work has been partially 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). Lines 483 5 1 Introduction 1 6 The question whether every C manifold admits a triangulation was of great importance to 7 topologists in the first half of the 20th century. This question was answered in the affirmative 8 by Cairns [11], see also Whitehead [26]. However the first proofs were complicated and not 9 very geometric, let alone algorithmic. It was Whitney [27, Chapter IV], who eventually gave 10 an insightful geometric constructive proof. Here, we will be reproving Theorem 12A of [27, 2 11 Section IV.12], in a more quantitative/algorithmic fashion for C manifolds: 2 d 12 I Theorem 1. Every compact n-dimensional C manifold M embedded in R with reach 13 rch(M) admits a triangulation. 14 By more quantitative, we mean that instead of being satisfied with the existence of constants 15 that are used in the construction, we want to provide explicit bounds in terms of the reach 16 of the manifold, which we shall assume to be positive. The reach was introduced by Federer 17 [19], as the minimal distance between a set M and its medial axis. The medial axis consists © Jean-Daniel Boissonnat, Siargey Kachanovich, and Mathijs Wintraecken; licensed under Creative Commons License CC-BY 35th International Symposium on Computational Geometry (SoCG 2019). Editors: Gill Barequet and Yusu Wang; Article No. 00; pp. 00:1–00:32 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 00:2 An elementary and quantified version of Whitney’s triangulation method 18 of points in ambient space that do not have a unique closest point on M. It is not too 1,1 19 difficult to generalize the precise quantities to the setting where the manifold is C at a 20 small cost, see Appendix B. 1 21 Note that Theorem 1 implies that any C manifolds admits a triangulation. This is 1 22 because any C manifold can be smoothed (see for example [20]) and Whitney’s own em- d 23 bedding theorem (see [27, Section IV.1]) gives an embedding in R . 24 Triangulations in computational geometry/topology are most often based on Voron- 25 oi/Delaunay triangulations of the input point set, see for example [3, 4, 9, 12, 15]. Whitney’s 26 construction is of a quite different nature. He uses an ambient triangulation and construct 27 the triangulation of the manifold M based on the intersections of M with this triangu- 28 lation. In this paper, we have chosen this ambient triangulation T˜ to be (a perturbation ˜ ˜ 29 of) a Coxeter triangulation T of type Ad. A Coxeter triangulation of type Ad is Delaunay 30 protected, a concept we’ll recall in detail in Section 4. Delaunay protection gives that the 31 triangulation is stable under perturbations. This property simplifies the proof, which in fact 32 was one of the motivations for our choice. Moreover, Coxeter triangulations can be stored 33 very compactly, in contrast with previous work [9, 12] on Delaunay triangulations. 34 The approach of the proof of correctness of the method, that we present in this paper, 35 focuses on proving that after perturbing the ambient triangulation the intersection of each d- 36 simplex in the triangulation T˜ with M is a slightly deformed n-dimensional convex polytope. 37 The triangulation K of M consists of a barycentric subdivision of a straightened version of 38 these polytopes. This may remind the reader of the general result on CW-complexes, see 39 [21], which was exploited by Edelsbrunner and Shah [18] for their triangulation result. This 40 interpretation of Whitney’s triangulation method is different from the Whitney’s original 41 proof where the homeomorphism is given by the closest point projection and uses techniques 42 which we also exploited in [8]. Here we construct ‘normals’ and a tubular neighbourhood 43 for K that is compatible with the ambient triangulation T˜ and prove that the projection 44 along these ‘normals’ is a homeomorphism. We believe that the tubular neighbourhood we 45 construct is of independent interest. Because we have a bound on the size of the tubular 46 neighbourhood of K and M lies in this neighbourhood, we automatically bound the Haus- 47 dorff distance between the two. A bound on the difference between the normals of K and 48 M is also provided. Thanks to our choice of ambient triangulation and our homeomorphism 49 proof, this entire paper is elementary in the sense that no topological results are needed, all 50 arguments are geometrical. 51 In addition to the more quantitative/algorithmic approach, the purely geometrical homeo- 52 morphism proof, the link with the closed ball property, the tubular neighbourhood for the 53 triangulation K, and a bound on the Hausdorff distance, we also give different proofs for a 54 fair number of Whitney’s intermediate results. 55 The algorithm described in this paper is being implemented at the moment. The results 56 will be reported in a companion paper. 57 All proofs of statements that are not recalled from other papers are given in the appendix. 58 2 The algorithm and overview 59 2.1 The algorithm 2 d 60 The algorithm takes as input an n-dimensional C manifold M ⊂ R with reach rch(M), 61 and outputs the triangulation K of M. The algorithm based on Whitney’s construction 62 consists of two parts. J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken 00:3 63 Figure 1 The two parts of the algorithm: Part 1, where we perturb the vertices of the ambient 64 triangulation, is depicted on top. Part 2, where the triangulation is constructed from the points of 65 intersection of M and the edges, is depicted below. 66 Part 1. This part of the algorithm handles the ambient triangulation and consists of 67 the following steps: ˜ d 68 Choose a Coxeter triangulation T of type Ad of R that is sufficiently fine (as determined 69 by the longest edge length L, see (8)). 70 Perturb the vertices of T slightly (see (14)) into a T˜ (with the same combinatorial 71 structure), such that all simplices in T˜ of dimension at most d − n − 1 are sufficiently 72 far away from the manifold, see (11). This is done as follows: One maintains a list of ˜ 73 vertices and simplices of Ti, starting with an empty list and adding perturbed vertices, 74 keeping the combinatorial structure of T intact. This means that if τ = {vj1 , . , vjk } 75 ˜ is a simplex in T and v˜j1 ,..., v˜jk ∈ Ti, where v˜i denotes the perturbed vertex vi, then 76 ˜ τ˜ = {v˜j1 ,..., v˜jk } is a simplex in Ti. To this list, one first adds all vertices of T that are 77 very far from M, as well as the simplices with these vertices (see Case 1 of Section 5.2). 78 For a vertex vi that is relatively close to M (Case 2), one goes through the following 79 procedure. We first pick a point p ∈ M that is not too far from vi. We then consider 0 ˜ 0 ˜ 80 all τj ⊂ Ti−1 of dimension at most d − n − 2, such that the join vi ∗ τj lies in Ti. For all 0 0 81 such τj we consider span(τj,TpM) and we pick our perturbed vi, that is v˜i, such that it 82 lies sufficiently far from the union of these spans (see (27)). S o C G 2 0 1 9 00:4 An elementary and quantified version of Whitney’s triangulation method 83 Note that we only require limited knowledge of the manifold. Given a vertex vi we need 84 to be able to find a point on M that is close to vi or know if vi is far from M and we 85 need access to T M in a finite sufficiently dense set of points (so that for every point vi that 86 is close to M we have a linear approximation of M).
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