Minimal Delaunay Triangulations of Hyperbolic Surfaces

Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we show that every hyperbolic surface of genus g has a simplicial Delaunay triangulation with O(g) vertices, where edges are given by distance paths. Then, we construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we show that the ?(?g) lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus g is tight for hyperbolic surfaces as well

Computing a Dirichlet Domain for a Hyperbolic Surface

This paper exhibits and analyzes an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm runs in polynomial time, in terms of the initial perimeter and the genus of the surface

Implementing Delaunay Triangulations of the Bolza Surface

The CGAL library offers software packages to compute Delaunay triangulations of the (flat) torus of genus one in two and three dimensions. To the best of our knowledge, there is no available software for the simplest possible extension, i.e., the Bolza surface, a hyperbolic manifold homeomorphic to a torus of genus two. In this paper, we present an implementation based on the theoretical results and the incremental algorithm proposed last year at SoCG by Bogdanov, Teillaud, and Vegter. We describe the representation of the triangulation, we detail the different steps of the algorithm, we study predicates, and report experimental results

Triangulations

The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods

Triangulations de Delaunay d'une famille de surfaces hyperboliques symétriques en pratique

The Bolza surface is the most symmetric compact orientable hyperbolic surface of genus 2. For any genus higher than 2, there exists one compact orientable surface constructed in a similar way as the Bolza surface having the same kind of symmetry. We refer to this family of surfaces as symmetric hyperbolic surfaces. This thesis deals with the computation of Delaunay triangulations of symmetric hyperbolic surfaces.Delaunay triangulations of compact surfaces can be seen as periodic Delaunay triangulations of their universal cover (in our case, the hyperbolic plane). A Delaunay triangulation is for us a simplicial complex. However, not all sets of points define a simplicial decomposition of a symmetric hyperbolic surface. In the literature, an algorithm has been proposed to deal with this issue by using so-called dummy points: initially a triangulation of the surface is constructed with a set of dummy points that defines a Delaunay triangulation of the surface, then input points are inserted with the well-known incremental algorithm by Bowyer, and finally the dummy points are removed, if the triangulation remains a simplicial complex after their removal. For the Bolza surface, the set of dummy points to initialize the triangulation is given. The existing algorithm computes a triangulation of the Bolza surface as a periodic triangulation of the hyperbolic plane and requires to identify a suitable subset of the hyperbolic plane in which to work.We study the properties of Delaunay triangulations of the Bolza surface defined by sets of points containing the proposed set of dummy points, and we describe in detail an implementation of the incremental algorithm for it. We begin by identifying a subset of the hyperbolic plane that contains at least one representative for each face of a Delaunay triangulation of the surface, which enables us to define a unique canonical representative in the hyperbolic plane for each face on the surface. We give a data structure to represent a Delaunay triangulation of the Bolza surface via the canonical representatives of its faces in the hyperbolic plane. We detail the construction of such a triangulation and additional operations that enable the location of points and the removal of vertices. We also report results on the algebraic degree of predicates needed for all operations. We provide a fully dynamic implementation for the Bolza surface, supporting insertion of new points, removal of existing vertices, point location, and construction of dual objects. Our implementation is based on CGAL, the Computational Geometry Algorithms Library, and is currently under revision for integration in the library. To incorporate our code into CGAL, all the objects that we introduce must be compatible with the existing framework and comply with the standards adopted by the library. We give a detailed description of the classes used to represent and handle periodic hyperbolic triangulations and related objects. Benchmarks and tests are performed to evaluate our implementation, and a simple application is given in the form of a CGAL demo.We discuss an extension of our implementation to symmetric hyperbolic surfaces of genus higher than 2. We propose three methods to generate sets of dummy points for each surface and present the advantages and shortcomings of each method. We identify a suitable subset of the hyperbolic plane that contains at least one representative for each face of a Delaunay triangulation of the surface, and we define a canonical representative in the hyperbolic plane for each face on the surface. We describe a data structure to represent such a triangulation via the canonical representatives of its faces, and give algorithms for the initialization of the triangulation with dummy points. Finally, we discuss a preliminary implementation in which we examine the difficulties of having efficient exact predicates for the construction of Delaunay triangulations of symmetric hyperbolic surfaces.La surface de Bolza est la surface hyperbolique orientable compacte la plus symétrique de genre 2. Pour tout genre supérieur à 2, il existe une surface orientable compacte construite de manière similaire à la surface de Bolza et ayant le même type de symétries. Nous appelons ces surfaces des surfaces hyperboliques symétriques. Cette thèse porte sur le calcul des triangulations de Delaunay (TD) de surfaces hyperboliques symétriques.Les TD de surfaces compactes peuvent être considérées comme des TD périodiques de leur revêtement universel (dans notre cas, le plan hyperbolique). Une TD est pour nous un complexe simplicial. Cependant, les ensembles de points ne définissent pas tous une décomposition simpliciale d'une surface hyperbolique symétrique. Dans la littérature, un algorithme a été proposé pour traiter ce problème avec l'utilisation de points factices : initialement une TD de la surface est construite avec un ensemble de points connu, puis des points d'entrée sont insérés avec le célèbre algorithme incrémental de Bowyer, et enfin les points factices sont supprimés, si la triangulation reste toujours un complexe simplicial. Pour la surface de Bolza, les points factices sont spécifiés. L'algorithme existant calcule une DT de la surface de Bolza comme une DT périodique du plan hyperbolique, ce qui nécessite de travailler dans un sous-ensemble approprié du plan hyperbolique.Nous étudions les propriétés des TD de la surface de Bolza définies par des ensembles de points contenants l'ensemble proposé de points factices, et nous décrivons en détail une implémentation de l'algorithme incrémentiel pour cette surface. Nous commençons par définir un représentant canonique unique qui est contenu dans un sous-ensemble borné du plan hyperbolique pour chaque face d'une TD de la surface. Nous donnons une structure de données pour représenter une TD de la surface de Bolza via les représentants canoniques de ses faces. Nous détaillons les étapes de la construction d'une telle triangulation et les opérations supplémentaires qui permettent de localiser les points et de retirer des sommets. Nous présentons également les résultats sur le degré algébrique des prédicats nécessaires pour toutes les opérations.Nous fournissons une implémentation entièrement dynamique pour la surface de Bolza, en offrant l'insertion de nouveaux points, la suppression des sommets existants, la localisation des points, et la construction d'objets duaux. Notre implémentation est basée sur la bibliothèque CGAL (Computational Geometry Algorithms Library), et est actuellement en cours de révision pour être intégrée dans la bibliothèque. L'intégration de notre code dans CGAL nécessite que tous les objets que nous introduisons soient compatibles avec le cadre existant et conformes aux standards adoptés par la bibliothèque. Nous donnons une description détaillée des classes utilisées pour représenter et traiter les triangulations hyperboliques périodiques et les objets associés. Des analyses comparatives et des tests sont effectués pour évaluer notre implémentation, et une application simple est donnée sous la forme d'une démonstration CGAL.Nous discutons une extension de notre implémentation à des surfaces hyperboliques symétriques de genre supérieur à 2. Nous proposons trois méthodes pour engendrer des ensembles de points factices pour chaque surface et présentons les avantages et les inconvénients de chaque méthode. Nous définissons un représentant canonique contenu dans un sous-ensemble borné du plan hyperbolique pour chaque face d'une TD de la surface. Nous décrivons une structure de données pour représenter une telle triangulation via les représentants canoniques de ses faces, et donnons des algorithmes pour l'initialisation de la triangulation. Enfin, nous discutons une implémentation préliminaire dans laquelle nous examinons les difficultés d'avoir des prédicats exacts efficaces pour la construction de TD de surfaces hyperboliques symétriques

Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field

Delaunay triangulations of regular hyperbolic surfaces

International audienceThe talk presents work on computing Delaunay triangulations of some symmetric hyperbolic surfaces of genus at least 2

Discrete Differential Geometry

Discrete Differential Geometry is a broad new area where differential geometry (studying smooth curves, surfaces and other manifolds) interacts with discrete geometry (studying polyhedral manifolds), using tools and ideas from all parts of mathematics. This report documents the 29 lectures at the first Oberwolfach workshop in this subject, with topics ranging from discrete integrable systems, polyhedra, circle packings and tilings to applications in computer graphics and geometry processing. It also includes a list of open problems posed at the problem session

A Growing Self-Organizing Network for Reconstructing Curves and Surfaces

Self-organizing networks such as Neural Gas, Growing Neural Gas and many others have been adopted in actual applications for both dimensionality reduction and manifold learning. Typically, in these applications, the structure of the adapted network yields a good estimate of the topology of the unknown subspace from where the input data points are sampled. The approach presented here takes a different perspective, namely by assuming that the input space is a manifold of known dimension. In return, the new type of growing self-organizing network presented gains the ability to adapt itself in way that may guarantee the effective and stable recovery of the exact topological structure of the input manifold