Computing Periodic Triangulations

International audienceThe talk presents work on periodic triangulations and meshes, in the Euclidean case as well as the hyperbolic case

Decoupling the CGAL 3D Triangulations from the Underlying Space

The {\em Computational Geometry Algorithms Library} {\sc Cgal} currently provides packages to compute triangulations in $\mathbb{R}^2$ and $\mathbb{R}^3$. In this paper we describe a new design for the 3D triangulation package that permits to easily add functionality to compute triangulations in other spaces. These design changes have been implemented, and validated on the case of the periodic space \T^3. We give a detailed description of the realized changes together with their motivation. Finally, we show benchmarks to prove that the new design does not affect the efficiency

Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web

We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, $\alpha$. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of $\alpha$, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution and scale-dependence of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web and yields a promising measure of cosmological parameters. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field.Comment: 42 pages, 14 figure

Robust and Efficient Delaunay Triangulations of Points on Or Close to a Sphere

International audienceWe propose two ways to compute the Delaunay triangulation of points on a sphere, or of rounded points close to a sphere, both based on the classic incremental algorithm initially designed for the plane. We use the so-called space of circles as mathematical background for this work. We present a fully robust implementation built upon existing generic algorithms provided by the Cgal library. The efficiency of the implementation is established by benchmarks

Cadmium (zinc) telluride 2D/3D spectrometers for scattering polarimetry

he semiconductor detectors technology has dramatically changed the broad field of x- and γ-rays spectroscopy and imaging. Semiconductor detectors, originally developed for particle physics applications, are now widely used for x/γ-rays spectroscopy and imaging in a large variety of fields, among which, for example, x-ray fluorescence, γ-ray monitoring and localization, noninvasive inspection and analysis, astronomy, and diagnostic medicine. The success of semiconductor detectors is due to several unique 242characteristics as the excellent energy resolution, the high detection efficiency, and the possibility of development of compact and highly segmented detection systems (i.e., spectroscopic imager). Among the semiconductor devices, silicon (Si) detectors are the key detectors in the soft x-ray band (15 keV) and will continue to be the first choice for laboratory-based high-performance spectrometers system (Eberth and Simpson 2006)

Robust and Efficient Delaunay triangulations of points on or close to a sphere

We propose two approaches for computing the Delaunay triangulation of points on a sphere, or of rounded points close to a sphere, both based on the classic incremental algorithm initially designed for the plane. The space of circles gives the mathematical background for this work. We implemented the two approaches in a fully robust way, building upon existing generic algorithms provided by the cgal library. The effciency and scalability of the method is shown by benchmarks

All-sky Medium Energy Gamma-ray Observatory: Exploring the Extreme Multimessenger Universe

The All-sky Medium Energy Gamma-ray Observatory (AMEGO) is a probe class mission concept that will provide essential contributions to multimessenger astrophysics in the late 2020s and beyond. AMEGO combines high sensitivity in the 200 keV to 10 GeV energy range with a wide field of view, good spectral resolution, and polarization sensitivity. Therefore, AMEGO is key in the study of multimessenger astrophysical objects that have unique signatures in the gamma-ray regime, such as neutron star mergers, supernovae, and flaring active galactic nuclei. The order-of-magnitude improvement compared to previous MeV missions also enables discoveries of a wide range of phenomena whose energy output peaks in the relatively unexplored medium-energy gamma-ray band

Triangulations d'ensembles de points dans des espaces quotients

In this work, we discuss triangulations of different topological spaces for given point sets. We propose both definitions and algorithms for different classes of spaces and provide an implementation for the specific case of the three-dimensional flat torus. The work is originally motivated by the need for software computing three-dimensional periodic Delaunay triangulations in numerous domains including astronomy, material engineering, biomedical computing, fluid dynamics etc. Periodic triangulations can be understood as triangulations of the flat torus. We provide a definition an develop an efficient incremental algorithm to compute Delaunay triangulations of the flat torus. The algorithm is a modification of the incremental algorithm for computing Delaunay triangulations in Ed. Unlike previous work on periodic triangulations we avoid maintaining several periodic copies of the input point set whenever possible. Also the output of our algorithm is guaranteed to always be a triangulation of the flat torus. We provide an implementation of our algorithm that has been made available to a broad public as a part of the Computational Geometry Algorithms Library CGAL. We generalize the work on the flat torus onto a more general class of flat orbit spaces as well as orbit spaces of constant negative curvature.Dans cette thèse, nous étudions les triangulations définies par un ensemble de points dans des espaces de topologies différentes. Nous proposons une définition générale de la triangulation de Delaunay, valide pour plusieurs classes d espaces, ainsi qu un algorithme de construction. Nous fournissons une implantation pour le cas particulier du tore plat tridimensionnel. Ce travail est motivé à l origine par le besoin de logiciels calculant des triangulations de Delaunay périodiques, dans de nombreux domaines dont l astronomie, l ingénierie des matériaux, le calcul biomédical, la dynamique des fluides, etc. Les triangulations périodiques peuvent être vues comme des triangulations du tore plat. Nous fournissons une définition et nous développons un algorithme incrémentiel efficace pour calculer la triangulation de Delaunay dans le tore plat. L algorithme est adapté de l algorithme incrémentiel usuel dans Rd. Au contraire des travaux antérieurs sur les triangulations périodiques, nous évitons de maintenir plusieurs copies périodiques des points, lorsque cela est possible. Le résultat fourni par l algorithme est toujours une triangulation du tore plat. Nous présentons une implantation de notre algorithme, à présent disponible publiquement comme un module de la bibliothèque d algorithmes géométriques CGAL. Nous généralisons les résultats à une classe plus générale d espaces quotients plats, ainsi qu à des espaces quotients de courbure constante positive. Enfin, nous considérons le cas du tore double, qui est un exemple de la classe beaucoup plus riche des espaces quotients de courbure négative constante.NICE-BU Sciences (060882101) / SudocSudocFranceF

Delaunay triangulations of closed Euclidean d-orbifolds

International audienceWe give a definition of the Delaunay triangulation of a point set in a closed Euclidean d-manifold, i.e. a compact quotient space of the Euclidean space for a discrete group of isometries (a so-called Bieberbach group or crystallographic group). We describe a geometric criterion to check whether a partition of the manifold actually forms a triangulation (which subsumes that it is a simplicial complex). We provide an incremental algorithm to compute the Delaunay triangulation of the manifold defined by a given set of input points, if it exists. Otherwise, the algorithm returns the Delaunay triangulation of a finite-sheeted covering space of the manifold. The algorithm has optimal randomized worst-case time and space complexity. It extends to closed Euclidean orbifolds. An implementation for the special case of the 3D flat torus has been released in Cgal 3.5. To the best of our knowledge, this is the first general result on this topic