Triangulating the Real Projective Plane

We consider the problem of computing a triangulation of the real projective plane P2, given a finite point set S={p1, p2,..., pn} as input. We prove that a triangulation of P2 always exists if at least six points in S are in general position, i.e., no three of them are collinear. We also design an algorithm for triangulating P2 if this necessary condition holds. As far as we know, this is the first computational result on the real projective plane

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

09111 Abstracts Collection -- Computational Geometry

From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 ``Computational Geometry \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

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

Generalizing CGAL Periodic Delaunay Triangulations

Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity

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

Union and Split Operations on Dynamic Trapezoidal Maps

We propose here algorithms to perform two new operations on an arrangement of line segments in the plane, represented by a trapezoidal map: the split of the map along a given vertical line $D$, and the union of two trapezoidal maps computed in two vertical slabs of the plane that are adjacent through a vertical line $D$. The data structure is a modified Influence graph, still allowing dynamic insertions and deletions of line segments in themap. The algorithms for both operations run in $O(s_D \log n +\log^2 n)$ time, where $n$ is the number of line segments in the map, and $s_D$ is the number of line segments intersected by $D$

Red de canales internos en el octocoral Muricea muricata, : correlación con sus patrones de crecimiento y ramificación

La estrategia de crecimiento del octocoral Muricea muricata (Plexauridae) y la caracterización de su red de canales inttra-coenquimal fueron determinados. Las colonias de M. muricata exhibieron un crecimiento promedio por rama de 0,12 ± 0,07 cm.mes-1 y generaron un promedio mensual de 1,46 ± 1,05 ramas nuevas. Las ramas que presentaron un mayor crecimiento fueron, en la jerarquía definida, las ramas más antiguas con un promedio de 0,39 ± 0,27 cm.mes-1. Las ramas terminales o de ultima generación fueron las que menos crecieron, con un promedio de 0,09 ± 0,05 cm.mes-1. La representación gráfica del Individual Based-Model (IBM) muestra como las ramas primordiales crecen y ganan altura en la columna de agua, mientras que las de generaciones más recientes rellenan el espacio por medio de una mayor ramificación. Esta organización permite a la especie tener su forma de abanico y crecer en un solo plano perpendicular a la corriente. El estudio de los canales internos mostró que estos se encuentran a lo largo de toda la colonia y se organizan paralelamente entre sí siguiendo longitudinalmente la dirección de las diferentes ramas de la colonia. El diámetro de los canales en M. muricata mide entre 0,053 y 0,238 mm. Las imágenes de tomografía axial computarizada de alta resolución (microTac) y la reconstrucción 3D de varios tramos de la colonia permitieron la visualización de la organización interna de la colonia y específicamente evidenciar como se realiza el proceso de ramificación. Este se produce por medio de la generación de un nuevo brote que hace recordar la formación de las yemas en los hidróides. Además, la reconstrucción de las imágenes 3D sugiere la existencia de diminutas conexiones presentes entre canales yuxtapuestos y conexiones entre algunos canales longitudinales y pólipos; así como la presencia, en donde terminan los canales en las zonas apicales de las ramas, de una masa volumétrica que los une a, todos. Por otro lado, se evidenció una relación lineal entre el número de canales alrededor del eje central y la longitud de las ramas. La base de la colonia es el sitio donde se encuentra el mayor número de canales y la extremidad de las ramas, el lugar donde existe un menor número de canales. Estas diferentes observaciones sugieren que este conjunto de canales y conexiones conforman una red tridimensional que une internamente toda la colonia. Por otra parte, el número de canales encontrados en la extremidad de las ramas, se aproxima curiosamente a un valor de 8. Este número podría representar un carácter ancestral dentro del grupo de los octocorales.Biólogo Marin