227 research outputs found
Maximizing Maximal Angles for Plane Straight-Line Graphs
Let be a plane straight-line graph on a finite point set
in general position. The incident angles of a vertex
of are the angles between any two edges of that appear consecutively in
the circular order of the edges incident to .
A plane straight-line graph is called -open if each vertex has an
incident angle of size at least . In this paper we study the following
type of question: What is the maximum angle such that for any finite set
of points in general position we can find a graph from a certain
class of graphs on that is -open? In particular, we consider the
classes of triangulations, spanning trees, and paths on and give tight
bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that
were omitted in the previous version are now include
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
On the Number of Pseudo-Triangulations of Certain Point Sets
We pose a monotonicity conjecture on the number of pseudo-triangulations of
any planar point set, and check it on two prominent families of point sets,
namely the so-called double circle and double chain. The latter has
asymptotically pointed pseudo-triangulations, which lies
significantly above the maximum number of triangulations in a planar point set
known so far.Comment: 31 pages, 11 figures, 4 tables. Not much technical changes with
respect to v1, except some proofs and statements are slightly more precise
and some expositions more clear. This version has been accepted in J. Combin.
Th. A. The increase in number of pages from v1 is mostly due to formatting
the paper with "elsart.cls" for Elsevie
On the number of pseudo-triangulations of certain point sets
We compute the exact number of pseudo-triangulations for two prominent point sets, namely the so-called double circle and the double chain. We also derive a new asymptotic lower bound for the maximal number of pseudotriangulations which lies significantly above the related bound for triangulations.Acciones Integradas 2003-2004Acción Integrada España-AustriaDirección General de Investigación CientÃfic
Multitriangulations, pseudotriangulations and primitive sorting networks
We study the set of all pseudoline arrangements with contact points which
cover a given support. We define a natural notion of flip between these
arrangements and study the graph of these flips. In particular, we provide an
enumeration algorithm for arrangements with a given support, based on the
properties of certain greedy pseudoline arrangements and on their connection
with sorting networks. Both the running time per arrangement and the working
space of our algorithm are polynomial.
As the motivation for this work, we provide in this paper a new
interpretation of both pseudotriangulations and multitriangulations in terms of
pseudoline arrangements on specific supports. This interpretation explains
their common properties and leads to a natural definition of
multipseudotriangulations, which generalizes both. We study elementary
properties of multipseudotriangulations and compare them to iterations of
pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of
presentatio
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
A Sweepline Algorithm for Generalized Delaunay Triangulations
We give a deterministic O(n log n) sweepline algorithm to construct the generalized Voronoi diagram for n points in the plane or rather its dual the generalized Delaunay triangulation. The algorithm uses no transformations and it is developed solely from the sweepline paradigm together with greediness. A generalized Delaunay triangulation can be based on an arbitrary strictly convex Minkowski distance function (including all L_p distance functions 1 < p < *) in contrast to ordinary Delaunay triangualations which are based on the Euclidean distance function
Dyck path triangulations and extendability
We introduce the Dyck path triangulation of the cartesian product of two
simplices . The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever , any triangulation of
extends to a unique triangulation of
. Moreover, with an explicit construction, we
prove that the bound is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.Comment: 15 pages, 14 figures. Comments very welcome
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