2,610 research outputs found
Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4
We present two versions of a method for generating all triangulations of any
punctured surface in each of these two families: (1) triangulations with inner
vertices of degree at least 4 and boundary vertices of degree at least 3 and
(2) triangulations with all vertices of degree at least 4. The method is based
on a series of reversible operations, termed reductions, which lead to a
minimal set of triangulations in each family. Throughout the process the
triangulations remain within the corresponding family. Moreover, for the family
(1) these operations reduce to the well-known edge contractions and removals of
octahedra. The main results are proved by an exhaustive analysis of all
possible local configurations which admit a reduction.Comment: This work has been partially supported by PAI FQM-164; PAI FQM-189;
MTM 2010-2044
Recent Results on Near-Best Spline Quasi-Interpolants
Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an
approximation operator of the form where the 's are B-splines and the 's
are linear discrete or integral forms acting on the given function . These
forms depend on a finite number of coefficients which are the components of
vectors for . The index refers to this sequence of
vectors. In order that for all polynomials belonging to some
subspace included in the space of splines generated by the 's, each
vector must lie in an affine subspace , i.e. satisfy some
linear constraints. However there remain some degrees of freedom which are used
to minimize for each . It is easy to
prove that is an upper bound of
: thus, instead of minimizing the infinite norm of
, which is a difficult problem, we minimize an upper bound of this norm,
which is much easier to do. Moreover, the latter problem has always at least
one solution, which is associated with a NB QI. In the first part of the paper,
we give a survey on NB univariate or bivariate spline QIs defined on uniform or
non-uniform partitions and already studied by the author and coworkers. In the
second part, we give some new results, mainly on univariate and bivariate
integral QIs on {\sl non-uniform} partitions: in that case, NB QIs are more
difficult to characterize and the optimal properties strongly depend on the
geometry of the partition. Therefore we have restricted our study to QIs having
interesting shape properties and/or infinite norms uniformly bounded
independently of the partition
Stabbing line segments with disks: complexity and approximation algorithms
Computational complexity and approximation algorithms are reported for a
problem of stabbing a set of straight line segments with the least cardinality
set of disks of fixed radii where the set of segments forms a straight
line drawing of a planar graph without edge crossings. Close
geometric problems arise in network security applications. We give strong
NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel
graphs and other subgraphs (which are often used in network design) for and some constant where and
are Euclidean lengths of the longest and shortest graph edges
respectively. Fast -time -approximation algorithm is
proposed within the class of straight line drawings of planar graphs for which
the inequality holds uniformly for some constant
i.e. when lengths of edges of are uniformly bounded from above by
some linear function of Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International
Conference on Analysis of Images, Social Networks and Texts (AIST-2017
Planar maps, circle patterns and 2d gravity
Via circle pattern techniques, random planar triangulations (with angle
variables) are mapped onto Delaunay triangulations in the complex plane. The
uniform measure on triangulations is mapped onto a conformally invariant
spatial point process. We show that this measure can be expressed as: (1) a sum
over 3-spanning-trees partitions of the edges of the Delaunay triangulations;
(2) the volume form of a K\"ahler metric over the space of Delaunay
triangulations, whose prepotential has a simple formulation in term of ideal
tessellations of the 3d hyperbolic space; (3) a discretized version (involving
finite difference complex derivative operators) of Polyakov's conformal
Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes,
thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17
figure
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
Uniform infinite planar triangulation and related time-reversed critical branching process
We establish a connection between the uniform infinite planar triangulation
and some critical time-reversed branching process. This allows to find a
scaling limit for the principal boundary component of a ball of radius R for
large R (i.e. for a boundary component separating the ball from infinity). We
show also that outside of R-ball a contour exists that has length linear in R.Comment: 27 pages, 5 figures, LaTe
Triangulated Surfaces in Twistor Space: A Kinematical Set up for Open/Closed String Duality
We exploit the properties of the three-dimensional hyperbolic space to
discuss a simplicial setting for open/closed string duality based on (random)
Regge triangulations decorated with null twistorial fields. We explicitly show
that the twistorial N-points function, describing Dirichlet correlations over
the moduli space of open N-bordered genus g surfaces, is naturally mapped into
the Witten-Kontsevich intersection theory over the moduli space of N-pointed
closed Riemann surfaces of the same genus. We also discuss various aspects of
the geometrical setting which connects this model to PSL(2,C) Chern-Simons
theory.Comment: 35 pages, references added, slightly revised introductio
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