1,275 research outputs found
How to Walk Your Dog in the Mountains with No Magic Leash
We describe a -approximation algorithm for computing the
homotopic \Frechet distance between two polygonal curves that lie on the
boundary of a triangulated topological disk. Prior to this work, algorithms
were known only for curves on the Euclidean plane with polygonal obstacles.
A key technical ingredient in our analysis is a -approximation
algorithm for computing the minimum height of a homotopy between two curves. No
algorithms were previously known for approximating this parameter.
Surprisingly, it is not even known if computing either the homotopic \Frechet
distance, or the minimum height of a homotopy, is in NP
Spectral Generalized Multi-Dimensional Scaling
Multidimensional scaling (MDS) is a family of methods that embed a given set
of points into a simple, usually flat, domain. The points are assumed to be
sampled from some metric space, and the mapping attempts to preserve the
distances between each pair of points in the set. Distances in the target space
can be computed analytically in this setting. Generalized MDS is an extension
that allows mapping one metric space into another, that is, multidimensional
scaling into target spaces in which distances are evaluated numerically rather
than analytically. Here, we propose an efficient approach for computing such
mappings between surfaces based on their natural spectral decomposition, where
the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS
procedure enables efficient embedding by implicitly incorporating smoothness of
the mapping into the problem, thereby substantially reducing the complexity
involved in its solution while practically overcoming its non-convex nature.
The method is compared to existing techniques that compute dense correspondence
between shapes. Numerical experiments of the proposed method demonstrate its
efficiency and accuracy compared to state-of-the-art approaches
Geodesics in Heat
We introduce the heat method for computing the shortest geodesic distance to
a specified subset (e.g., point or curve) of a given domain. The heat method is
robust, efficient, and simple to implement since it is based on solving a pair
of standard linear elliptic problems. The method represents a significant
breakthrough in the practical computation of distance on a wide variety of
geometric domains, since the resulting linear systems can be prefactored once
and subsequently solved in near-linear time. In practice, distance can be
updated via the heat method an order of magnitude faster than with
state-of-the-art methods while maintaining a comparable level of accuracy. We
provide numerical evidence that the method converges to the exact geodesic
distance in the limit of refinement; we also explore smoothed approximations of
distance suitable for applications where more regularity is required
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
Transforming triangulations on non planar-surfaces
We consider whether any two triangulations of a polygon or a point set on a
non-planar surface with a given metric can be transformed into each other by a
sequence of edge flips. The answer is negative in general with some remarkable
exceptions, such as polygons on the cylinder, and on the flat torus, and
certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM
Journal on Discrete Mathematics. Keywords: Graph of triangulations,
triangulations on surfaces, triangulations of polygons, edge fli
- …