1,951 research outputs found
Geodesic-Preserving Polygon Simplification
Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon by a polygon such that (1)
contains , (2) has its reflex
vertices at the same positions as , and (3) the number of vertices
of is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
and , our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of
Monotonicity Analysis over Chains and Curves
Chains are vector-valued signals sampling a curve. They are important to
motion signal processing and to many scientific applications including location
sensors. We propose a novel measure of smoothness for chains curves by
generalizing the scalar-valued concept of monotonicity. Monotonicity can be
defined by the connectedness of the inverse image of balls. This definition is
coordinate-invariant and can be computed efficiently over chains. Monotone
curves can be discontinuous, but continuous monotone curves are differentiable
a.e. Over chains, a simple sphere-preserving filter shown to never decrease the
degree of monotonicity. It outperforms moving average filters over a synthetic
data set. Applications include Time Series Segmentation, chain reconstruction
from unordered data points, Optical Character Recognition, and Pattern
Matching.Comment: to appear in Proceedings of Curves and Surfaces 200
Optimizing the geometrical accuracy of curvilinear meshes
This paper presents a method to generate valid high order meshes with
optimized geometrical accuracy. The high order meshing procedure starts with a
linear mesh, that is subsequently curved without taking care of the validity of
the high order elements. An optimization procedure is then used to both
untangle invalid elements and optimize the geometrical accuracy of the mesh.
Standard measures of the distance between curves are considered to evaluate the
geometrical accuracy in planar two-dimensional meshes, but they prove
computationally too costly for optimization purposes. A fast estimate of the
geometrical accuracy, based on Taylor expansions of the curves, is introduced.
An unconstrained optimization procedure based on this estimate is shown to
yield significant improvements in the geometrical accuracy of high order
meshes, as measured by the standard Haudorff distance between the geometrical
model and the mesh. Several examples illustrate the beneficial impact of this
method on CFD solutions, with a particular role of the enhanced mesh boundary
smoothness.Comment: Submitted to JC
On the negative spectrum of the Robin Laplacian in corner domains
For a bounded corner domain , we consider the Robin Laplacian in
with large Robin parameter. Exploiting multiscale analysis and a
recursive procedure, we have a precise description of the mechanism giving the
ground state of the spectrum. It allows also the study of the bottom of the
essential spectrum on the associated tangent structures given by cones. Then we
obtain the asymptotic behavior of the principal eigenvalue for this singular
limit in any dimension, with remainder estimates. The same method works for the
Schr\"odinger operator in with a strong attractive
delta-interaction supported on . Applications to some Erhling's
type estimates and the analysis of the critical temperature of some
superconductors are also provided
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