1,951 research outputs found

    Geodesic-Preserving Polygon Simplification

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    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 P\mathcal{P} by a polygon P′\mathcal{P}' such that (1) P′\mathcal{P}' contains P\mathcal{P}, (2) P′\mathcal{P}' has its reflex vertices at the same positions as P\mathcal{P}, and (3) the number of vertices of P′\mathcal{P}' 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 P\mathcal{P} and P′\mathcal{P}', 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 P\mathcal{P}

    Monotonicity Analysis over Chains and Curves

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    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

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    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

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    For a bounded corner domain Ω\Omega, we consider the Robin Laplacian in Ω\Omega 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 Rn\mathbb{R}^n with a strong attractive delta-interaction supported on ∂Ω\partial\Omega. Applications to some Erhling's type estimates and the analysis of the critical temperature of some superconductors are also provided
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