2,892 research outputs found
Speeding up Simplification of Polygonal Curves using Nested Approximations
We develop a multiresolution approach to the problem of polygonal curve
approximation. We show theoretically and experimentally that, if the
simplification algorithm A used between any two successive levels of resolution
satisfies some conditions, the multiresolution algorithm MR will have a
complexity lower than the complexity of A. In particular, we show that if A has
a O(N2/K) complexity (the complexity of a reduced search dynamic solution
approach), where N and K are respectively the initial and the final number of
segments, the complexity of MR is in O(N).We experimentally compare the
outcomes of MR with those of the optimal "full search" dynamic programming
solution and of classical merge and split approaches. The experimental
evaluations confirm the theoretical derivations and show that the proposed
approach evaluated on 2D coastal maps either shows a lower complexity or
provides polygonal approximations closer to the initial curves.Comment: 12 pages + figure
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
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 Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages
We extend our study of Motion Planning via Manifold Samples (MMS), a general
algorithmic framework that combines geometric methods for the exact and
complete analysis of low-dimensional configuration spaces with sampling-based
approaches that are appropriate for higher dimensions. The framework explores
the configuration space by taking samples that are entire low-dimensional
manifolds of the configuration space capturing its connectivity much better
than isolated point samples. The contributions of this paper are as follows:
(i) We present a recursive application of MMS in a six-dimensional
configuration space, enabling the coordination of two polygonal robots
translating and rotating amidst polygonal obstacles. In the adduced experiments
for the more demanding test cases MMS clearly outperforms PRM, with over
20-fold speedup in a coordination-tight setting. (ii) A probabilistic
completeness proof for the most prevalent case, namely MMS with samples that
are affine subspaces. (iii) A closer examination of the test cases reveals that
MMS has, in comparison to standard sampling-based algorithms, a significant
advantage in scenarios containing high-dimensional narrow passages. This
provokes a novel characterization of narrow passages which attempts to capture
their dimensionality, an attribute that had been (to a large extent) unattended
in previous definitions.Comment: 20 page
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