3,833 research outputs found
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
Total Curvature of Graphs after Milnor and Euler
We define a new notion of total curvature, called net total curvature, for
finite graphs embedded in Rn, and investigate its properties. Two guiding
principles are given by Milnor's way of measuring the local crookedness of a
Jordan curve via a Crofton-type formula, and by considering the double cover of
a given graph as an Eulerian circuit. The strength of combining these ideas in
defining the curvature functional is (1) it allows us to interpret the
singular/non-eulidean behavior at the vertices of the graph as a superposition
of vertices of a 1-dimensional manifold, and thus (2) one can compute the total
curvature for a wide range of graphs by contrasting local and global properties
of the graph utilizing the integral geometric representation of the curvature.
A collection of results on upper/lower bounds of the total curvature on
isotopy/homeomorphism classes of embeddings is presented, which in turn
demonstrates the effectiveness of net total curvature as a new functional
measuring complexity of spatial graphs in differential-geometric terms.Comment: Most of the results contained in "Total curvature and isotopy of
graphs in ."(arXiv:0806.0406) have been incorporated into the current
articl
On comparing the writhe of a smooth curve to the writhe of an inscribed polygon
We find bounds on the difference between the writhing number of a smooth
curve, and the writhing number of a polygon inscribed within. The proof is
based on an extension of Fuller's difference of writhe formula to the case of
polygonal curves. The results establish error bounds useful in the computation
of writhe.Comment: 16 pages, 5 figure
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
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
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