1,040 research outputs found
Gauss images of hyperbolic cusps with convex polyhedral boundary
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral
boundary is uniquely determined by its Gauss image. Furthermore, any spherical
metric on the torus with cone singularities of negative curvature and all
closed contractible geodesics of length greater than is the metric of
the Gauss image of some convex polyhedral cusp. This result is an analog of the
Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in
terms of their Gauss images.
The proof uses a variational method. Namely, a cusp with a given Gauss image
is identified with a critical point of a functional on the space of cusps with
cone-type singularities along a family of half-lines. The functional is shown
to be concave and to attain maximum at an interior point of its domain. As a
byproduct, we prove rigidity statements with respect to the Gauss image for
cusps with or without cone-type singularities.
In a special case, our theorem is equivalent to existence of a circle pattern
on the torus, with prescribed combinatorics and intersection angles. This is
the genus one case of a theorem by Thurston. In fact, our theorem extends
Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's
theorem on compact convex polyhedra with non-obtuse dihedral angles.
The functional used in the proof is the sum of a volume term and curvature
term. We show that, in the situation of Thurston's theorem, it is the potential
for the combinatorial Ricci flow considered by Chow and Luo.
Our theorem represents the last special case of a general statement about
isometric immersions of compact surfaces.Comment: 55 pages, 17 figure
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
Design and loading of dragline buckets
Draglines are an expensive and essential part of open cut coal mining. Small improvements in performance can produce substantial savings. The design of the bucket and the way in which it fills with overburden are very important to the overall dragline performance. Here we use a numerical model to simulate this filling process and to differentiate between the flow patterns of two different buckets. Extensions to the model are explored
Local geometry of random geodesics on negatively curved surfaces
It is shown that the tessellation of a compact, negatively curved surface
induced by a typical long geodesic segment, when properly scaled, looks locally
like a Poisson line process. This implies that the global statistics of the
tessellation -- for instance, the fraction of triangles -- approach those of
the limiting Poisson line process.Comment: This version extends the results of the previous version to surfaces
with possibly variable negative curvatur
VoroCrust: Voronoi Meshing Without Clipping
Polyhedral meshes are increasingly becoming an attractive option with
particular advantages over traditional meshes for certain applications. What
has been missing is a robust polyhedral meshing algorithm that can handle broad
classes of domains exhibiting arbitrarily curved boundaries and sharp features.
In addition, the power of primal-dual mesh pairs, exemplified by
Voronoi-Delaunay meshes, has been recognized as an important ingredient in
numerous formulations. The VoroCrust algorithm is the first provably-correct
algorithm for conforming polyhedral Voronoi meshing for non-convex and
non-manifold domains with guarantees on the quality of both surface and volume
elements. A robust refinement process estimates a suitable sizing field that
enables the careful placement of Voronoi seeds across the surface circumventing
the need for clipping and avoiding its many drawbacks. The algorithm has the
flexibility of filling the interior by either structured or random samples,
while preserving all sharp features in the output mesh. We demonstrate the
capabilities of the algorithm on a variety of models and compare against
state-of-the-art polyhedral meshing methods based on clipped Voronoi cells
establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed
images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf.
Supplemental materials available on
https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd
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