56,654 research outputs found
A generalisation of the deformation variety
Given an ideal triangulation of a connected 3-manifold with non-empty
boundary consisting of a disjoint union of tori, a point of the deformation
variety is an assignment of complex numbers to the dihedral angles of the
tetrahedra subject to Thurston's gluing equations. From this, one can recover a
representation of the fundamental group of the manifold into the isometries of
3-dimensional hyperbolic space. However, the deformation variety depends
crucially on the triangulation: there may be entire components of the
representation variety which can be obtained from the deformation variety with
one triangulation but not another. We introduce a generalisation of the
deformation variety, which again consists of assignments of complex variables
to certain dihedral angles subject to polynomial equations, but together with
some extra combinatorial data concerning degenerate tetrahedra. This "extended
deformation variety" deals with many situations that the deformation variety
cannot. In particular we show that for any ideal triangulation of a small
orientable 3-manifold with a single torus boundary component, we can recover
all of the irreducible non-dihedral representations from the associated
extended deformation variety. More generally, we give an algorithm to produce a
triangulation of a given orientable 3-manifold with torus boundary components
for which the same result holds. As an application, we show that this extended
deformation variety detects all factors of the PSL(2,C) A-polynomial associated
to the components consisting of the representations it recovers.Comment: 47 pages, 26 figures. Rewrote introduction and added motivation
section based on referee's comments. Rewrote the section on retriangulation,
and added new result on small manifolds with a single cus
SurfelMeshing: Online Surfel-Based Mesh Reconstruction
We address the problem of mesh reconstruction from live RGB-D video, assuming
a calibrated camera and poses provided externally (e.g., by a SLAM system). In
contrast to most existing approaches, we do not fuse depth measurements in a
volume but in a dense surfel cloud. We asynchronously (re)triangulate the
smoothed surfels to reconstruct a surface mesh. This novel approach enables to
maintain a dense surface representation of the scene during SLAM which can
quickly adapt to loop closures. This is possible by deforming the surfel cloud
and asynchronously remeshing the surface where necessary. The surfel-based
representation also naturally supports strongly varying scan resolution. In
particular, it reconstructs colors at the input camera's resolution. Moreover,
in contrast to many volumetric approaches, ours can reconstruct thin objects
since objects do not need to enclose a volume. We demonstrate our approach in a
number of experiments, showing that it produces reconstructions that are
competitive with the state-of-the-art, and we discuss its advantages and
limitations. The algorithm (excluding loop closure functionality) is available
as open source at https://github.com/puzzlepaint/surfelmeshing .Comment: Version accepted to IEEE Transactions on Pattern Analysis and Machine
Intelligenc
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
Computing trisections of 4-manifolds
Algorithms that decompose a manifold into simple pieces reveal the geometric
and topological structure of the manifold, showing how complicated structures
are constructed from simple building blocks. This note describes a way to
algorithmically construct a trisection, which describes a -dimensional
manifold as a union of three -dimensional handlebodies. The complexity of
the -manifold is captured in a collection of curves on a surface, which
guide the gluing of the handelbodies. The algorithm begins with a description
of a manifold as a union of pentachora, or -dimensional simplices. It
transforms this description into a trisection. This results in the first
explicit complexity bounds for the trisection genus of a -manifold in terms
of the number of pentachora (-simplices) in a triangulation.Comment: 15 pages, 9 figure
Obtaining Self-similar Scalings in Focusing Flows
The surface structure of converging thin fluid films displays self-similar
behavior, as was shown in the work by Diez et al [Q. Appl. Math 210, 155,
1990]. Extracting the related similarity scaling exponents from either
numerical or experimental data is non-trivial. Here we provide two such
methods. We apply them to experimental and numerical data on converging fluid
films driven by both surface tension and gravitational forcing. In the limit of
pure gravitational driving, we recover Diez' semi-analytic result, but our
methods also allow us to explore the entire regime of mixed capillary and
gravitational driving, up to entirely surface tension driven flows. We find
scaling forms of smoothly varying exponents up to surprisingly small Bond
numbers. Our experimental results are in reasonable agreement with our
numerical simulations, which confirm theoretically obtained relations between
the scaling exponents.Comment: 11 pages, 11 figures, accepted for Phys Rev
Holography in the EPRL Model
In this research announcement, we propose a new interpretation of the EPR
quantization of the BC model using a functor we call the time functor, which is
the first example of a CLa-ren functor. Under the hypothesis that the universe
is in the Kodama state, we construct a holographic version of the model.
Generalisations to other CLa-ren functors and connections to model category
theory are considered.Comment: research announcement. Latex fil
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