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Triangulation Made Easy
We describe a simple and efficient algorithm for two-view triangulation of 3D points from approximate 2D matches based on minimizing the L2 reprojection error. Our iterative algorithm improves on the one by Kanatani et al. by ensuring that in each iteration the epipolar constraint is satisfied. In the case where the two cameras are pointed in the same direction, the method provably converges to an optimal solution in exactly two iterations. For more general camera poses, two iterations are sufficient to achieve convergence to machine precision, which we exploit to devise a fast, non-iterative method. The resulting algorithm amounts to little more than solving a quadratic equation, and involves a fixed, small number of simple matrixvector operations and no conditional branches. We demonstrate that the method computes solutions that agree to very high precision with those of Hartley and Sturm's original polynomial method, though achieves higher numerical stability and 1-4 orders of magnitude greater speed
The Impact of Service Quality on Cooperative Customer Satisfaction (Case Study: of Jakarta Cooperatives)
Cooperatives have provided quality service for their members but the steps taken by cooperatives for members are not very easy, over time it has provided the best service benefits for members, and some members disagree with providing services and some agree to give service. The research method used in this study is a qualitative method which is useful for providing facts and data. Then the techniques used by researchers are source triangulation techniques, data collection technique triangulation, and time triangulation. In this study, the Jakarta Cooperative of educators and education staff has explained the analysis of service quality and satisfaction of cooperative members, expert judgment on service quality must be made on all services available at cooperatives, service treatment and satisfaction of cooperative members must be (equalized) both Honorary, civil servants as well as government employees with work agreements. Meanwhile, all members, employees, and administrators to further enhance the cohesiveness of togetherness to advance the cooperation of educators and education staff in Jakarta
A calculus for ideal triangulations of three-manifolds with embedded arcs
Refining the notion of an ideal triangulation of a compact three-manifold, we
provide in this paper a combinatorial presentation of the set of pairs (M,a),
where M is a three-manifold and a is a collection of properly embedded arcs. We
also show that certain well-understood combinatorial moves are sufficient to
relate to each other any two refined triangulations representing the same
(M,a). Our proof does not assume the Matveev-Pergallini calculus for ideal
triangulations, and actually easily implies this calculus.Comment: 32 pages, 30 figure
Non ambiguous structures on 3-manifolds and quantum symmetry defects
The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic
oriented cusped -manifolds can be split in a "symmetrization" factor and a
"reduced" state sum. We show that these factors are invariants on their own,
that we call "symmetry defects" and "reduced QHI", provided the manifolds are
endowed with an additional "non ambiguous structure", a new type of
combinatorial structure that we introduce in this paper. A suitably normalized
version of the symmetry defects applies to compact -manifolds endowed with
-characters, beyond the case of cusped manifolds. Given a
manifold with non empty boundary, we provide a partial "holographic"
description of the non-ambiguous structures in terms of the intrinsic geometric
topology of . Special instances of non ambiguous structures can be
defined by means of taut triangulations, and the symmetry defects have a
particularly nice behaviour on such "taut structures". Natural examples of taut
structures are carried by any mapping torus with punctured fibre of negative
Euler characteristic, or by sutured manifold hierarchies. For a cusped
hyperbolic -manifold which fibres over , we address the question of
determining whether the fibrations over a same fibered face of the Thurston
ball define the same taut structure. We describe a few examples in detail. In
particular, they show that the symmetry defects or the reduced QHI can
distinguish taut structures associated to different fibrations of . To
support the guess that all this is an instance of a general behaviour of state
sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we
describe similar results about reduced Turaev-Viro invariants.Comment: 58 pages, 32 figures; exposition improved, ready for publicatio
A glimpse of the conformal structure of random planar maps
We present a way to study the conformal structure of random planar maps. The
main idea is to explore the map along an SLE (Schramm--Loewner evolution)
process of parameter and to combine the locality property of the
SLE_{6} together with the spatial Markov property of the underlying lattice in
order to get a non-trivial geometric information. We follow this path in the
case of the conformal structure of random triangulations with a boundary. Under
a reasonable assumption called (*) that we have unfortunately not been able to
verify, we prove that the limit of uniformized random planar triangulations has
a fractal boundary measure of Hausdorff dimension almost surely.
This agrees with the physics KPZ predictions and represents a first step
towards a rigorous understanding of the links between random planar maps and
the Gaussian free field (GFF).Comment: To appear in Commun. Math. Phy
Extremal properties for dissections of convex 3-polytopes
A dissection of a convex d-polytope is a partition of the polytope into
d-simplices whose vertices are among the vertices of the polytope.
Triangulations are dissections that have the additional property that the set
of all its simplices forms a simplicial complex. The size of a dissection is
the number of d-simplices it contains. This paper compares triangulations of
maximal size with dissections of maximal size. We also exhibit lower and upper
bounds for the size of dissections of a 3-polytope and analyze extremal size
triangulations for specific non-simplicial polytopes: prisms, antiprisms,
Archimedean solids, and combinatorial d-cubes.Comment: 19 page
On canonical triangulations of once-punctured torus bundles and two-bridge link complements
We prove the hyperbolization theorem for punctured torus bundles and
two-bridge link complements by decomposing them into ideal tetrahedra which are
then given hyperbolic structures, following Rivin's volume maximization
principle.Comment: This is the version published by Geometry & Topology on 16 September
2006. Appendix by David Fute
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