4,558 research outputs found
The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory
We derive several results in classical Euclidean elementary geometry using
the steering ellipsoid formalism from quantum mechanics. This gives a
physically motivated derivation of very non-trivial geometric results, some of
which are entirely new. We consider a sphere of radius contained inside
another sphere of radius , with the sphere centres separated by distance
. When does there exist a nested tetrahedron circumscribed about the smaller
sphere and inscribed in the larger? We derive the Grace-Danielsson inequality
as the sole necessary and sufficient condition for the
existence of a nested tetrahedron. Our method also gives the condition for the existence of a nested triangle in the analogous
2-dimensional scenario. These results imply the Euler inequality in 2 and 3
dimensions. Furthermore, we formulate a new inequality that applies to the more
general case of ellipses and ellipsoids.Comment: 8 pages, 1 figure. Published versio
Chaining, Interpolation, and Convexity
We show that classical chaining bounds on the suprema of random processes in
terms of entropy numbers can be systematically improved when the underlying set
is convex: the entropy numbers need not be computed for the entire set, but
only for certain "thin" subsets. This phenomenon arises from the observation
that real interpolation can be used as a natural chaining mechanism. Unlike the
general form of Talagrand's generic chaining method, which is sharp but often
difficult to use, the resulting bounds involve only entropy numbers but are
nonetheless sharp in many situations in which classical entropy bounds are
suboptimal. Such bounds are readily amenable to explicit computations in
specific examples, and we discover some old and new geometric principles for
the control of chaining functionals as special cases.Comment: 21 pages; final version, to appear in J. Eur. Math. So
Relative Critical Points
Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are
critical points of appropriate scalar functions parametrized by the Lie algebra
(or its dual) of the symmetry group. Setting aside the structures - symplectic,
Poisson, or variational - generating dynamical systems from such functions
highlights the common features of their construction and analysis, and supports
the construction of analogous functions in non-Hamiltonian settings. If the
symmetry group is nonabelian, the functions are invariant only with respect to
the isotropy subgroup of the given parameter value. Replacing the parametrized
family of functions with a single function on the product manifold and
extending the action using the (co)adjoint action on the algebra or its dual
yields a fully invariant function. An invariant map can be used to reverse the
usual perspective: rather than selecting a parametrized family of functions and
finding their critical points, conditions under which functions will be
critical on specific orbits, typically distinguished by isotropy class, can be
derived. This strategy is illustrated using several well-known mechanical
systems - the Lagrange top, the double spherical pendulum, the free rigid body,
and the Riemann ellipsoids - and generalizations of these systems
Real-Time Hand Tracking Using a Sum of Anisotropic Gaussians Model
Real-time marker-less hand tracking is of increasing importance in
human-computer interaction. Robust and accurate tracking of arbitrary hand
motion is a challenging problem due to the many degrees of freedom, frequent
self-occlusions, fast motions, and uniform skin color. In this paper, we
propose a new approach that tracks the full skeleton motion of the hand from
multiple RGB cameras in real-time. The main contributions include a new
generative tracking method which employs an implicit hand shape representation
based on Sum of Anisotropic Gaussians (SAG), and a pose fitting energy that is
smooth and analytically differentiable making fast gradient based pose
optimization possible. This shape representation, together with a full
perspective projection model, enables more accurate hand modeling than a
related baseline method from literature. Our method achieves better accuracy
than previous methods and runs at 25 fps. We show these improvements both
qualitatively and quantitatively on publicly available datasets.Comment: 8 pages, Accepted version of paper published at 3DV 201
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