3,021 research outputs found
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
Simple Wriggling is Hard unless You Are a Fat Hippo
We prove that it is NP-hard to decide whether two points in a polygonal
domain with holes can be connected by a wire. This implies that finding any
approximation to the shortest path for a long snake amidst polygonal obstacles
is NP-hard. On the positive side, we show that snake's problem is
"length-tractable": if the snake is "fat", i.e., its length/width ratio is
small, the shortest path can be computed in polynomial time.Comment: A shorter version is to be presented at FUN 201
Liftings and stresses for planar periodic frameworks
We formulate and prove a periodic analog of Maxwell's theorem relating
stressed planar frameworks and their liftings to polyhedral surfaces with
spherical topology. We use our lifting theorem to prove deformation and
rigidity-theoretic properties for planar periodic pseudo-triangulations,
generalizing features known for their finite counterparts. These properties are
then applied to questions originating in mathematical crystallography and
materials science, concerning planar periodic auxetic structures and ultrarigid
periodic frameworks.Comment: An extended abstract of this paper has appeared in Proc. 30th annual
Symposium on Computational Geometry (SOCG'14), Kyoto, Japan, June 201
Travel time stability in weakly range-dependent sound channels
Travel time stability is investigated in environments consisting of a
range-independent background sound-speed profile on which a highly structured
range-dependent perturbation is superimposed. The stability of both
unconstrained and constrained (eigenray) travel times are considered. Both
general theoretical arguments and analytical estimates of time spreads suggest
that travel time stability is largely controlled by a property of the background sound speed profile. Here, is
the range of a ray double loop and is the ray action variable. Numerical
results for both volume scattering by internal waves in deep ocean environments
and rough surface scattering in upward refracting environments are shown to
confirm the expectation that travel time stability is largely controlled by
.Comment: Submitted to J. Acoust. Soc. Am., 30 June 200
Holographic Holes and Differential Entropy
Recently, it has been shown by Balasubramanian et al. and Myers et al. that
the Bekenstein-Hawking entropy formula evaluated on certain closed surfaces in
the bulk of a holographic spacetime has an interpretation as the differential
entropy of a particular family of intervals (or strips) in the boundary theory.
We first extend this construction to bulk surfaces which vary in time. We then
give a general proof of the equality between the gravitational entropy and the
differential entropy. This proof applies to a broad class of holographic
backgrounds possessing a generalized planar symmetry and to certain classes of
higher-curvature theories of gravity. To apply this theorem, one can begin with
a bulk surface and determine the appropriate family of boundary intervals by
considering extremal surfaces tangent to the given surface in the bulk.
Alternatively, one can begin with a family of boundary intervals; as we show,
the differential entropy then equals the gravitational entropy of a bulk
surface that emerges from the intersection of the neighboring entanglement
wedges, in a continuum limit.Comment: 62 pages; v2: minor improvements to presentation, references adde
Goal Set Inverse Optimal Control and Iterative Re-planning for Predicting Human Reaching Motions in Shared Workspaces
To enable safe and efficient human-robot collaboration in shared workspaces
it is important for the robot to predict how a human will move when performing
a task. While predicting human motion for tasks not known a priori is very
challenging, we argue that single-arm reaching motions for known tasks in
collaborative settings (which are especially relevant for manufacturing) are
indeed predictable. Two hypotheses underlie our approach for predicting such
motions: First, that the trajectory the human performs is optimal with respect
to an unknown cost function, and second, that human adaptation to their
partner's motion can be captured well through iterative re-planning with the
above cost function. The key to our approach is thus to learn a cost function
which "explains" the motion of the human. To do this, we gather example
trajectories from pairs of participants performing a collaborative assembly
task using motion capture. We then use Inverse Optimal Control to learn a cost
function from these trajectories. Finally, we predict reaching motions from the
human's current configuration to a task-space goal region by iteratively
re-planning a trajectory using the learned cost function. Our planning
algorithm is based on the trajectory optimizer STOMP, it plans for a 23 DoF
human kinematic model and accounts for the presence of a moving collaborator
and obstacles in the environment. Our results suggest that in most cases, our
method outperforms baseline methods when predicting motions. We also show that
our method outperforms baselines for predicting human motion when a human and a
robot share the workspace.Comment: 12 pages, Accepted for publication IEEE Transaction on Robotics 201
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