1,139 research outputs found
Polygon Exploration with Time-Discrete Vision
With the advent of autonomous robots with two- and three-dimensional scanning
capabilities, classical visibility-based exploration methods from computational
geometry have gained in practical importance. However, real-life laser scanning
of useful accuracy does not allow the robot to scan continuously while in
motion; instead, it has to stop each time it surveys its environment. This
requirement was studied by Fekete, Klein and Nuechter for the subproblem of
looking around a corner, but until now has not been considered in an online
setting for whole polygonal regions.
We give the first algorithmic results for this important algorithmic problem
that combines stationary art gallery-type aspects with watchman-type issues in
an online scenario: We demonstrate that even for orthoconvex polygons, a
competitive strategy can be achieved only for limited aspect ratio A (the ratio
of the maximum and minimum edge length of the polygon), i.e., for a given lower
bound on the size of an edge; we give a matching upper bound by providing an
O(log A)-competitive strategy for simple rectilinear polygons, using the
assumption that each edge of the polygon has to be fully visible from some scan
point.Comment: 28 pages, 17 figures, 2 photographs, 3 tables, Latex. Updated some
details (title, figures and text) for final journal revision, including
explicit assumption of full edge visibilit
Diophantine approximations for translation surfaces and planar resonant sets
We consider Teichm\"uller geodesics in strata of translation surfaces. We
prove lower and upper bounds for the Hausdorff dimension of the set of
parameters generating a geodesic bounded in some compact part of the stratum.
Then we compute the dimension of those parameters generating geodesics that
make excursions to infinity at a prescribed rate. Finally we compute the
dimension of the set of directions in a rational billiard having fast
recurrence, which corresponds to a dynamical version of a classical result of
Jarn\'ik and Besicovich. Our main tool are planar resonant sets arising from a
given translation surface, that is the countable set of directions of its
saddle connections or of its closed geodesics, filtered according to length. In
an abstract setting, and assuming specific metric properties on a general
planar resonant set, we prove a dichotomy for the Hausdorff measure of the set
of directions which are well approximable by directions in the resonant set,
and we give an estimate on the dimension of the set of badly approximable
directions. Then we prove that the resonant sets arising from a translation
surface satisfy the required metric properties.Comment: Added appendix B, which provides a proof for a statement in Remark
1.10 of the previous version. Minor changes in the rest of the paper. 53
page
A (7/2)-Approximation Algorithm for Guarding Orthogonal Art Galleries with Sliding Cameras
Consider a sliding camera that travels back and forth along an orthogonal
line segment inside an orthogonal polygon with vertices. The camera
can see a point inside if and only if there exists a line segment
containing that crosses at a right angle and is completely contained in
. In the minimum sliding cameras (MSC) problem, the objective is to guard
with the minimum number of sliding cameras. In this paper, we give an
-time -approximation algorithm to the MSC problem on any
simple orthogonal polygon with vertices, answering a question posed by Katz
and Morgenstern (2011). To the best of our knowledge, this is the first
constant-factor approximation algorithm for this problem.Comment: 11 page
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