4,392 research outputs found
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
The Stretch Factor of the Delaunay Triangulation Is Less Than 1.998
Let be a finite set of points in the Euclidean plane. Let be a
Delaunay triangulation of . The {\em stretch factor} (also known as {\em
dilation} or {\em spanning ratio}) of is the maximum ratio, among all
points and in , of the shortest path distance from to in
over the Euclidean distance . Proving a tight bound on the stretch
factor of the Delaunay triangulation has been a long standing open problem in
computational geometry.
In this paper we prove that the stretch factor of the Delaunay triangulation
of a set of points in the plane is less than , improving the
previous best upper bound of 2.42 by Keil and Gutwin (1989). Our bound 1.998 is
better than the current upper bound of 2.33 for the special case when the point
set is in convex position by Cui, Kanj and Xia (2009). This upper bound breaks
the barrier 2, which is significant because previously no family of plane
graphs was known to have a stretch factor guaranteed to be less than 2 on any
set of points.Comment: 41 pages, 16 figures. A preliminary version of this paper appeared in
the Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG
2011). This is a revised version of the previous preprint [v1
Quadrotor control for persistent surveillance of dynamic environments
Thesis (M.S.)--Boston UniversityThe last decade has witnessed many advances in the field of small scale unmanned aerial vehicles (UAVs). In particular, the quadrotor has attracted significant attention. Due to its ability to perform vertical takeoff and landing, and to operate in cluttered spaces, the quadrotor is utilized in numerous practical applications, such as reconnaissance and information gathering in unsafe or otherwise unreachable environments.
This work considers the application of aerial surveillance over a city-like environment. The thesis presents a framework for automatic deployment of quadrotors to monitor and react to dynamically changing events. The framework has a hierarchical structure. At the top level, the UAVs perform complex behaviors that satisfy high- level mission specifications. At the bottom level, low-level controllers drive actuators on vehicles to perform the desired maneuvers.
In parallel with the development of controllers, this work covers the implementation of the system into an experimental testbed. The testbed emulates a city using physical objects to represent static features and projectors to display dynamic events occurring on the ground as seen by an aerial vehicle. The experimental platform features a motion capture system that provides position data for UAVs and physical features of the environment, allowing for precise, closed-loop control of the vehicles. Experimental runs in the testbed are used to validate the effectiveness of the developed control strategies
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