5,788 research outputs found
Limits of Voronoi Diagrams
In this thesis we study sets of points in the plane and their Voronoi
diagrams, in particular when the points coincide. We bring together two ways of
studying point sets that have received a lot of attention in recent years:
Voronoi diagrams and compactifications of configuration spaces. We study moving
and colliding points and this enables us to introduce `limit Voronoi diagrams'.
We define several compactifications by considering geometric properties of
pairs and triples of points. In this way we are able to define a smooth, real
version of the Fulton-MacPherson compactification. We show how to define
Voronoi diagrams on elements of these compactifications and describe the
connection with the limit Voronoi diagrams.Comment: PhD thesis, 132 pages, lots of figure
Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions
Let be a set of points and a convex -gon in .
We analyze in detail the topological (or discrete) changes in the structure of
the Voronoi diagram and the Delaunay triangulation of , under the convex
distance function defined by , as the points of move along prespecified
continuous trajectories. Assuming that each point of moves along an
algebraic trajectory of bounded degree, we establish an upper bound of
on the number of topological changes experienced by the
diagrams throughout the motion; here is the maximum length of an
-Davenport-Schinzel sequence, and is a constant depending on the
algebraic degree of the motion of the points. Finally, we describe an algorithm
for efficiently maintaining the above structures, using the kinetic data
structure (KDS) framework
A Voronoi poset
Given a set S of n points in general position, we consider all k-th order
Voronoi diagrams on S, for k=1,...,n, simultaneously. We deduce symmetry
relations for the number of faces, number of vertices and number of circles of
certain orders. These symmetry relations are independent of the position of the
sites in S. As a consequence we show that the reduced Euler characteristic of
the poset of faces equals zero whenever n odd.Comment: 14 pages 4 figure
Vesicle computers: Approximating Voronoi diagram on Voronoi automata
Irregular arrangements of vesicles filled with excitable and precipitating
chemical systems are imitated by Voronoi automata --- finite-state machines
defined on a planar Voronoi diagram. Every Voronoi cell takes four states:
resting, excited, refractory and precipitate. A resting cell excites if it has
at least one excited neighbour; the cell precipitates if a ratio of excited
cells in its neighbourhood to its number of neighbours exceed certain
threshold. To approximate a Voronoi diagram on Voronoi automata we project a
planar set onto automaton lattice, thus cells corresponding to data-points are
excited. Excitation waves propagate across the Voronoi automaton, interact with
each other and form precipitate in result of the interaction. Configuration of
precipitate represents edges of approximated Voronoi diagram. We discover
relation between quality of Voronoi diagram approximation and precipitation
threshold, and demonstrate feasibility of our model in approximation Voronoi
diagram of arbitrary-shaped objects and a skeleton of a planar shape.Comment: Chaos, Solitons & Fractals (2011), in pres
Farthest-Polygon Voronoi Diagrams
Given a family of k disjoint connected polygonal sites in general position
and of total complexity n, we consider the farthest-site Voronoi diagram of
these sites, where the distance to a site is the distance to a closest point on
it. We show that the complexity of this diagram is O(n), and give an O(n log^3
n) time algorithm to compute it. We also prove a number of structural
properties of this diagram. In particular, a Voronoi region may consist of k-1
connected components, but if one component is bounded, then it is equal to the
entire region
Empirical characteristics of different types of pedestrian streams
Reliable empirical data and proper understanding of pedestrian dynamics are
necessary for fire safety design. However, specifications and data in different
handbooks as well as experimental studies differ considerably. In this study,
series of experiments under laboratory conditions were carried out to study the
characteristics of uni- and bidirectional pedestrian streams in straight
corridor. The Voronoi method is used to resolve the fine structure of the
resulting velocity-density relations and spatial dependence of the
measurements. The result shows differences in the shape of the relation for
\rho > 1.0 m-2. The maximal specific flow of unidirectional streams is
significantly larger than that of all bidirectional streams examined
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