107 research outputs found
Distance k-Sectors Exist
The bisector of two nonempty sets P and Q in a metric space is the set of all
points with equal distance to P and to Q. A distance k-sector of P and Q, where
k is an integer, is a (k-1)-tuple (C_1, C_2, ..., C_{k-1}) such that C_i is the
bisector of C_{i-1} and C_{i+1} for every i = 1, 2, ..., k-1, where C_0 = P and
C_k = Q. This notion, for the case where P and Q are points in Euclidean plane,
was introduced by Asano, Matousek, and Tokuyama, motivated by a question of
Murata in VLSI design. They established the existence and uniqueness of the
distance trisector in this special case. We prove the existence of a distance
k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in
Euclidean spaces of any (finite) dimension, or more generally, in proper
geodesic spaces (uniqueness remains open). The core of the proof is a new
notion of k-gradation for P and Q, whose existence (even in an arbitrary metric
space) is proved using the Knaster-Tarski fixed point theorem, by a method
introduced by Reem and Reich for a slightly different purpose.Comment: 10 pages, 5 figure
Zone Diagrams in Euclidean Spaces and in Other Normed Spaces
Zone diagram is a variation on the classical concept of a Voronoi diagram.
Given n sites in a metric space that compete for territory, the zone diagram is
an equilibrium state in the competition. Formally it is defined as a fixed
point of a certain "dominance" map.
Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone
diagram for point sites in Euclidean plane, and Reem and Reich showed existence
for two arbitrary sites in an arbitrary metric space. We establish existence
and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary
(finite) dimension, and more generally, in a finite-dimensional normed space
with a smooth and rotund norm. The proof is considerably simpler than that of
Asano et al. We also provide an example of non-uniqueness for a norm that is
rotund but not smooth. Finally, we prove existence and uniqueness for two point
sites in the plane with a smooth (but not necessarily rotund) norm.Comment: Title page + 16 pages, 20 figure
Voronoi Diagrams for Parallel Halflines and Line Segments in Space
We consider the Euclidean Voronoi diagram for a set of parallel halflines in 3-space.
A relation of this diagram to planar power diagrams is shown, and is used to
analyze its geometric and topological properties. Moreover, an easy-to-implement
space sweep algorithm is proposed that computes the Voronoi diagram for parallel halflines
at logarithmic cost per face. Previously only an approximation algorithm for this problem was known.
Our method of construction generalizes to Voronoi diagrams for parallel line segments,
and to higher dimensions
Global Topology of 3D Symmetric Tensor Fields
There have been recent advances in the analysis and visualization of 3D
symmetric tensor fields, with a focus on the robust extraction of tensor field
topology. However, topological features such as degenerate curves and neutral
surfaces do not live in isolation. Instead, they intriguingly interact with
each other. In this paper, we introduce the notion of {\em topological graph}
for 3D symmetric tensor fields to facilitate global topological analysis of
such fields. The nodes of the graph include degenerate curves and regions
bounded by neutral surfaces in the domain. The edges in the graph denote the
adjacency information between the regions and degenerate curves. In addition,
we observe that a degenerate curve can be a loop and even a knot and that two
degenerate curves (whether in the same region or not) can form a link. We
provide a definition and theoretical analysis of individual degenerate curves
in order to help understand why knots and links may occur. Moreover, we
differentiate between wedges and trisectors, thus making the analysis more
detailed about degenerate curves. We incorporate this information into the
topological graph. Such a graph can not only reveal the global structure in a
3D symmetric tensor field but also allow two symmetric tensor fields to be
compared. We demonstrate our approach by applying it to solid mechanics and
material science data sets.Comment: IEEE VIS 202
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