1,536 research outputs found
Convex-Arc Drawings of Pseudolines
A weak pseudoline arrangement is a topological generalization of a line
arrangement, consisting of curves topologically equivalent to lines that cross
each other at most once. We consider arrangements that are outerplanar---each
crossing is incident to an unbounded face---and simple---each crossing point is
the crossing of only two curves. We show that these arrangements can be
represented by chords of a circle, by convex polygonal chains with only two
bends, or by hyperbolic lines. Simple but non-outerplanar arrangements
(non-weak) can be represented by convex polygonal chains or convex smooth
curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was
made as a poster at the 21st International Symposium on Graph Drawing,
Bordeaux, France, September 2013, and published in Lecture Notes in Computer
Science 8242, Springer, 2013, pp. 522--52
Grid Orientations, (d,d + 2)-Polytopes, and Arrangements of Pseudolines
We investigate the combinatorial structure of linear programs on simple d-polytopes with d + 2 facets. These can be encoded by admissible grid orientations. Admissible grid orientations are also obtained through orientation properties of a planar point configuration or by the dual line arrangement. The point configuration and the polytope corresponding to the same grid are related through an extended Gale transform. The class of admissible grid orientations is shown to contain nonrealizable examples, i.e., there are admissible grid orientations which cannot be obtained from a polytope or a point configuration. It is shown, however, that every admissible grid orientation is induced by an arrangement of pseudolines. This later result is used to prove several nontrivial facts about admissible grid orientation
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
Multitriangulations, pseudotriangulations and primitive sorting networks
We study the set of all pseudoline arrangements with contact points which
cover a given support. We define a natural notion of flip between these
arrangements and study the graph of these flips. In particular, we provide an
enumeration algorithm for arrangements with a given support, based on the
properties of certain greedy pseudoline arrangements and on their connection
with sorting networks. Both the running time per arrangement and the working
space of our algorithm are polynomial.
As the motivation for this work, we provide in this paper a new
interpretation of both pseudotriangulations and multitriangulations in terms of
pseudoline arrangements on specific supports. This interpretation explains
their common properties and leads to a natural definition of
multipseudotriangulations, which generalizes both. We study elementary
properties of multipseudotriangulations and compare them to iterations of
pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of
presentatio
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