19,192 research outputs found
On Grids in Point-Line Arrangements in the Plane
The famous Szemer\'{e}di-Trotter theorem states that any arrangement of
points and lines in the plane determines incidences, and this
bound is tight. In this paper, we prove the following Tur\'an-type result for
point-line incidence. Let and be two sets of
lines in the plane and let be the set of intersection points
between and . We say that forms a \emph{natural grid} if , and
does not contain the intersection point of some two lines in
for For fixed , we show that any arrangement
of points and lines in the plane that does not contain a natural
grid determines incidences, where
. We also provide a construction of points
and lines in the plane that does not contain a natural grid
and determines at least incidences.Comment: 13 pages, 5 figure
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
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
Intake ground vortex characteristics
The development of ground vortices when an intake operates in close proximity to the ground has been studied computationally for several configurations including front and rear quarter approaching flows as well as tailwind arrangements. The investigations have been conducted at model scale using a generic intake geometry. Reynolds Averaged Navier–Stokes calculations have been used and an initial validation of the computational model has been carried out against experimental data. The computational method has subsequently been applied to configurations that are difficult to test experimentally by including tailwind and rear quarter flows. The results, along with those from a previous compatible study of headwind and pure cross-wind configurations, have been used to assess the ground vortex behaviour under a broad range of velocity ratios and approaching wind angles. The characteristics provide insights on the influence of the size and strength of ground vortices on the overall quality of the flow ingested by the intake
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
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