19,192 research outputs found

    On Grids in Point-Line Arrangements in the Plane

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    The famous Szemer\'{e}di-Trotter theorem states that any arrangement of nn points and nn lines in the plane determines O(n4/3)O(n^{4/3}) incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for point-line incidence. Let L1\mathcal{L}_1 and L2\mathcal{L}_2 be two sets of tt lines in the plane and let P={ℓ1∩ℓ2:ℓ1∈L1,ℓ2∈L2}P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\} be the set of intersection points between L1\mathcal{L}_1 and L2\mathcal{L}_2. We say that (P,L1∪L2)(P, \mathcal{L}_1 \cup \mathcal{L}_2) forms a \emph{natural t×tt\times t grid} if ∣P∣=t2|P| =t^2, and conv(P)conv(P) does not contain the intersection point of some two lines in Li,\mathcal{L}_i, for i=1,2.i = 1,2. For fixed t>1t > 1, we show that any arrangement of nn points and nn lines in the plane that does not contain a natural t×tt\times t grid determines O(n43−ε)O(n^{\frac{4}{3}- \varepsilon}) incidences, where ε=ε(t)\varepsilon = \varepsilon(t). We also provide a construction of nn points and nn lines in the plane that does not contain a natural 2×22 \times 2 grid and determines at least Ω(n1+114)\Omega({n^{1+\frac{1}{14}}}) incidences.Comment: 13 pages, 5 figure

    Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

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    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

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    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

    Why are most buildings rectangular?

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    Intake ground vortex characteristics

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    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

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    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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