121 research outputs found

    Surface discretisation with rectifying strips on Geodesics

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    The use of geodesic curves of surfaces has enormous potential in architecture due to their multiple properties and easy geometric control using digital graphic tools. Among their numerous properties, the geodesic curves of a surface are the paths along which straight strips can be placed tangentially to the surface. On this basis, a graphical method is proposed to discretize surfaces using straight strips, which optimizes material consumption since rectangular straight strips take advantage of 100% of the material in the cutting process. The contribution of the article consists of presenting the geometric constraints that characterize this type of panelling from the idea of “rectifying surface”, considering the material inextensible. Experimental prototypes that have been part of the research are also described and the final theoretical results are presented

    Hyperbolic intersection graphs and (quasi)-polynomial time

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    We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in dd-dimensional hyperbolic space, which we denote by Hd\mathbb{H}^d. Using a new separator theorem, we show that unit ball graphs in Hd\mathbb{H}^d enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in 2O(n11/(d1))2^{O(n^{1-1/(d-1)})} time for any fixed d3d\geq 3, while the same problems need 2O(n11/d)2^{O(n^{1-1/d})} time in Rd\mathbb{R}^d. We also show that these algorithms in Hd\mathbb{H}^d are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in H2\mathbb{H}^2, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial (nO(logn)n^{O(\log n)}) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and 33-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in H2\mathbb{H}^2 have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require 2Ω(n)2^{\Omega(\sqrt{n})} time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching nΩ(logn)n^{\Omega(\log n)} lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

    Contracting Boundaries of CAT(0) Spaces

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    As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like properties. We prove that these properties are all equivalent and that the contracting boundary is a quasi-isometry invariant. We use this invariant to distinguish the quasi-isometry classes of certain right-angled Coxeter groups.Comment: 27 pages, 8 figure
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