656 research outputs found

    Survey of two-dimensional acute triangulations

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    AbstractWe give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual

    Generalizations of the Kolmogorov-Barzdin embedding estimates

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    We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.Comment: 45 page

    Linear Complexity Hexahedral Mesh Generation

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    We show that any polyhedron forming a topological ball with an even number of quadrilateral sides can be partitioned into O(n) topological cubes, meeting face to face. The result generalizes to non-simply-connected polyhedra satisfying an additional bipartiteness condition. The same techniques can also be used to reduce the geometric version of the hexahedral mesh generation problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at the 12th ACM Symp. on Computational Geometry. This is the final version, and will appear in a special issue of Computational Geometry: Theory and Applications for papers from SCG '9

    Schnyder woods for higher genus triangulated surfaces, with applications to encoding

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    Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus gg and compute a so-called gg-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus gg and nn vertices in 4n+O(glog(n))4n+O(g \log(n)) bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time O((n+g)g)O((n+g)g), hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometr
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