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A constructive approach to constrained hexahedral mesh generation
 In Proceedings, 15th International Meshing Roundtable
, 2006
"... S. Mitchell proved that a necessary and sufficient condition for the existence of a topological hexahedral mesh constrained to a quadrilateral mesh on the sphere is that the constraining quadrilateral mesh contains an even number of elements. S. Mitchell’s proof depends on S. Smale’s theorem on the ..."
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S. Mitchell proved that a necessary and sufficient condition for the existence of a topological hexahedral mesh constrained to a quadrilateral mesh on the sphere is that the constraining quadrilateral mesh contains an even number of elements. S. Mitchell’s proof depends on S. Smale’s theorem on the regularity of curves on compact manifolds. Although the question of the existence of constrained hexahedral meshes has been solved, the known solution is not easily programmable; indeed, there are cases, such as Schneider’s pyramid, that are not easily solved. D. Eppstein later utilized portions of S. Mitchell’s existence proof to demonstrate that hexahedral mesh generation has linear complexity. In this paper, we demonstrate a constructive proof to the existence theorem for the sphere, as well as assign an upperbound to the constant of the linear term in the asymptotic complexity measure provided by D. Eppstein. Our construction generates 76*n hexahedra elements within the solid where n is the number of quadrilaterals on the boundary. The construction presented is used to solve some open problems posed by R. Schneiders and D. Eppstein. We will also use the results provided in this paper, in conjunction with S. Mitchell’s GeodeTemplate, to create an alternative way of creating a constrained hexahedral mesh. The construction utilizing the GeodeTemplate requires 130*n hexahedra, but will have fewer topological irregularities in the final mesh. 1
Topology and intelligent data analysis
 IDA03 (International Symposium on Intelligent Data Analysis
, 2003
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Representing Topological Structures Using CellChains ⋆
"... Abstract. A new topological representation of surfaces in higher dimensions, “cellchains ” is developed. The representation is a generalization of Brisson’s celltuple data structure. Cellchains are identical to celltuples when there are no degeneracies: cells or simplices with identified vertices ..."
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Abstract. A new topological representation of surfaces in higher dimensions, “cellchains ” is developed. The representation is a generalization of Brisson’s celltuple data structure. Cellchains are identical to celltuples when there are no degeneracies: cells or simplices with identified vertices. The proof of correctness is based on axioms true for maps, such as those in Brisson’s celltuple representation. A critical new condition (axiom) is added to those of Lienhardt’s nGmaps to give “cellmaps”. We show that cellmaps and cellchains characterize the same topological representations.
Published In Representing Topological Structures Using CellChains⋆
"... Abstract. A new topological representation of surfaces in higher dimensions, “cellchains ” is developed. The representation is a generalization of Brisson’s celltuple data structure. Cellchains are identical to celltuples when there are no degeneracies: cells or simplices with identified verti ..."
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Abstract. A new topological representation of surfaces in higher dimensions, “cellchains ” is developed. The representation is a generalization of Brisson’s celltuple data structure. Cellchains are identical to celltuples when there are no degeneracies: cells or simplices with identified vertices. The proof of correctness is based on axioms true for maps, such as those in Brisson’s celltuple representation. A critical new condition (axiom) is added to those of Lienhardt’s nGmaps to give “cellmaps”. We show that cellmaps and cellchains characterize the same topological representations.
Efficient Computation of Proximity Graphs (Short Abstract)
"... The problem of efficiently reconstructing the shape from a set of scattered points on the plane has been recently mentioned as one of the emerging challenges in the computational geometry and computational topology fields [2]. Namely, efficient algorithms that can extract a shape from a discrete sa ..."
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The problem of efficiently reconstructing the shape from a set of scattered points on the plane has been recently mentioned as one of the emerging challenges in the computational geometry and computational topology fields [2]. Namely, efficient algorithms that can extract a shape from a discrete sample of points have immediate applications in areas like visual perception, computer vision and pattern recognition, geography and cartography, data mining, and biology, to cite a few (see, e.g., [9]). A typical approach to extracting a shape from a given set P of points is to compute a proximity graph of P , i.e. a geometric graph whose vertices are elements of P and where two points are connected by an edge if they are deemed close according to some proximity measure. It is the definition of closeness which determines different types of proximity graphs on the same point set. We give here only a few examples. The Gabriel graph [3] is a proximi...