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Optimal Möbius Transformations for Information Visualization and Meshing
 Meshing, WADS 2001, Lecture Notes in Computer Science 2125
, 2001
"... . We give lineartime quasiconvex programming algorithms for ..."
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. We give lineartime quasiconvex programming algorithms for
Shell maps
 ACM TRANSACTIONS ON GRAPHICS
, 2005
"... A shell map is a bijective mapping between shell space and texture space that can be used to generate smallscale features on surfaces using a variety of modeling techniques. The method is based upon the generation of an offset surface and the construction of a tetrahedral mesh that fills the space ..."
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A shell map is a bijective mapping between shell space and texture space that can be used to generate smallscale features on surfaces using a variety of modeling techniques. The method is based upon the generation of an offset surface and the construction of a tetrahedral mesh that fills the space between the base surface and its offset. By identifying a corresponding tetrahedral mesh in texture space, the shell map can be implemented through a straightforward barycentric coordinate map between corresponding tetrahedra. The generality of shell maps allows texture space to contain geometric objects, procedural volume textures, scalar fields, or other shellmapped objects.
Centroidal Voronoi Diagrams for Isotropic Surface Remeshing
, 2005
"... This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a userspecified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image ..."
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This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a userspecified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly lowpass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.
Sliverfree Three Dimensional Delaunay Mesh Generation
 PH.D THESIS, UIUC
, 2000
"... A key step in the nite element method is to generate wellshaped meshes in 3D. A mesh is wellshaped if every tetrahedron element has a small aspect ratio. It is an old outstanding problem to generate wellshaped Delaunay meshes in three or more dimensions. Existing algorithms do not completely solv ..."
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A key step in the nite element method is to generate wellshaped meshes in 3D. A mesh is wellshaped if every tetrahedron element has a small aspect ratio. It is an old outstanding problem to generate wellshaped Delaunay meshes in three or more dimensions. Existing algorithms do not completely solve this problem, primarily because they can not eliminate all slivers. A sliver is a tetrahedron whose vertices are almost coplanar and whose circumradius is not much larger than its shortest edge length. We present two new algorithms to generate sliverfree Delaunay meshes. The rst algorithm locally moves the vertices of an almostgood mesh, whose tetrahedra have small circumradius to shortest edge length ratio. We show that the Delaunay triangulation of the perturbed mesh vertices is still almost good. Furthermore, most slivers disappear after a mild perturbation of the mesh vertices. The remaining slivers migrate to the boundary where they can be peeled o or can be treated with boundary enforcement heuristics. The second algorithm adds points to generate wellshaped meshes. It is based on the following observations. Any tetrahedron will disappear from the Delaunay triangulation if a point is added inside the circumsphere of the tetrahedron. Among the tetrahedra created by
On Structural and Graph Theoretic Properties of Higher Order Delaunay Graphs
"... Given a set P of n points in the plane, the orderk Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order ..."
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Given a set P of n points in the plane, the orderk Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an orderk Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the orderk Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the orderk Delaunay graph is connected for all k ≥ 0. We show that the orderk Gabriel graph, a subgraph of the orderk Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the orderk Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.
ForceDirected Methods For Smoothing Unstructured Triangular And Tetrahedral Meshes
 In Proceedings of the 9th International Meshing Roundtable
, 2000
"... We develop and implement new algorithms for smoothing triangular and tetrahedral unstructured meshes. Our approach is based on a variation of the forcedirected method used in graph drawing. This method assumes that on each vertex a certain force is applied that moves the vertex relative to its neig ..."
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We develop and implement new algorithms for smoothing triangular and tetrahedral unstructured meshes. Our approach is based on a variation of the forcedirected method used in graph drawing. This method assumes that on each vertex a certain force is applied that moves the vertex relative to its neighbors so that the shapes of its incident elements are improved. The final stable configuration often corresponds to a graph with good global properties. In this paper we show that this method can be successfully applied to mesh smoothing and describe some details of our implementation and test results.
Optimization for first order Delaunay triangulations
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 2009
"... ..."
Emerging Challenges in Computational Topology
 Results of the NFS Workshop on Computational Topology
, 1999
"... Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc. ..."
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Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc.xerox.com. David Eppstein, Univ. of California, Irvine, Dept. of Information & Computer Science, eppstein@ics.uci.edu. Pankaj K. Agarwal, Duke Univ., Dept. of Computer Science, pankaj@cs.duke.edu. Nina Amenta, Univ. of Texas, Austin, Dept. of Computer Sciences, amenta@cs.utexas.edu. Paul Chew, Cornell Univ., Dept. of Computer Science, chew@cs.cornell.edu. Tamal Dey, Ohio State Univ., Dept. of Computer and Information Science, tamaldey@cis.ohiostate.edu. David P. Dobkin, Princeton Univ., Dept. of Computer Science, dpd@cs.princeton.edu. Herbert Edelsbrunner, Duke Univ., Dept. of Computer Science, edels@cs.duke.edu. Cindy Grimm, Brown Univ., Dept. of Computer Science, cmg@cs.brown.edu. Leonid...