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General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Jonathan Richard Shewchuk
2007 Discrete & Computational Geometry  
The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained  ...  Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely  ...  Acknowledgments I thank Dafna Talmor and Herbert Edelsbrunner for helpful discussions.  ... 
doi:10.1007/s00454-006-1236-0 fatcat:umbgwlojgrbb3g7yc44gxnivkm

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

Jonathan Richard Shewchuk
2008 Discrete & Computational Geometry  
The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained  ...  Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely  ...  Acknowledgments I thank Dafna Talmor and Herbert Edelsbrunner for helpful discussions.  ... 
doi:10.1007/s00454-008-9060-3 fatcat:js3sg5axlzfvzk4gt6swpqxxcm

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties [chapter]

Jonathan Richard Shewchuk
Twentieth Anniversary Volume:  
The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained  ...  Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely  ...  Acknowledgments I thank Dafna Talmor and Herbert Edelsbrunner for helpful discussions.  ... 
doi:10.1007/978-0-387-87363-3_28 fatcat:xzgp2ig33rhsfgex3koh7etoze

Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions

Evan VanderZee, Anil N. Hirani, Damrong Guoy, Vadim Zharnitsky, Edgar A. Ramos
2013 Computational geometry  
Hirani and Evan VanderZee was supported by an NSF Grant No. DMS-0645604.  ...  Evan VanderZee was also partially supported by a fellowship jointly funded by the Computational Science and Engineering Program and the Applied Mathematics Program of the University of Illinois at Urbana-Champaign  ...  It can be shown that an n-dimensional triangulation consisting of n-well-centered simplices is a Delaunay triangulation, however n-well-centeredness is stronger than Delaunay and is different from other  ... 
doi:10.1016/j.comgeo.2012.11.003 fatcat:ojvekzpx7veg5ouk2j6b7ft5vu

Triangulations in CGAL

Jean-Daniel Boissonnat, Olivier Devillers, Sylvain Pion, Monique Teillaud, Mariette Yvinec
2002 Computational geometry  
This paper presents the main algorithmic and design choices that have been made to implement triangulations in the computational geometry algorithms library CGAL.   ...  Acknowledgements The authors would like to thank Carine Bonetto, Hervé Brönnimann, Frank Da, Andreas Fabri, Frédéric Fichel and François Rebufat for contributing to the CGAL triangulation package.  ...  In the plane, CGAL offers basic, Delaunay and regular triangulations, as well as constrained triangulations and constrained Delaunay triangulations.  ... 
doi:10.1016/s0925-7721(01)00054-2 fatcat:bgfipdphuzdytlcvjwabwhyumm

Sphere packings, I

T. C. Hales
1997 Discrete & Computational Geometry  
Yt) to be the (ordered) simplex whose i th edge has length yi. If S is a Delaunay simplex in a fixed Delaunay star, then it has a distinguished vertex, the vertex common to all simplices in the star.  ...  To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture. S(yl .....  ...  In this section we define the score of a quasi-regular tetrahedron and describe the properties that the score should have in general. Let S be a quasi-regular tetrahedron.  ... 
doi:10.1007/bf02770863 fatcat:6du2crv7v5hvdayynr4wv76m5e

Updating and constructing constrained delaunay and constrained regular triangulations by flips

Jonathan Richard Shewchuk
2003 Proceedings of the nineteenth conference on Computational geometry - SCG '03  
I discuss algorithms based on bistellar flips for inserting and deleting constraining (d − 1)-facets in d-dimensional constrained Delaunay triangulations (CDTs) and weighted CDTs, also known as constrained  ...  regular triangulations.  ...  [2] have shown that regular triangulations are useful in three-dimensional mesh generation.  ... 
doi:10.1145/777819.777821 fatcat:jejpsxe3aveirlp46g7jma3pcq

Parametrization of Generalized Primal-Dual Triangulations [chapter]

Pooran Memari, Patrick Mullen, Mathieu Desbrun
2011 Proceedings of the 20th International Meshing Roundtable  
Using algebraic and computational geometry results, we show that compatible dual complexes exist only for a particular type of triangulation known as weakly regular.  ...  Motivated by practical numerical issues in a number of modeling and simulation problems, we introduce the notion of a compatible dual complex to a primal triangulation, such that a simplicial mesh and  ...  Partial funding was generously provided by the National Science Foundation through grants CCF-1011944, CCF-0811373, and CMMI-0757106.  ... 
doi:10.1007/978-3-642-24734-7_13 dblp:conf/imr/MemariMD11 fatcat:pjbzq7fhebenvmfvj4x7slpnb4

Sphere packings I [article]

Thomas C. Hales
1998 arXiv   pre-print
To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.  ...  We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture.  ...  In this section we define the score of a quasi-regular tetrahedron and describe the properties that the score should have in general. Let S be a quasi-regular tetrahedron.  ... 
arXiv:math/9811073v1 fatcat:p6bk76vqurhvxn6wqjjaxr2jvu

Pre-Triangulations and Liftable Complexes

Oswin Aichholzer, Franz Aurenhammer, Thomas Hackl
2007 Discrete & Computational Geometry  
With the advent of computational geometry, it soon became apparent that combinatorial and geometric properties of certain polygonal complexes prove useful for structuring geometric data and designing efficient  ...  Keywords Pre-triangulations · Pseudo-triangulations · Liftable complexes Introduction Polygonal complexes in the plane have been objects of interest in combinatorial geometry from various points of view  ...  Acknowledgements The authors would like to thank an anonymous referee for his various constructive and clarifying comments.  ... 
doi:10.1007/s00454-007-9032-z fatcat:kbnzhxg4bjhatdvey5hmerdn7y

An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Nikolay Martynchuk
2017 Discrete & Computational Geometry  
Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space R m triangulates the convex hull of P , provided that P satisfies a mild genericity property.  ...  Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds.  ...  We also thank David Cohen-Steiner and Mathijs Wintraecken for illuminating discussions.  ... 
doi:10.1007/s00454-017-9908-5 fatcat:xbcayt3btjg43py6kg264b67dq

An obstruction to Delaunay triangulations in Riemannian manifolds [article]

Jean-Daniel Boissonnat and Ramsay Dyer and Arijit Ghosh and Nikolay Martynchuk
2016 arXiv   pre-print
Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space R^m triangulates the convex hull of P, provided that P satisfies a mild genericity property.  ...  Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds.  ...  We also thank David Cohen-Steiner and Mathijs Wintraecken for illuminating discussions.  ... 
arXiv:1612.02905v1 fatcat:ta7p4jsk2bahvouo2ajrov3jp4

Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

Zhonggui Chen, Wenping Wang, Bruno Lévy, Ligang Liu, Feng Sun
2014 SIAM Journal on Scientific Computing  
In this paper, we study the optimal Delaunay triangulation (ODT) for three-dimensional (3D) graded mesh generation. ODT is defined as the minimizer of an objective function [2] .  ...  This paper proposes a new algorithm to generate a graded three-dimensional tetrahedral mesh.  ...  We use the constrained Delaunay triangulation in the ODT optimization. However, the constrained Delaunay triangulation may not exist in some extreme cases.  ... 
doi:10.1137/120875132 fatcat:yav6t7xsnfeh3p2edojsmhsdhm

Geometric bistellar flips. The setting, the context and a construction [article]

Francisco Santos
2006 arXiv   pre-print
As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations.  ...  We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results  ...  Let t be a sufficiently small and positive constant. Then, 1. There are triangulations of A(t) containing the simplicial complex K.  ... 
arXiv:math/0601746v1 fatcat:o5s3kbdzfrgz3egmngf3s2oloe

Approximation and geometric modeling with simplex B-splines associated with irregular triangles

S. Auerbach, R.H.J. Gmelig Meyling, M. Neamtu, H. Schaeben
1991 Computer Aided Geometric Design  
Bivariate quadratic simplicial B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a Cl-smooth surface  ...  We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide winth the abscissae of the data.  ...  Boehm, Braunschweig, for his introduction into the world of CAGD, and Prof. W. Dahmen, Berlin, Prof. K. Scherer, Bonn, and Prof. C.R.  ... 
doi:10.1016/0167-8396(91)90050-l fatcat:gbafi4wvenh3relzkmlwmk4a2e
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