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On the average length of Delaunay triangulations

R. C. Chang, R. C. T. Lee
1984 BIT Numerical Mathematics  
We shall show that on the average, the total length of a Delaunay triangulation is of the same order as that of a minimum triangulation, under the assumption that our points are drawn from a homogeneous  ...  planar Poisson point distribution.  ...  This means that the expected number of points in each domain X 1 ..... X m is a random variable depending only on the area IX~l (Lebesgue measure), neither on the shape nor the position of the X[s.  ... 
doi:10.1007/bf02136025 fatcat:cjbps7sqnjbsdfj2dh755bsvum

Page 5720 of Mathematical Reviews Vol. , Issue 92j [page]

1992 Mathematical Reviews  
Summary: “An algorithm is presented for constructing a domain Delaunay triangulation (DDT) of an arbitrarily shaped, multiply connected (manifold or nonmanifold), planar domain.  ...  shaped planar domains.  ... 

Large mesh generation from boundary models with parametric face representation

Reinhard Klein, Wolfgang Straber
1995 Proceedings of the third ACM symposium on Solid modeling and applications - SMA '95  
Instead of triangulating each face seperately, the faces of the solid are glued and triangulated together according to the neighbourhood information of the BRep.  ...  We therefore turn our attention to a new incremental algorithm for the accurate triangulation of meshes of trimmed parametric surfaces.  ...  Hüttner for his efforts on all the major implementation issues and the fruitful discussions that led to the realization of this algorithm.  ... 
doi:10.1145/218013.218097 dblp:conf/sma/KleinS95 fatcat:5sbgllvptjcanhyguolztsegha

Unstructured Mesh Generation [chapter]

Jonathan Shewchuk
2012 Combinatorial Scientific Computing  
(In that case, the Delaunay triangulation is the planar dual of the well-known Voronoi diagram.)  ...  A conforming Delaunay triangulation is a Delaunay mesh such that every domain edge is a union of mesh edges, and every domain face is a union of mesh faces.  ...  An algorithm by Labelle [67] is a true hybrid of the Delaunay and octree methods: it refines a Delaunay triangulation by inserting vertices at lattice points near the circumcenters of skinny tetrahedra  ... 
doi:10.1201/b11644-11 fatcat:zjfel6436jhwlidqrtkvjt3ezy

CAD Tools for Creating Space-filing 3D Escher Tiles

Mark Howison, Carlo H. Séquin
2009 Computer-Aided Design and Applications  
Delaunay triangulation library and a mesh-cutting algorithm used in layering extruded tiles to create more intricate designs.  ...  Starting with the simplest case of extruded 2D tilings, we describe geometric algorithms used for maintaining boundary representations of 3D tiles, including a Java implementation of an interactive constrained  ...  Such tilings can also be created in non-planar domains.  ... 
doi:10.3722/cadaps.2009.737-748 fatcat:gd4sqn6d45eidobmidszjjlana

Effective Gouge-Free Tool Selection for Free-Form Surface Machining

Daoshan OuYang, Hsi-Yung Feng, Benjamin A. Van Nest, Ralph O. Buchal
2009 Computer-Aided Design and Applications  
Delaunay triangulation library and a mesh-cutting algorithm used in layering extruded tiles to create more intricate designs.  ...  Starting with the simplest case of extruded 2D tilings, we describe geometric algorithms used for maintaining boundary representations of 3D tiles, including a Java implementation of an interactive constrained  ...  Such tilings can also be created in non-planar domains.  ... 
doi:10.3722/cadaps.2009.839-849 fatcat:zbjcz2fxkraktg25hyjpfs3yg4

ANISOTROPIC TRIANGULATION OF PARAMETRIC SURFACES VIA CLOSE PACKING OF ELLIPSOIDS

KENJI SHIMADA, ATSUSHI YAMADA, TAKAYUKI ITOH
2000 International journal of computational geometry and applications  
The centers of the ellipsoids are then connected by anisotropic Delaunay triangulation for a complete mesh topology.  ...  This paper describes a new computational method of fully automated anisotropic triangulation of a trimmed parametric surface.  ...  However, there are several unique characteristics that make this method particularly suitable for FEM mesh generation: • A bubble system can triangulate a curved domain, a planar domain, a surface domain  ... 
doi:10.1142/s0218195900000243 fatcat:7ckd2uobcbb7vphux7nzwi7eai

MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1992 Lecture Notes Series on Computing  
An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles.  ...  We especially focus on optimal triangulations of geometric domains in two-and three-dimensions.  ...  This material spans a spectrum from purely theoretical results (for example, Chazelle's linear-time triangulation algorithm), through a middle ground (our own work on Steiner triangulations), to practical  ... 
doi:10.1142/9789814355858_0002 fatcat:zjlfgakee5df7nihosmpeuhco4

MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1995 Lecture Notes Series on Computing  
An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles.  ...  We especially focus on optimal triangulations of geometric domains in two-and three-dimensions.  ...  This material spans a spectrum from purely theoretical results (for example, Chazelle's linear-time triangulation algorithm), through a middle ground (our own work on Steiner triangulations), to practical  ... 
doi:10.1142/9789812831699_0003 fatcat:ec6436f5i5bszheuiherz452ae

Delaunay Mesh Construction [article]

Ramsay Dyer, Hao Zhang, Torsten Moeller
2007 Symposium on geometry processing : [proceedings]. Symposium on Geometry Processing  
We further examine the usefulness of trading off the geometry-preserving feature of our algorithm with the ability to create fewer triangles.  ...  We demonstrate the performance of our algorithms through several experiments.  ...  Acknowledgments: We are grateful to John Li for providing the implementation of his nonobtuse decimation algorithm, which we modified to produce Delaunay meshes.  ... 
doi:10.2312/sgp/sgp07/273-282 fatcat:v7ovrydo6rd65juc7cuf2eceny

Well-Centered Triangulation

Evan VanderZee, Anil N. Hirani, Damrong Guoy, Edgar A. Ramos
2010 SIAM Journal on Scientific Computing  
We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations.  ...  We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.  ...  Lastly, we thank the reviewers; their comments led to improvement of the paper.  ... 
doi:10.1137/090748214 fatcat:35lftukgqjhm5m2ikc2dhaetoy

Delaunay Quadrangulation by Two-coloring Vertices

Scott A. Mitchell, Mohammed A. Mohammed, Ahmed H. Mahmoud, Mohamed S. Ebeida
2014 Procedia Engineering  
We introduce a bichromatic Delaunay quadrangulation principle by assigning the vertices of a Delaunay triangulation one of two colors, then discarding edges between vertices of the same color.  ...  We present two new sphere-packing algorithms for generating the colored triangulation, and we may also take as input a Delaunay refinement mesh and color it arbitrarily.  ...  Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.  ... 
doi:10.1016/j.proeng.2014.10.397 fatcat:zsaeqhuqjfbpdl44nksijs5i4m

From Segmented Images to Good Quality Meshes Using Delaunay Refinement [chapter]

Jean-Daniel Boissonnat, Jean-Philippe Pons, Mariette Yvinec
2009 Lecture Notes in Computer Science  
Moreover, it offers extensive control over the size and shape of mesh elements, for instance through a (possibly non-uniform) sizing field.  ...  This paper surveys Delaunay-based meshing techniques for curved objects, and their application in medical imaging and in computer vision to the extraction of geometric models from segmented images.  ...  Fig. 3 . 3 The medial axis of a planar curve (only the portion inside the domain bounded by the curve is shown). The thin curves are parallel to the boundary of the domain.  ... 
doi:10.1007/978-3-642-00826-9_2 fatcat:raeabxptmfhhpis3irgby5d4ri

Variational Delaunay approach to the generation of tetrahedral finite element meshes

Petr Krysl, Michael Ortiz
2001 International Journal for Numerical Methods in Engineering  
To determine if a constraining facet is present in the unconstrained Delaunay triangulation of the current vertex set, we use the results of Rajan which re-formulate Delaunay triangulation as a linear  ...  The vertex set positions are randomized to guarantee existence of a unique triangulation which satisfies the Delaunay empty-sphere property.  ...  Acknowledgements We are grateful for support from the Department of Energy through Caltech's ASCI Center of Excellence for Simulating Dynamic Response of Materials.  ... 
doi:10.1002/nme.91 fatcat:2mhfj64seneodg742vc7udyfnm

On the number of higher order Delaunay triangulations

Dieter Mitsche, Maria Saumell, Rodrigo I. Silveira
2011 Theoretical Computer Science  
Higher order Delaunay triangulations are a generalization of the Delaunay triangulation that provides a class of well-shaped triangulations, over which extra criteria can be optimized.  ...  We show that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delaunay triangulations, the maximum number is 2 n−3 .  ...  Acknowledgements We are grateful to Nick Wormald for his help on dealing with the integrals of Section 4. We would also like to thank Christiane Schmidt and Kevin Verbeek for helpful discussions.  ... 
doi:10.1016/j.tcs.2011.03.005 fatcat:iflyhd4tqnbi3gpbgd7pochsyi
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