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Delaunay triangulations approximate anchor hulls

Tamal K. Dey, Joachim Giesen, Samrat Goswami
2007 Computational geometry  
We show that, for two-dimensional shapes, the Delaunay triangulation approximates a specific structure which we call anchor hulls.  ...  The corresponding question for the dual Delaunay triangulation is not addressed in the literature.  ...  We establish that a subset of the Delaunay triangulation approximates the anchor hulls of the shape.  ... 
doi:10.1016/j.comgeo.2006.01.001 fatcat:y6rd56iayba2xc4reqannqwhba

MESH GENERATION AND OPTIMAL TRIANGULATION [chapter]

MARSHALL BERN, DAVID EPPSTEIN
1992 Lecture Notes Series on Computing  
Proof: We use the lifting relation between DTs and convex hulls.  ...  Under the mapping that takes (x; y) to (x; y; x 2 + y 2 ), the DT of the input vertices lifts to the lower convex hull, and|due to the input edges|the CDT lifts to a surface above the lower convex hull  ...  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  
Proof: We use the lifting relation between DTs and convex hulls.  ...  Under the mapping that takes (x; y) to (x; y; x 2 + y 2 ), the DT of the input vertices lifts to the lower convex hull, and|due to the input edges|the CDT lifts to a surface above the lower convex hull  ...  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

Meshing skin surfaces with certified topology

N.G.H. Kruithof, G. Vegter
2007 Computational geometry  
The complexity of the mesh is linear in the size of the Delaunay triangulation of the balls, which is worst case optimal.  ...  With each face of this complex we associate an anchor point, which plays a crucial role in the meshing algorithm.  ...  The affine hulls of a Delaunay simplex δ X and its dual Voronoi cell ν X are complementary and orthogonal.  ... 
doi:10.1016/j.comgeo.2006.01.003 fatcat:2gwxoem54ba6ffe6l3ai5ayw3i

Improvement of volume estimation of stockpile of earthworks using a concave hull- footprint

David N. SIRIBA, Sammy M. MATARA, Sammy M. MUSYOKA
2015 Micro, Macro & Mezzo Geoinformation  
Several algorithms for creating concave hulls are suggested, in this paper an algorithm based on Delaunay triangulation and linear referencing was used to create the concave hull.  ...  A comparison of volume estimations of stockpiles taking into consideration the footprint via convex hull, concave hull and manually delineated outline showed that volumes based on the concave hull are  ...  Alpha shapes are considered as a generalization of the convex hull and a sub-graph of the Delaunay triangulation.  ... 
doaj:41b5cf23fee94e209b7ecfb8e64b8f14 fatcat:u6mzxwagzrhibk6r4u2vwvxhkm

Design and Implementation of a Practical Parallel Delaunay Algorithm

G. E. Blelloch, G. L. Miller, J. C. Hardwick, D. Talmor
1999 Algorithmica  
We use the well known reduction of 2D Delaunay triangulation to find the 3D convex hull of points on a paraboloid.  ...  This paper describes the design and implementation of a practical parallel algorithm for Delaunay triangulation that works well on general distributions.  ...  Return DELAUNAY(P L , B L ) ∪ DELAUNAY(P R , B R ) Fig. 3. The projection-based parallel Delaunay triangulation algorithm. Initially B is the convex hull of P.  ... 
doi:10.1007/pl00008262 fatcat:fz4tkkuicnb75b467cmparbccy

A Short Proof of the Toughness of Delaunay Triangulations [article]

Ahmad Biniaz
2019 arXiv   pre-print
An important implication of this result is that Delaunay triangulations have perfect matchings.  ...  Another implication of our result is a proof of the conjecture of Aichholzer et al. (2010) that at least n points are required to block any n-vertex Delaunay triangulation  ...  This Delaunay triangulation can be blocked by n points that are placed outside the convex hull: two points are placed very close to p and n − 2 points are placed very close to the n − 2 convex hull edges  ... 
arXiv:1907.01617v2 fatcat:k5bqub5e5vbhzhdvblruomcrva

The greedy triangulation can be computed from the Delaunay triangulation in linear time

Christos Levcopoulos, Drago Krznaric
1999 Computational geometry  
In this paper we show that for any planar point set S, if the Delaunay triangulation of S is given, then the greedy triangulation of S can be computed in linear worst-case time (and linear space).  ...  The Delaunay triangulation, which is the straight-line dual of the Voronoi diagram, can be produced in O(n log n) worst-case time, and often even faster, by several practical algorithms.  ...  of its convex hull has already been fully triangulated.  ... 
doi:10.1016/s0925-7721(99)00037-1 fatcat:wuhsbwhrqja73ozv255zvsklhe

Adaptive skin meshes coarsening for biomolecular simulation

Xinwei Shi, Patrice Koehl
2011 Computer Aided Geometric Design  
Firstly, the quality of both the surface triangulations and tetrahedral meshes is guaranteed. Secondly, the interface in the tetrahedral mesh is an accurate approximation of the molecular boundary.  ...  Thirdly, our meshes are Delaunay meshes. Finally, the meshes are adaptive to the geometry.  ...  We note that the restricted Delaunay triangulation differs from the constrained Delaunay triangulation as follows: the restricted Delaunay triangulation is the subset of the Delaunay triangulation of a  ... 
doi:10.1016/j.cagd.2011.04.001 pmid:21779137 pmcid:PMC3138625 fatcat:vnf4s6wgo5d57ilcnjcd6juy2i

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  
Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision  ...  The Delaunay triangulation of V has the same combinatorial structure as the underside of the convex hull of V + .  ...  Delaunay triangulations and convex hulls are connected through the wellknown parabolic lifting map of Seidel [13] . Let V be a set of vertices in R 3 for which a Delaunay triangulation is sought.  ... 
doi:10.1145/777792.777821 dblp:conf/compgeom/Shewchuk03 fatcat:kvstswvk35elzpb7jsmezdig4i

Incrementally Constructing and Updating Constrained Delaunay Tetrahedralizations with Finite Precision Coordinates [chapter]

Hang Si, Jonathan Richard Shewchuk
2013 Proceedings of the 21st International Meshing Roundtable  
Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision  ...  The Delaunay triangulation of V has the same combinatorial structure as the underside of the convex hull of V + .  ...  Delaunay triangulations and convex hulls are connected through the wellknown parabolic lifting map of Seidel [13] . Let V be a set of vertices in R 3 for which a Delaunay triangulation is sought.  ... 
doi:10.1007/978-3-642-33573-0_11 dblp:conf/imr/SiS12 fatcat:2uxlroulmfeqflrikys3x47fvy

Experimental results on quadrangulations of sets of fixed points

Prosenjit Bose, Suneeta Ramaswami, Godfried Toussaint, Alain Turki
2002 Computer Aided Geometric Design  
on the dual graph of the triangulations.  ...  Therefore we propose and experimentally investigate a heuristic approach to solve this problem by converting "nice" triangulations to the desired quadrangulations with the aid of maximum matchings computed  ...  new half-ray anchored at the second point of the spiral chain, and so on until all the points in S are marked.  ... 
doi:10.1016/s0167-8396(02)00133-4 fatcat:cisaq4vpjzhg5dcocuv6psbs64

Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates

Hang Si, Jonathan Richard Shewchuk
2013 Engineering with Computers  
Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision  ...  The Delaunay triangulation of V has the same combinatorial structure as the underside of the convex hull of V + .  ...  Delaunay triangulations and convex hulls are connected through the wellknown parabolic lifting map of Seidel [13] . Let V be a set of vertices in R 3 for which a Delaunay triangulation is sought.  ... 
doi:10.1007/s00366-013-0331-0 fatcat:pwtguahmgfcxxmrqbl3n4u364q

Natural neighbor extrapolation using ghost points

Tom Bobach, Gerald Farin, Dianne Hansford, Georg Umlauf
2009 Computer-Aided Design  
However, its restriction to the convex hull of the data sites is a limitation that has not yet been satisfyingly overcome.  ...  Similar methods addressing the extension of Delaunay triangulations past their convex hull have been discussed in [29] , Section 6.3, where it was concluded that the quality of the resulting triangulation  ...  interpolants can be computed directly from the Delaunay triangulation of the data sites, such that the preprocessing amounts to the construction of the Delaunay triangulation and a possible generation  ... 
doi:10.1016/j.cad.2008.08.007 fatcat:flraoqmz35h7tfesm3okbd2fty

Bounded-degree plane geometric spanners in practice [article]

Frederick Anderson, Anirban Ghosh, Matthew Graham, Lucas Mougeot, David Wisnosky
2022 arXiv   pre-print
This was the first algorithm that can construct boundeddegree plane spanners using the classic L 2 -Delaunay triangulation. First, a Delaunay triangulation DT of P is constructed.  ...  Eight of the eleven algorithms use L 2 -Delaunay triangulation as the starting point. The remaining three use L ∞ or T D-Delaunay triangulations, as described later in this section.  ... 
arXiv:2205.03204v1 fatcat:jfklsbxq7vflbj2kpzjuhrfizm
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