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Voronoi Diagrams and Convex Hulls of Random Moving Points

2000
*
Discrete & Computational Geometry
*

This report considers the expected combinatorial complexity

doi:10.1007/pl00009505
fatcat:dbuxuqav3ve7rnjwbpdwyyle2m
*of*the Euclidean*Voronoi**diagram**and*the*convex**hull**of*sets*of*n independent*random**points**moving*in unit time between two positions drawn independently ... for the*convex**hull*. ...*Voronoi**Diagrams**and**Convex**Hulls**of**Random**Moving**Points*345*Voronoi**Diagrams**and**Convex**Hulls**of**Random**Moving**Points*351 364 R. A. ...##
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Intersections with random geometric objects

1998
*
Computational geometry
*

tree, a Gabriel graph, a relative neighborhood graph, a Hamiltonian circuit, a furthest

doi:10.1016/s0925-7721(98)00004-2
fatcat:fpbk62ycsbbd7ahj6hebtsfscm
*point**Voronoi**diagram*, a*convex**hull*, a k-dimensional tree,*and*a rectangular grid. ¢ 1998 Elsevier Science B.V. ... We first study the expected size*of*the intersection between a*random**Voronoi**diagram**and*a generic geometric object that consists*of*a finite collection*of*line segments in the plane. ... It remains unchanged if we remove all*points*that are not on the*convex**hull*. Consider n*points*that gives rise to both the*Voronoi**diagram*V*and*the furthest-*point**Voronoi**diagram*G. ...##
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Optimal Compression of a Polyline with Segments and Arcs
[article]

2017
*
arXiv
*
pre-print

The approach described in this paper finds a compressed polyline with a minimum number

arXiv:1604.07476v5
fatcat:43y4b3ulyvgnbmtjaunzoynbou
*of*segments*and*arcs. ... This paper describes an efficient approach to constructing a resultant polyline with a minimum number*of*segments*and*arcs. ... The closest*Voronoi**diagram*for each*point*has a cell; however, that is not the case for the farthest*Voronoi**diagram*. Only*points**of*the*convex**hull*have cells. ...##
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Computing Voronoi Treemaps: Faster, Simpler, and Resolution-independent

2012
*
Computer graphics forum (Print)
*

J.:

doi:10.1111/j.1467-8659.2012.03078.x
fatcat:mc33x44dxzfsngifrufaci5awq
*Convex**hulls**of*finite [BDL05] BALZER M., D EUSSEN O., L EWERENTZ C.:*Voronoi*sets*of**points*in two*and*three dimensions. ...*moved*to the average*of*all sample*points*chical partition, subdivide this region by a*Voronoi**diagram*closest to them. ...##
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Distributed k-Coverage Decision Scheme for System Deployment in Mobile Sensor Networks

2013
*
International Journal of Distributed Sensor Networks
*

In our schemes, the th order

doi:10.1155/2013/485250
fatcat:d66fmuqokzc5da5wbzteqymlfm
*Voronoi**diagram*is used to discover the regions that do not meet the -coverage requirement. ... This is because an efficient topology structure significantly affects the quality*of*service*and*lifetime*of*WSNs. ... In particular, the*point*set contains one*point*; the*convex**hull**of*the set , ( ) = { 1 , 2 , . . . , }*and*sort the*points*in anti-clock order; (2) Divide the*convex**hull*with sides into − 2 triangles ...##
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Voronoi Diagrams and Delaunay Triangulations
[chapter]

2004
*
Handbook of Discrete and Computational Geometry, Second Edition
*

6 Further reading Survey papers by Aurenhammer 5 a n d F ortune 15 cover many aspects

doi:10.1201/9781420035315.ch23
fatcat:n7e7x75e4rahta6fzawznhw3pe
*of*Delaunay triangulations*and**Voronoi**diagrams*. ... The book by O k abe, Boots,*and*Sugihara 24 i s entirely devoted to*Voronoi**diagrams*,*and*has an extensive discussion*of*applications. Basic reference for geometric algorithms are 13, 2 5 . ...*of*the*convex**hull**of*T . ...##
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VORONOI DIAGRAMS and DELAUNAY TRIANGULATIONS
[chapter]

1992
*
Lecture Notes Series on Computing
*

6 Further reading Survey papers by Aurenhammer 5 a n d F ortune 15 cover many aspects

doi:10.1142/9789814355858_0006
fatcat:l6z2qh3jlbdp7ld27tvxq35hbu
*of*Delaunay triangulations*and**Voronoi**diagrams*. ... The book by O k abe, Boots,*and*Sugihara 24 i s entirely devoted to*Voronoi**diagrams*,*and*has an extensive discussion*of*applications. Basic reference for geometric algorithms are 13, 2 5 . ...*of*the*convex**hull**of*T . ...##
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VORONOI DIAGRAMS and DELAUNAY TRIANGULATIONS
[chapter]

1995
*
Lecture Notes Series on Computing
*

6 Further reading Survey papers by Aurenhammer 5 a n d F ortune 15 cover many aspects

doi:10.1142/9789812831699_0007
fatcat:hrohranyebethhmym46wlqumtu
*of*Delaunay triangulations*and**Voronoi**diagrams*. ... The book by O k abe, Boots,*and*Sugihara 24 i s entirely devoted to*Voronoi**diagrams*,*and*has an extensive discussion*of*applications. Basic reference for geometric algorithms are 13, 2 5 . ...*of*the*convex**hull**of*T . ...##
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Java Applets for the Dynamic Visualization of Voronoi Diagrams
[chapter]

2003
*
Lecture Notes in Computer Science
*

We discuss the design

doi:10.1007/3-540-36477-3_14
fatcat:jxtmkiuhzzeavbmqdsfpijxkua
*of*Java applets that visualize how the*Voronoi**diagram**of*n*points*continuously changes as individual*points*are*moved*across the plane, or as the underlying distance function is ... Moreover, we report on some experiences made in using these applets in teaching*and*research. The applets can be found*and*tried out at http://wwwpi6.fernuni-hagen.de/GeomLab/. ... Theorem 1 A*point*site's*Voronoi*region is unbounded if*and*only if the*point*lies on the*convex**hull**of*the set*of*all sites. ...##
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Recent Developments and Open Problems in Voronoi Diagrams

2006
*
2006 3rd International Symposium on Voronoi Diagrams in Science and Engineering
*

, defines a -simplex if that is the

doi:10.1109/isvd.2006.30
dblp:conf/isvd/Bereg06
fatcat:vhn4kobn3zblzj4tcspaq2jcgu
*convex**hull**of*. ... The expected complexity*of*the*Voronoi**diagram**of**random**points*in the threedimensional cube is [4] . ...##
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Voronoi diagrams over dynamic scenes

1993
*
Discrete Applied Mathematics
*

Given a finite set S

doi:10.1016/0166-218x(93)90115-5
fatcat:7ee4cbi6fvgy3iykorky4a676a
*of*n*points*in the Euclidean plane [E', we investigate the change*of*the*Voronoi**diagram*VD(S)*and*its dual, the Delaunay triangulation DT(S), under continuous motions*of*the underlying ... There are a lot*of*related problems in computational geometry, as for example the dynamic*convex**hull**and*the dynamic nearest neighbor problem, but also applications in motion planning*and*pattern recognition ... Acknowledgement The author wishes to thank Professor Hartmut Noltemeier for his support*and*Gerhard Albers for the nice implementation. ...##
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The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms

2020
*
Mathematics
*

However, we proved that there always exists a

doi:10.3390/math8040645
fatcat:65rqvub4qzgfzjfy6g5gobhnuq
*convex*configuration in the three-dimensional case. In the application, we can imply the existence*of*the non-empty spherical Laguerre*Voronoi**diagram*. ... Given a set*of*constrained vertex norms, we proved the existence*of*a*convex*configuration with respect to the set*of*distinct constrained vertex norms in the two-dimensional case when the constrained ... Acknowledgments: We would like to thank Masaki Moriguchi*and*Vorapong Suppakitpaisan for their discussions. ...##
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An introduction to randomization in computational geometry

1996
*
Theoretical Computer Science
*

This paper is not a complete survey on

doi:10.1016/0304-3975(95)00174-3
fatcat:bla4hiwv7rfabnd6bfokzsqole
*randomized*algorithms in computational geometry, but an introduction to this subject providing intuitions*and*references. ... First, some basic ideas are illustrated by the sorting problem,*and*then a few results on computational geometry are briefly explained. * ...*Voronoi**diagrams**Points**Voronoi**diagrams*We recall the definition*of*the*Voronoi**diagram*in the plane. ...##
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Average case analysis of dynamic geometric optimization

1996
*
Computational geometry
*

We use as subroutines dynamic algorithms for two other geometric graphs: the farthest neighbor forest

doi:10.1016/0925-7721(95)00018-6
fatcat:tmaeehsaongfdlnh3zxnsihtsm
*and*the rotating caliper graph related to an algorithm for static computation*of**point*set widths*and*... We also use the rotating caliper graph to maintain the diameter, width"*and*minimum enclosing rectangle*of*a*point*set in expected time O(log n) per update. ...*Voronoi**diagram*itself, we keep track*of*the set*of*input*points*within each*diagram*cell, using a*convex**hull*data structure (Fig. 4) . ...##
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An efficient algorithm for the three-dimensional diameter problem

2001
*
Discrete & Computational Geometry
*

We explore a new approach for computing the diameter

doi:10.1007/s004540010086
fatcat:yfeaxdqzm5cbdkng3s5rkqbn4q
*of*n*points*in R3 that is based on the restriction*of*the furthest-*point**Voronoi**diagram*to the*convex**hull*. ... We show that the restricted*Voronoi**diagram*has linear complexity. We present a deterministic algorithm with O(n log2 n) running time. ... By Lemma 6, T'(N) = 0(N log N) + T'(N1) + T'(N2), (1) producing T'(N) = , CH(A) denotes the*convex**hull**of*A,*and*Vor(A) denotes the furthest-*point**Voronoi**diagram**of*ig. 1. ...
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