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On levels in arrangements and voronoi diagrams

1991
*
Discrete & Computational Geometry
*

*Voronoi*

*diagrams*of order 1 to k

*in*any dimension. ... This paper gives efficient, randomized algorithms for the following problems: (1) construction of

*levels*of order 1 to k

*in*an

*arrangement*of hyperplanes

*in*any dimension

*and*(2) construction of higher-order ... Hiyher-Order

*Voronoi*

*Diagrams*The construction of higher-order

*Voronoi*

*diagrams*is,

*in*fact, a special case of the construction of higher-order

*levels*

*in*

*arrangements*. ...

##
###
Constructing Levels in Arrangements and Higher Order Voronoi Diagrams

1998
*
SIAM journal on computing (Print)
*

Our approach can also compute the k-

doi:10.1137/s0097539795281840
fatcat:wgncoxmqcrfqvaesiwhdcqllkq
*level*; this yields a randomized algorithm for computing the order-k*Voronoi**diagram*of n points*in*the plane*in*expected time O(k(n − k) log n + n log 3 n). ... We give simple randomized incremental algorithms for computing the ≤k-*level**in*an*arrangement*of n lines*in*the plane or*in*an*arrangement*of n planes*in*R 3 . ... order*Voronoi**diagram**in*the plane)? ...##
###
Constructing levels in arrangements and higher order Voronoi diagrams

1994
*
Proceedings of the tenth annual symposium on Computational geometry - SCG '94
*

Our approach can also compute the k-

doi:10.1145/177424.177521
dblp:conf/compgeom/AgarwalBMS94
fatcat:sfmgnseoqjcdhdnchmaqqh2tmu
*level*; this yields a randomized algorithm for computing the order-k*Voronoi**diagram*of n points*in*the plane*in*expected time O(k(n − k) log n + n log 3 n). ... We give simple randomized incremental algorithms for computing the ≤k-*level**in*an*arrangement*of n lines*in*the plane or*in*an*arrangement*of n planes*in*R 3 . ... order*Voronoi**diagram**in*the plane)? ...##
###
Fast Approximation of High-Order Voronoi Diagrams and Distance Transforms on the GPU

2006
*
Journal of Graphics Tools
*

The kth

doi:10.1080/2151237x.2006.10129229
fatcat:ldwpyxjnxjg5ljxk7k6gftcytu
*level*of an*arrangement*is the set of points at*level*k. A specific*arrangement*of planes*in*3D can be used to compute the kth-order planar*Voronoi**diagram*. ... kth-Nearest Point*Diagram*The kth-order*Voronoi**diagram*is formed by compositing (summing) the top k*levels*of the*arrangement*of tangent planes, as discussed*in*the previous section. ...##
###
Voronoi diagrams and arrangements

1986
*
Discrete & Computational Geometry
*

*In*this framework a

*Voronoi*

*diagram*is a partition of a domain D induced by a finite number of real valued functions

*on*D. ... We propose a uniform

*and*general framework for defining

*and*dealing with

*Voronoi*

*diagrams*. ... We also want to thank Bennett Battaile, Gianfranco Bilardi, Joseph O'Rourke,

*and*Chee Yap for their comments. ...

##
###
Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

2019
*
International Symposium on Algorithms and Computation
*

We study the behavior at infinity of the farthest

doi:10.4230/lipics.isaac.2019.62
dblp:conf/isaac/BarequetPS19
fatcat:qud5yqocmzcdlemmhmxbt7mlwy
*and*the higher-order*Voronoi**diagram*of n line segments or lines*in*a d-dimensional Euclidean space. ... All the d-dimensional cells of the farthest*Voronoi**diagram*are unbounded, its (d−1)-skeleton is connected,*and*it does not have tunnels. ... Edelsbrunner*and*Seidel [13] pointed out that the order-k*Voronoi**diagram*of points*in*R d can be derived from the ≤ k-*level*of an*arrangement*of hyperplanes*in*R d+1 . ...##
###
On Higher Order Voronoi Diagrams of Line Segments
[chapter]

2012
*
Lecture Notes in Computer Science
*

We analyze structural properties of the order-k

doi:10.1007/978-3-642-35261-4_21
fatcat:perl6rf7tfad3a47xpw5x7riia
*Voronoi**diagram*of line segments, which surprisingly has not received any attention*in*the computational geometry literature. ... We show that order-k*Voronoi*regions of line segments may be disconnected;*in*fact a single orderk*Voronoi*region may consist of Ω(n) disjoint faces. ... Then*in*Lemma 7 we analyze the number of unbounded faces of the order-k*Voronoi**diagram**in*dual setting using results*on**arrangements*of wedges [3, 7]*and*(≤k)-*level**in**arrangements*of Jordan-curves ...##
###
Output sensitive and dynamic constructions of higher order voronoi diagrams and levels in arrangements

1993
*
Journal of computer and system sciences (Print)
*

We give efficient, output sensitive algorithms to construct

doi:10.1016/0022-0000(93)90041-t
fatcat:vkmzrakm25ffbibfr6fehij5cq
*Voronoi**diagrams*of order*one*to k of a given collection of sites*in*Rd*and**levels*of order*one*to k*in*a nonredundant*arrangement*of hyperplanes ... We also give efficient dynamic algorithms for the same problems that allow the user to add or delete an object-the object being a site*in*the case of*Voronoi**diagrams**and*a nonredundant hyperplane*in*the ...*Levels**in**arrangements*become important*in*connection with*Voronoi**diagrams*because the construction of a kth order*Voronoi**diagram**in*R" can be reduced to the construction of a k-*level**in*Rd+' [9] . ...##
###
Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space
[chapter]

2010
*
Lecture Notes in Computer Science
*

The framework is sufficiently general to support

doi:10.1007/978-3-642-16007-3_1
fatcat:6yorbvghgfdsle3celaedfdn2i
*diagrams*embedded*on*a family of two-dimensional parametric surfaces*in*R^3. ... A straightforward application of the divide-*and*-conquer approach for computing*Voronoi**diagrams*yields algorithms that are inefficient*in*the worst case. ...*and*support of this work. random point sets it is known that the expected number of intersections between the farthest*and*the nearest*Voronoi**diagrams*is linear. ...##
###
Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

2009
*
2009 Sixth International Symposium on Voronoi Diagrams
*

A straightforward application of the divide-

doi:10.1109/isvd.2009.20
dblp:conf/isvd/SetterSH09
fatcat:pexovgtyq5e4fazlbxxp4ylnpa
*and*-conquer approach for*Voronoi**diagrams*yields algorithms that are inefficient*in*the worst case. ... However, the framework is sufficiently general to support*diagrams*embedded*on*a family of two-dimensional parametric surfaces*in*R 3 . ...*and*support of this work. random point sets it is known that the expected number of intersections between the farthest*and*the nearest*Voronoi**diagrams*is linear. ...##
###
Dilation, smoothed distance, and minimization diagrams of convex functions
[article]

2010
*
arXiv
*
pre-print

Additionally, the

arXiv:0812.0607v2
fatcat:ifbgwo3ggvfe3ncramvb2zashy
*level*sets of the distances from the sites form a family of pseudocircles*in*the plane, all cells*in*the*Voronoi**diagram*are connected,*and*the set of bisectors separating any*one*cell ...*in*the*diagram*from each of the others forms an*arrangement*of pseudolines*in*the plane. ... ACKNOWLEDGEMENTS This work was supported*in*part by NSF grant 0830403*and*by the Office of Naval Research under grant N00014-08-1-1015. ...##
###
Further results on the hyperbolic Voronoi diagrams
[article]

2014
*
arXiv
*
pre-print

*In*Euclidean geometry, it is well-known that the k-order

*Voronoi*

*diagram*

*in*R^d can be computed from the vertical projection of the k-

*level*of an

*arrangement*of hyperplanes tangent to a convex potential ... We investigate the hyperbolic

*Voronoi*

*diagram*

*in*the hyperboloid model

*and*show how it reduces to a Klein-type model using central projections. ... We report further novel results for constructing hyperbolic

*Voronoi*

*diagrams*(HVDs)

*in*Klein model [1]

*and*present yet another approach to get Klein-type affine bisectors/

*diagrams*from the hyperboloid ...

##
###
Arrangements on Parametric Surfaces II: Concretizations and Applications

2010
*
Mathematics in Computer Science
*

We also demonstrate how such

doi:10.1007/s11786-010-0043-4
fatcat:v2tmz3bacva25mrylhvbk64er4
*arrangements*can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces,*and**Voronoi**diagrams*... We describe the algorithms*and*implementation details involved*in*the concretizations of a generic framework that enables exact construction, maintenance,*and*manipulation of*arrangements*embedded*on*certain ... We thank Pavel Emeliyanenko for his work*on*visualizing*arrangements*embedded into algebraic surfaces*in*three-dimensional space, which is utilized*in*Section 3. ...##
###
Role of cell regularity and relative density on elasto-plastic compression response of random honeycombs generated using Voronoi diagrams

2014
*
International Journal of Solids and Structures
*

The

doi:10.1016/j.ijsolstr.2014.07.009
fatcat:poiizj4i6zcitii4inw6kwupmq
*Voronoi*tessellation technique*and*solid modeling methods are used*in*this work to create virtual random structures*and*link cell morphology with the mechanical behavior. ... The effect of relative density*on*structures with different*levels*of randomness is also studied. ...*in**Voronoi**diagrams*shown*in*Fig. 2. ...##
###
Page 10287 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

*In*this paper the author continues his own work

*on*simple abstract

*Voronoi*

*diagrams*(1997). ... Two open problems are stated: 1. to give a tight bound

*on*the complexity of the abstract

*Voronoi*

*diagram*

*and*2. to find an efficient algorithm to compute the general abstract

*Voronoi*

*diagram*. ...

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