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On levels in arrangements and voronoi diagrams
1991
Discrete & Computational Geometry
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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. ...
doi:10.1007/bf02574692
fatcat:tjwkejyi45evroz5kaz5xqggca
Constructing Levels in Arrangements and Higher Order Voronoi Diagrams
1998
SIAM journal on computing (Print)
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Our approach can also compute the k-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)? ...
doi:10.1137/s0097539795281840
fatcat:wgncoxmqcrfqvaesiwhdcqllkq
Constructing levels in arrangements and higher order Voronoi diagrams
1994
Proceedings of the tenth annual symposium on Computational geometry - SCG '94
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Our approach can also compute the k-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)? ...
doi:10.1145/177424.177521
dblp:conf/compgeom/AgarwalBMS94
fatcat:sfmgnseoqjcdhdnchmaqqh2tmu
Fast Approximation of High-Order Voronoi Diagrams and Distance Transforms on the GPU
2006
Journal of Graphics Tools
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The kth 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. ...
doi:10.1080/2151237x.2006.10129229
fatcat:ldwpyxjnxjg5ljxk7k6gftcytu
Voronoi diagrams and arrangements
1986
Discrete & Computational Geometry
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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. ...
doi:10.1007/bf02187681
fatcat:qaoojhms2va7ve7u7ejuzmkqwe
Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions
2019
International Symposium on Algorithms and Computation
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We study the behavior at infinity of the farthest 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 . ...
doi:10.4230/lipics.isaac.2019.62
dblp:conf/isaac/BarequetPS19
fatcat:qud5yqocmzcdlemmhmxbt7mlwy
On Higher Order Voronoi Diagrams of Line Segments
[chapter]
2012
Lecture Notes in Computer Science
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We analyze structural properties of the order-k 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 ...
doi:10.1007/978-3-642-35261-4_21
fatcat:perl6rf7tfad3a47xpw5x7riia
Output sensitive and dynamic constructions of higher order voronoi diagrams and levels in arrangements
1993
Journal of computer and system sciences (Print)
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We give efficient, output sensitive algorithms to construct 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] . ...
doi:10.1016/0022-0000(93)90041-t
fatcat:vkmzrakm25ffbibfr6fehij5cq
Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space
[chapter]
2010
Lecture Notes in Computer Science
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The framework is sufficiently general to support 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. ...
doi:10.1007/978-3-642-16007-3_1
fatcat:6yorbvghgfdsle3celaedfdn2i
Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space
2009
2009 Sixth International Symposium on Voronoi Diagrams
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A straightforward application of the divide-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. ...
doi:10.1109/isvd.2009.20
dblp:conf/isvd/SetterSH09
fatcat:pexovgtyq5e4fazlbxxp4ylnpa
Dilation, smoothed distance, and minimization diagrams of convex functions
[article]
2010
arXiv
pre-print
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Additionally, the 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. ...
arXiv:0812.0607v2
fatcat:ifbgwo3ggvfe3ncramvb2zashy
Further results on the hyperbolic Voronoi diagrams
[article]
2014
arXiv
pre-print
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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 ...
arXiv:1410.1036v1
fatcat:zcovdun6pnd6fbijdkffogpney
Arrangements on Parametric Surfaces II: Concretizations and Applications
2010
Mathematics in Computer Science
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We also demonstrate how such 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. ...
doi:10.1007/s11786-010-0043-4
fatcat:v2tmz3bacva25mrylhvbk64er4
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
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The 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. ...
doi:10.1016/j.ijsolstr.2014.07.009
fatcat:poiizj4i6zcitii4inw6kwupmq
Page 10287 of Mathematical Reviews Vol. , Issue 2004m
[page]
2004
Mathematical Reviews
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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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