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### Voronoi Diagrams and Convex Hulls of Random Moving Points

R. A. Dwyer
2000 Discrete & Computational Geometry
This report considers the expected combinatorial complexity 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.  ...

### Intersections with random geometric objects

Prosenjit Bose, Luc Devroye
1998 Computational geometry
tree, a Gabriel graph, a relative neighborhood graph, a Hamiltonian circuit, a furthest 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.  ...

### Optimal Compression of a Polyline with Segments and Arcs [article]

Alexander Gribov
2017 arXiv   pre-print
The approach described in this paper finds a compressed polyline with a minimum number 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.  ...

### Computing Voronoi Treemaps: Faster, Simpler, and Resolution-independent

Arlind Nocaj, Ulrik Brandes
2012 Computer graphics forum (Print)
J.: 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.  ...

### Distributed k-Coverage Decision Scheme for System Deployment in Mobile Sensor Networks

Guofang Nan, Huidong Lian, Minqiang Li
2013 International Journal of Distributed Sensor Networks
In our schemes, the th order 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  ...

### Voronoi Diagrams and Delaunay Triangulations [chapter]

Steven Fortune
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 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 .  ...

### VORONOI DIAGRAMS and DELAUNAY TRIANGULATIONS [chapter]

STEVEN FORTUNE
1992 Lecture Notes Series on Computing
6 Further reading Survey papers by Aurenhammer 5 a n d F ortune 15 cover many aspects 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 .  ...

### VORONOI DIAGRAMS and DELAUNAY TRIANGULATIONS [chapter]

STEVEN FORTUNE
1995 Lecture Notes Series on Computing
6 Further reading Survey papers by Aurenhammer 5 a n d F ortune 15 cover many aspects 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 .  ...

### Java Applets for the Dynamic Visualization of Voronoi Diagrams [chapter]

Christian Icking, Rolf Klein, Peter Köllner, Lihong Ma
2003 Lecture Notes in Computer Science
We discuss the design 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.  ...

### Recent Developments and Open Problems in Voronoi Diagrams

Sergey Bereg
2006 2006 3rd International Symposium on Voronoi Diagrams in Science and Engineering
, defines a -simplex if that is the convex hull of .  ...  The expected complexity of the Voronoi diagram of random points in the threedimensional cube is  .  ...

### Voronoi diagrams over dynamic scenes

Thomas Roos
1993 Discrete Applied Mathematics
Given a finite set S 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.  ...

### The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms

Supanut Chaidee, Kokichi Sugihara
2020 Mathematics
However, we proved that there always exists a 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.  ...

### An introduction to randomization in computational geometry

Olivier Devillers
1996 Theoretical Computer Science
This paper is not a complete survey on 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.  ...

### Average case analysis of dynamic geometric optimization

David Eppstein
1996 Computational geometry
We use as subroutines dynamic algorithms for two other geometric graphs: the farthest neighbor forest 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) .  ...

### An efficient algorithm for the three-dimensional diameter problem

S. Bespamyatnikh
2001 Discrete & Computational Geometry
We explore a new approach for computing the diameter 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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