On the number of line separations of a finite set in the plane
Herbert Edelsbrunner,Emo Welzl +1 more
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TLDR
The asymptotic behaviour of f k ( n) is studied as this function has applications to a number of problems in computational geometry and a lower and an upper bound on fk ( n ) is established.About:
This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1985-01-01 and is currently open access. It has received 90 citations till now. The article focuses on the topics: Upper and lower bounds & Function (mathematics).read more
Citations
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Voronoi diagrams—a survey of a fundamental geometric data structure
TL;DR: The Voronoi diagram as discussed by the authors divides the plane according to the nearest-neighbor points in the plane, and then divides the vertices of the plane into vertices, where vertices correspond to vertices in a plane.
Book
Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Book
Research Problems in Discrete Geometry
TL;DR: This chapter discusses 100 Research Problems in Discrete Geometry from the Facsimile edition of the World Classics in Mathematics Series, vol.
Journal ArticleDOI
Power diagrams: properties, algorithms and applications
TL;DR: The close relationship to convex hulls and arrangements of hyperplanes is investigated and exploited, and efficient algorithms that compute the power diagram and its order-k modifications are obtained.
Journal ArticleDOI
New applications of random sampling in computational geometry
TL;DR: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
References
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Proceedings ArticleDOI
Closest-point problems
Michael Ian Shamos,Dan Hoey +1 more
TL;DR: The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.
Journal ArticleDOI
Data Structures for Range Searching
TL;DR: The purpose of this paper is to acquaint the reader with the structures currently avadable for solving the particular problem of range searching, and to display a set of general methods for attacking multikey searching problems.
MonographDOI
Arrangements and Spreads
TL;DR: In this paper, the pseudolines and arrangements of curves spreads of curves have been used to create pseudoline-based arrangements of lines and pseudoline pseudoline spreads.
Journal ArticleDOI
On k-Nearest Neighbor Voronoi Diagrams in the Plane
TL;DR: It is shown that the k-nearest neighbor problem and other seemingly unrelated problems can be solved efficiently with the Voronoi diagram.
Book ChapterDOI
Dissection Graphs of Planar Point Sets
TL;DR: In this paper, the authors discuss the dissection graphs of planar point sets and discuss the general properties of the graphs G k, which can be constructed as follows: any oriented line is to be translated to its left until it meets a point p 1 of S. This line will be called l (0).