Arrangements of curves in the plane — topology, combinatorics, and algorithms
 Herbert Edelsbrunner,
 Leonidas Guibas,
 Janos Pach,
 Richard Pollack,
 Raimund Seidel,
 Micha Sharir
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Abstract
Arrangements of curves in the plane are of fundamental significance in many problems of computational and combinatorial geometry (e.g. motion planning, algebraic cell decomposition, etc.). In this paper we study various topological and combinatorial properties of such arrangements under some mild assumptions on the shape of the curves, and develop basic tools for the construction, manipulation, and analysis of these arrangements. Our main results include a generalization of the zone theorem of [EOS], [CGL] to arrangements of curves (in which we show that the combinatorial complexity of the zone of a curve is nearly linear in the number of curves), and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.
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 Title
 Arrangements of curves in the plane — topology, combinatorics, and algorithms
 Book Title
 Automata, Languages and Programming
 Book Subtitle
 15th International Colloquium Tampere, Finland, July 11–15, 1988 Proceedings
 Pages
 pp 214229
 Copyright
 1988
 DOI
 10.1007/3540194886_118
 Print ISBN
 9783540194880
 Online ISBN
 9783540392910
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 317
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
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 Editors
 Authors

 Herbert Edelsbrunner ^{(1)}
 Leonidas Guibas ^{(2)} ^{(3)}
 Janos Pach ^{(4)} ^{(5)}
 Richard Pollack ^{(5)}
 Raimund Seidel ^{(6)}
 Micha Sharir ^{(4)} ^{(7)}
 Author Affiliations

 1. Computer Science Department, University of Illinois at Urbana, USA
 2. DEC Systems Research Center, USA
 3. Computer Science Department, Stanford University, USA
 4. Courant Institute of Mathematical Sciences, New York University, USA
 5. Mathematical Institute of the Hungarian Academy of Sciences, Hungary
 6. Computer Science Department, University of California at Berkeley, USA
 7. School of Mathematical Sciences, Tel Aviv University, Israel
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