Arrangements of curves in the plane — topology, combinatorics, and algorithms
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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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- 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
- pp 214-229
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- Series Title
- Lecture Notes in Computer Science
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- Springer Berlin Heidelberg
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- 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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