Arrangements of curves in the plane - topology, combinatorics, and algorithms
Edelsbrunner H, Guibas L, Pach J, Pollack R, Seidel R, Sharir M. 1992. Arrangements of curves in the plane - topology, combinatorics, and algorithms. Theoretical Computer Science. 92(2), 319–336.
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Author
Edelsbrunner, HerbertISTA 
;
Guibas, Leonidas;
Pach, János;
Pollack, Richard;
Seidel, Raimund;
Sharir, Micha
;
Guibas, Leonidas;
Pach, János;
Pollack, Richard;
Seidel, Raimund;
Sharir, MichaAbstract
Arrangements of curves in the plane are fundamental to many problems in 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 Edelsbrunner (1986) and Chazelle (1985) 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 that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.
Publishing Year
Date Published
1992-01-20
Journal Title
Theoretical Computer Science
Publisher
Elsevier
Acknowledgement
Work on this paper by the first author has been supported by the National Science Foundation under grant CCR-8714565. Work by the third and sixth authors has been supported by Office of Naval Research Grant NOOOl4-82-K-0381, by National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the sixth author has also been supported by a research grant from the NCRD- the Israeli National Council for Research and Development. Work by the fourth author has been supported by National Science Foundation Grant DMS-8501947.
Volume
92
Issue
2
Page
319 - 336
ISSN
IST-REx-ID
Cite this
Edelsbrunner H, Guibas L, Pach J, Pollack R, Seidel R, Sharir M. Arrangements of curves in the plane - topology, combinatorics, and algorithms. Theoretical Computer Science. 1992;92(2):319-336. doi:10.1016/0304-3975(92)90319-B
Edelsbrunner, H., Guibas, L., Pach, J., Pollack, R., Seidel, R., & Sharir, M. (1992). Arrangements of curves in the plane - topology, combinatorics, and algorithms. Theoretical Computer Science. Elsevier. https://doi.org/10.1016/0304-3975(92)90319-B
Edelsbrunner, Herbert, Leonidas Guibas, János Pach, Richard Pollack, Raimund Seidel, and Micha Sharir. “Arrangements of Curves in the Plane - Topology, Combinatorics, and Algorithms.” Theoretical Computer Science. Elsevier, 1992. https://doi.org/10.1016/0304-3975(92)90319-B.
H. Edelsbrunner, L. Guibas, J. Pach, R. Pollack, R. Seidel, and M. Sharir, “Arrangements of curves in the plane - topology, combinatorics, and algorithms,” Theoretical Computer Science, vol. 92, no. 2. Elsevier, pp. 319–336, 1992.
Edelsbrunner H, Guibas L, Pach J, Pollack R, Seidel R, Sharir M. 1992. Arrangements of curves in the plane - topology, combinatorics, and algorithms. Theoretical Computer Science. 92(2), 319–336.
Edelsbrunner, Herbert, et al. “Arrangements of Curves in the Plane - Topology, Combinatorics, and Algorithms.” Theoretical Computer Science, vol. 92, no. 2, Elsevier, 1992, pp. 319–36, doi:10.1016/0304-3975(92)90319-B.
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