Arrangements of curves in the plane — topology, combinatorics, and algorithms

  • Herbert Edelsbrunner
  • Leonidas Guibas
  • Janos Pach
  • Richard Pollack
  • Raimund Seidel
  • Micha Sharir
Conference paper

DOI: 10.1007/3-540-19488-6_118

Volume 317 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Edelsbrunner H., Guibas L., Pach J., Pollack R., Seidel R., Sharir M. (1988) Arrangements of curves in the plane — topology, combinatorics, and algorithms. In: Lepistö T., Salomaa A. (eds) Automata, Languages and Programming. ICALP 1988. Lecture Notes in Computer Science, vol 317. Springer, Berlin, Heidelberg

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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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Herbert Edelsbrunner
    • 1
  • Leonidas Guibas
    • 2
    • 3
  • Janos Pach
    • 4
    • 5
  • Richard Pollack
    • 5
  • Raimund Seidel
    • 6
  • Micha Sharir
    • 4
    • 7
  1. 1.Computer Science DepartmentUniversity of Illinois at UrbanaUSA
  2. 2.DEC Systems Research CenterUSA
  3. 3.Computer Science DepartmentStanford UniversityUSA
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityUSA
  5. 5.Mathematical Institute of the Hungarian Academy of SciencesHungary
  6. 6.Computer Science DepartmentUniversity of California at BerkeleyUSA
  7. 7.School of Mathematical SciencesTel Aviv UniversityIsrael