Lines in Space: Combinatorics and Algorithms

B. Chazelle, H. Edelsbrunner, L. J. Guibas, M. Sharir, J. Stolfi

Research output: Contribution to journalArticlepeer-review

43 Scopus citations


Questions about lines in space arise frequently as subproblems in three-dimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in three-dimensional space. Our main results include: 1. A tight Θ(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to the n given lines. 2. A similar bound of Θ(n3) for the complexity of the set of all lines passing above the n given lines. 3. A preprocessing procedure using O(n2+ε) time and storage, for any ε > 0, that builds a structure supporting O(logn)-time queries for testing if a line lies above all the given lines. 4. An algorithm that tests the "towering property" in O(n4/3+ε) time, for any ε > 0: do n given red lines lie all above n given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space and ε-nets for various geometric range spaces.

Original languageEnglish (US)
Pages (from-to)428-447
Number of pages20
JournalAlgorithmica (New York)
Issue number5
StatePublished - 1996
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Computer Science
  • Applied Mathematics
  • Computer Science Applications


  • Computational geometry
  • Lines in space
  • Plücker coordinates
  • ε-Nets


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