Advertisement

Discrete & Computational Geometry

, Volume 19, Issue 3, pp 315–331 | Cite as

On Levels in Arrangements of Lines, Segments, Planes, and Triangles%

  • P. K. Agarwal
  • B. Aronov
  • T. M. Chan
  • M. Sharir

Abstract.

We consider the problem of bounding the complexity of the k th level in an arrangement of n curves or surfaces, a problem dual to, and an extension of, the well-known k-set problem. Among other results, we prove a new bound, O(nk 5/3 ) , on the complexity of the k th level in an arrangement of n planes in R 3 , or on the number of k -sets in a set of n points in three dimensions, and we show that the complexity of the k th level in an arrangement of n line segments in the plane is \(O(n\sqrt{k}\alpha(n/k))\) , and that the complexity of the k th level in an arrangement of n triangles in 3-space is O(n 2 k 5/6 α(n/k)) . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p315.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>

Keywords

Line Segment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • P. K. Agarwal
    • 1
  • B. Aronov
    • 2
  • T. M. Chan
    • 3
  • M. Sharir
    • 4
  1. 1.Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA {pankaj@cs.duke.edu} US
  2. 2.Department of Computer and Information Science, Polytechnic University, Brooklyn, NY 11201-3840, USA {aronov@ziggy.poly.edu} US
  3. 3.Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124-4250, USA tchan@cs.miami.edu US
  4. 4.School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA sharir@math.tau.ac.ilIL

Personalised recommendations