Discrete & Computational Geometry

, Volume 28, Issue 4, pp 585–592

Untangling a Polygon

  • Pach
  • Tardos

DOI: 10.1007/s00454-002-2889-y

Cite this article as:
Pach & Tardos Discrete Comput Geom (2002) 28: 585. doi:10.1007/s00454-002-2889-y


The following problem was raised by M. Watanabe. Let P be a self-intersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P , i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in some cases one has to move all but at most O((n log n)2/3) vertices. On the other hand, every polygon P can be untangled in at most
steps. Some related questions are also considered.

Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Pach
    • 1
  • Tardos
    • 1
  1. 1.Courant Institute, New York University, New York, NY 10012, USA pach@cims.nyu.edu USA
  2. 2.Rényi Institute of the Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary pach@renyi.hu, tardos@renyi.huHungary

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