Convexifying Monotone Polygons

  • Therese C. Biedl
  • Erik D. Demaine
  • Sylvain Lazard
  • Steven M. Robbins
  • Michael A. Soss
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1741)


This paper considers reconfigurations of polygons, where each polygon edge is a rigid link, no two of which can cross during the motion. We prove that one can reconfigure any monotone polygon into a convex polygon; a polygon is monotone if any vertical line intersects the interior at a (possibly empty) interval. Our algorithm computes in O(n 2) time a sequence of O(n 2) moves, each of which rotates just four joints at once.


Joint Angle Convex Polygon Vertical Edge Link Length Virtual Link 
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  1. 1.
    T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O’Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint, and S. Whitesides. Locked and unlocked polygonal chains in 3D. Manuscript in preparation, 1999. A preliminary version appeared in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, 1999, 866–867.Google Scholar
  2. 2.
    Therese Biedl, Erik Demaine, Martin Demaine, Sylvain Lazard, Anna Lubiw, Joseph O’Rourke, Steve Robbins, Ileana Streinu, Godfried Toussaint, and Sue Whitesides. On reconfiguring tree linkages: Trees can lock. In Proc. 10th Canadian Conf. Comput. Geom., Montréal, Aug. 1998.Google Scholar
  3. 3.
    Prosenjit Bose, William Lenhart, and Giuseppe Liotta. Personal comm., 1999.Google Scholar
  4. 4.
    Roxana Cocan and Joseph O’Rourke. Polygonal chains cannot lock in 4D. In Proc. 11th Canadian Conf. Comput. Geom., Vancouver, Aug. 1999.Google Scholar
  5. 5.
    H. Everett, S. Lazard, S. Robbins, H. Schröder, and S. Whitesides. Convexifying star-shaped polygons. In Proc. 10th Canadian Conf. Comput. Geom., Montr_eal, Aug. 1998.Google Scholar
  6. 6.
    W. J. Lenhart and S. H. Whitesides. Reconfiguring closed polygonal chains in Euclidean d-space. Discrete Comput. Geom., 13:123–140, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Joseph O’Rourke. Folding and unfolding in computational geometry. In Proc. Japan Conf. Discrete and Computational Geometry, Tokyo, Dec. 1998. To appear.Google Scholar
  8. 8.
    Godfried Toussaint. The Erdos-Nagy theorem and its ramifications. In Proc. 11th Canadian Conf. Comput. Geom., Vancouver, Aug. 1999.Google Scholar
  9. 9.
    Sue Whitesides. Algorithmic issues in the geometry of planar linkage movement. Australian Computer Journal, 24(2):42–50, May 1992.Google Scholar
  10. 10.
    Sue Whitesides. Personal communication, Oct. 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Therese C. Biedl
    • 1
  • Erik D. Demaine
    • 1
  • Sylvain Lazard
    • 2
  • Steven M. Robbins
    • 3
  • Michael A. Soss
    • 3
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.INRIA Lorraine — LORIA, Projet ISAVillers les NancyFrance
  3. 3.School of Computer ScienceMcGill UniversityMontréalCanada

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