Paper by Erik D. Demaine

Reference:

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”, in Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'99), Baltimore, Maryland, January 17–19, 1999, pages 866–867.

Abstract:

In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D with a polynomial number of moves.

Comments:

This paper is also available as arXiv:cs.CG/9811019 of the Computing Research Repository (CoRR).

Updates:

The pocket-flipping algorithm described in this paper has been implemented by Jean-Philippe Cote and Marc-Andre Sauve as a course project for Godfried Toussaint. Their project web page includes the detailed history of the problem, and an applet to demonstrate the motion.

Length:

The paper is 2 pages and the talk is 20 minutes.

Availability:

The paper is available in PostScript (165k).

See information on file formats.

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Related papers:

3DChains_DCG2001 (Locked and Unlocked Polygonal Chains in Three Dimensions)

3DChainsTR (Locked and Unlocked Polygonal Chains in 3D)