Locked and Unlocked Polygonal Chains in 3D
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.
 Publication:

arXiv eprints
 Pub Date:
 November 1998
 DOI:
 10.48550/arXiv.cs/9811019
 arXiv:
 arXiv:cs/9811019
 Bibcode:
 1998cs.......11019B
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Robotics;
 F.2.2;
 I.2.9
 EPrint:
 To appear in Proc. 10th ACMSIAM Sympos. Discrete Algorithms, Jan. 1999