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# Title: 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. All our algorithms require only O(n) basic moves'' and run in linear time.
 Comments: 29 pages; This is a revised and expanded version of an abstract that appeared in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms (SODA '98), Jan. 1998, pp. 866-867 Subjects: Computational Geometry (cs.CG); Discrete Mathematics (cs.DM) ACM classes: F.2.2 Report number: Smith Tech. Rep. 060 Cite as: arXiv:cs/9910009 [cs.CG] (or arXiv:cs/9910009v1 [cs.CG] for this version)

## Submission history

From: Joseph O'Rourke [view email]
[v1] Fri, 8 Oct 1999 12:04:16 GMT (153kb)