Locked and Unlocked Polygonal Chains in 3D

Therese Biedl, University of Waterloo
Erik D. Demaine, University of Waterloo
Martin L. Demaine, University of Waterloo
Sylvain Lazard, Institut National de Recherche en Informatique et en Automatique, Lorraine
Anna Lubiw, University of Waterloo
Joseph O'Rourke, Smith College
Mark Overmars, Utrecht University
Steve Robbins, McGill University
Ileana Streinu, Smith College
Godfried Toussaint, McGill University
Sue Whitesides, McGill University

This document has been relocated to https://scholarworks.smith.edu/csc_facpubs/84/

There were 37 downloads as of 2 Nov 2022.


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.