Locked and Unlocked Polygonal Chains in 3D

TL;DR

This paper investigates the reconfiguration of 3D polygonal chains, demonstrating conditions for straightening and convexifying chains, and providing efficient algorithms for these transformations.

Contribution

It introduces linear-time algorithms for straightening open chains with planar projections and convexifying planar simple polygons in 3D, including locked chain examples.

Findings

Open chains with planar projections can be straightened.

Unknotted closed chains can be locked and not convexified.

Algorithms operate in linear time with O(n) moves.

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.