# Locked and unlocked polygonal chains in 3D

@article{Biedl1998LockedAU, title={Locked and unlocked polygonal chains in 3D}, author={Therese C. Biedl and Erik D. Demaine and Martin L. Demaine and Sylvain Lazard and Anna Lubiw and Joseph O'Rourke and Mark H. Overmars and Steven M. Robbins and Ileana Streinu and Godfried T. Toussaint and Sue Whitesides}, journal={ArXiv}, year={1998}, volume={cs.CG/9910009} }

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…

## 55 Citations

### C G ] 8 O ct 1 99 9 Locked and Unlocked Polygonal Chains in 3 D ∗ T . Biedl

- Mathematics, Computer Science
- 2008

It is said 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.

### Computational Polygonal Entanglement Theory

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### Locked and Unlocked Polygonal Chains in Three Dimensions

- Mathematics, Computer ScienceDiscret. Comput. Geom.
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The main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions.

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It is proved that the producible chains are rare in the following technical sense: if a class of chains has a locked configuration, then the probability that a random configuration of a random chain is producible approaches zero geometrically as n → ∞.

### Straightening polygonal arcs and convexifying polygonal cycles

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It is proved that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths.

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This work shows that a number of grasp points that is linear in the number of crossings in a knot diagram is sufficient to immobilize string in a polygonal shape with the topology of an arbitrary knot, or to fold or unfold the knot from a straight configuration.

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