# On reconfiguring tree linkages: Trees can lock

@article{Biedl1998OnRT, title={On reconfiguring tree linkages: Trees can lock}, author={Therese C. Biedl and Erik D. Demaine and Martin L. Demaine and Sylvain Lazard and Anna Lubiw and Joseph O'Rourke and Steven M. Robbins and Ileana Streinu and Godfried T. Toussaint and Sue Whitesides}, journal={ArXiv}, year={1998}, volume={cs.CG/9910024} }

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an N-link tree can have 2 (N) equivalence classes of configurations.

#### 19 Citations

A note on reconfiguring tree linkages: trees can lock

- Computer Science, Mathematics
- Discret. Appl. Math.
- 2002

It is proved that an N-link tree can have 2 Ω(N) equivalence classes of configurations, and it is shown that there are trees with two configurations that are not connected by a motion. Expand

C G ] 2 9 Se p 20 00 On Reconfiguring Tree Linkages : Trees can Lock

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It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex.… Expand

Polygonal chains cannot lock in 4d

- Computer Science, Mathematics
- CCCG
- 1999

We prove that, in all dimensions d⩾4, every simple open polygonal chain and every tree may be straightened, and every simple closed polygonal chain may be convexified. These reconfigurations can be… Expand

Convexifying Monotone Polygons

- Mathematics, Computer Science
- ISAAC
- 1999

It is proved that one can reconfigure any monotone polygon into a convex polygon; a polygon is monot one if any vertical line intersects the interior at a (possibly empty) interval. Expand

Straightening polygonal arcs and convexifying polygonal cycles

- Mathematics, Computer Science
- Proceedings 41st Annual Symposium on Foundations of Computer Science
- 2000

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. Expand

Computational Polygonal Entanglement Theory

- Mathematics
- 1999

In this paper we are concerned with motions for untangling polygonal linkages (chains, polygons and trees) in 2 and 3 dimensions. We say that an open, simple polygonal chain can be straightened if it… Expand

Geometric Restrictions on Producible Polygonal Protein Chains

- Mathematics, Computer Science
- ISAAC
- 2003

It is proved that two seemingly disparate classes of chains are in fact identical, and an algorithm that reconfigures between any two flat states of a nonacute chain in O(n) “moves,” improving the O( n 2)-move algorithm in [ADD + 02]. Expand

Chain Reconfiguration. The INs and Outs, Ups and Downs of Moving Polygons and Polygonal Linkages

- Computer Science
- ISAAC
- 2001

This paper reviews some results in chain reconfiguration and highlights several open problems in polygonal linkage or chain Reconfiguration. Expand

Geometric Restrictions on Producible Polygonal Protein Chains

- Mathematics, Computer Science
- Algorithmica
- 2005

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 → ∞. Expand

Angles and Lengths in Reconfigurations of Polygons and Polyhedra

- Mathematics, Computer Science
- MFCS
- 2004

This work explores the possibility of reconfiguring, or “morphing”, one simple polygon to another, maintaining simplicity, and preserving some properties of angles and edge lengths, and shows that monotone morphs exist for parallel pairs of polygons that are either convex; or orthogonally convex. Expand

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