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

Abstract

A polygonal linkage or chain is a sequence of segments of fixed lengths, free to turn about their endpoints, which act as joints. This paper reviews some results in chain reconfiguration and highlights several open problems 1 We consider a sequence of closed straight line segments [A 0 , A 1 ], [A 1 , A 2 ],. .. [A n−1 , A n ] of fixed lengths l 1 , l 2 ,. .. l n , respectively, imagining that these line segments are mechanical objects such as rods, and that their endpoints are joints about which these rods are free to turn. We ask how and whether such a chain can be moved from one given configuration to another under various assumptions or "rules of the game". The chain may be confined to the plane throughout its motions; it may be supposed to start and finish in the plane, with motion into 3D allowed for intermediate configurations; its motions in arbitrary dimensional space may be considered. The chain may consist of an open or closed sequence of segments. The links may be allowed or forbidden to cross over or to pass through one another. All of these models are of interest to us. When the chain consists of a closed sequence of links, we say that the chain is polygonal, or that it is a polygon. Consequently, it is natural to use both the language of geometry and mechanics when describing chains. The terms node, vertex and joint are used interchangeably, as are the terms rod, link, edge, and segment. The term "polygon" may refer either to a planar object or to a cyclic sequence of links in arbitrary dimension. In case the links are not allowed to intersect except at shared endpoints, we say that the polygon must remain simple, i.e., it is not allowed to intersect itself either at rest or during motion. Polygonal chains are interesting for several reasons. First, there are aesthetic reasons. These very basic objects exhibit surprising behaviors and pose challenging, easily stated questions that arouse our natural curiosity as mathematicians, algorithm designers, and problem solvers. Second, chains can model physical objects such as robot arms and molecules. Here, a word of caution is in order. In Research supported by FCAR and NSERC. 1 A preliminary version of this paper was presented to AWOCA '92, the 12th Australasian Workshop on Combinatorial Algorithms, Lembang, Indonesia, July 14-17, hosted by the Bandung Institute of Technology.

On Three-Dimensional Graph Drawing and Embedding

“…Additional reading and references on linkages are provided by [6]; reference [5] relates the so-called HP model for protein folding to the reconfiguration of chains of links.…”