Reconfiguring Convex Polygons

Overview

Polygons are used to model many modern-day problems touching the fields of computer vision, robotics, physics, biology, and chemistry, among others, and their use in each is motivated by the mathematical properties and analysis that their structure entails.  Computational geometry methods can be applied to such models in order to derive properties and solutions to problems in those fields, yielding inisght into problems that might otherwise be considered unsolvable.

The study of linkages represented by polygons whose vertices act as hinges and whose edges serve as rigid bars has been covered in numerous publications, from which a unversality result has emerged: that every polygon can be reached by another polygon with the same sequence of edge lengths through motions that preserve edge lengths.  The proof of this result in papers covering the topic involves proving that every polygon can be convexified while preserving edge lengths, and employs the reconfiguration of the polygon into a "canonical triangle" with the crossing of edges allowed.

First, it will be demonstrated that any convex polygon can be reconfigured into any other polygon with the same counterclockwise sequence of edge lengths without the use of a "canonical triangle" intermediate state.  The method described here will also be shown to be the simplest one possible as each vertex angle decreases or increases monotonically with time as the polygon approaches the desired configuration.

It will then be shown that the reconfiguration of a polygon into any other with the same sequence of edge lengths is possible using only pivots , the simplest possible move in 3 dimensions, and thus that the restrictions on motions in 3D do not limit the ability to reconfigure a polygon in the desired fashion.  All proofs, lemmas and propositions that follow are taken from [1] unless otherwise specified.