Description
A mechanical linkage is a device consisting of various links joined together to manage forces and movement. In the 19th century the question arose whether linkages could be used to trace a straight line, a problem that was eventually solved positively by Peaucellier and Lipkin. See the wikipedia page for more information and a nice animation. More surprisingly, Kempe showed soon afterwards that any algebraic curve in the plane could be traced by an appropriate linkage.Mathematically, a linkage can be described by a graph, where edges are given a length, and some of the vertices may be fixed. A realization would then be an embedding into Euclidean space which preserves lengths and fixed vertices. The collection of such realizations is the configuration space, a space which usually has interesting topology. In fact, in the 1970's Thurston gave arguments that any compact real algebraic set in some Euclidean space can be realized as a configuration space of a planar linkage.
In this project, we will apply some of these methods to study the configuration spaces of linkages. Rather than looking at linkages one can also look at linkages on the sphere or in the hyperbolic plane. Not much is known in these cases.