Abstract

For about 20 years, a handful of mathematicians and engineers have been developing a powerful method of solving systems of coupled, nonlinear equations called numerical continuation. In particular, this method lends itself to the solution of polynomial equations, where it comes under the heading of polynomial continuation. Coupled systems of polynomial equations often have more than one, and in some cases can have tens of thousands of perfectly valid solutions. It is now possible to find computationally every one of those solutions, and to be confident that no others exist. From a design point of view, this means that every feasible option for a problem posed in polynomial form can be determined at once. A broad variety of engineering design problems or models can be posed in polynomial form, but continuation finds a natural home in the field of kinematics and mechanism/linkage analysis (a topic of study at Cambridge since the 1870’s). Some of the most interesting results of polynomial continuation arise when applied to overconstrained mechanisms; those which are mobile only because of some quirk of geometry. This talk takes its name from a feature of polynomial continuation known as a cheater’s homotopy, in which all the solutions of a system of equations are found quickly and easily using only a little knowledge about the equations’ structure. As the name suggests, using continuation to design linkages might be almost too simple!