Using Algebra to Detect Flexible Positions of Bar and Joint Frameworks in the Plane

Abstract

Bar and joint rigidity theory in the plane is the study of frameworks and their embeddings into two-dimensional space. Rigidity theory in the plane is concerned with deciding whether a given planar embedding of a framework will be rigid or will allow motion about one or more of its joints. There are some frameworks that are generically rigid, but at certain special positions are flexible, and vice versa. In this thesis we convert graphical frameworks into algebraic systems of polynomial equations in variables and parameters where the parameters control edge lengths and the variables locate each vertex in the plane. However, some parameter values will yield different sets of points satisfying our equations and we may thus end up with some embeddings that are rigid and others that are flexible. In this thesis we use Groebner covers as in automatic geometric theorem proving to find conditions under which generically rigid frameworks are flexible. We show how we can combine Groebner covers and rigidity theory to explore special positions of three frameworks under particular assumptions.