ABSTRACT
Given a framework determined by a system of geometric constraints, a fundamental question is to determine whether the framework is rigid or flexible. This chapter introduces the Grassmann-Cayley algebra, which can be used to give a geometric interpretation of the vanishing of bracket polynomials in order to better understand when a generically rigid framework admits nontrival internal motions. It lays out the mathematical framework used to describe invariant conditions, a topic that, perhaps surprisingly, links back up to Grassmannians and relations among the Plucker coordinates. The Plucker relations all come from Cramer's rule, a well-known result about determinants. The bracket polynomials that occur as the pure condition of a body-and-bar framework are homogeneous and multilinear. Neil White's algorithm implements Grassmann-Cayley factorization for multilinear expressions in the bracket algebra. It remains an open problem to develop an algorithm that produces a Grassmann-Cayley factorization of an arbitrary bracket polynomial.
