The Algebraic Geometry of Stresses in Frameworks

Abstract

A bar-and-joint framework, with rigid bars and flexible joints, is said to be generically isostatic if it has just enough bars to be infinitesimally rigid in some realization in Euclidean n-space. We determine the equation that must be satisfied by the coordinates of the joints in a given realization in order to have a nonzero stress, and hence an infinitesimal motion, in the framework. This equation, called the pure condition, is expressed in terms of certain determinants, called brackets. The pure condition is obtained by choosing a way to tie down the framework to eliminate the Euclidean motions, computing a bracket expression by a method due to Rosenberg and then factoring out part of the expression related to the tie-down. A major portion of this paper is devoted to proving that the resulting pure condition is independent of the tie-down chosen. We then catalog a number of small examples and their pure conditions, along with the geometric conditions for the existence of a stress which are equivalent to the algebraic pure conditions. We also explain our methods for calculating these conditions and determining their factorization. We then use the pure conditions to investigate stresses and tension-compression splits in 1-overbraced frameworks. Finally we touch briefly upon some of the problems arising when multiple factors occur in the pure condition.