Infinitesimal motions of a bipartite framework

A recently established criterion for stresses on any bar and joint framework with a complete bipartite graph is converted into explicit criteria for infinitesimal motions of the framework. These criteria, based on quadric surfaces through the joints, include and complete those developed in recent years in the study of geodesy with range satellite networks. The infinitesimal motions are displayed in a simple geometric form, appropriate to any dimension. This geometric description is used to establish necessary and sufficient conditions, based on ruled quadric surfaces, for which bars may be added to a bipartite framework without removing a prior infinitesimal motion. These criteria are applied to the behaviour of structural engineering space frames.