Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry

Abstract

Several aspects of the infinitesimal rigidity of frameworks in Euclidean, hyperbolic, and spherical geometry are presented. First, we prove the equivalence of the static and kinematic formulations of infinitesimal rigidity. By means of the projective interpretation of statics (representing forces as bivectors), this allows us to prove the projective invariance of infinitesimal rigidity and to establish a correspondence between infinitesimal motions of a Euclidean framework and of its geodesic image in any other geometry of constant curvature. We also describe the Maxwell–Cremona correspondence between equilibrium loads and polyhedral lifts, both for Euclidean and for non-Euclidean frameworks.