A Characterization of Generically Rigid Frameworks on Surfaces of Revolution

A foundational theorem of Laman provides a counting characterisation of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterisation was obtained for frameworks in three dimensions whose vertices are constrained to concentric spheres or to concentric cylinders. Noting that the plane and the sphere have 3 independent locally tangential infinitesimal motions while the cylinder has 2, we obtain here a Laman-type theorem for frameworks on algebraic surfaces with a 1-dimensional space of tangential motions. Such surfaces include the torus, helicoids and surfaces of revolution. The relevant class of graphs are the (2, 1)-tight graphs, in contrast to (2, 3)-tightness for the plane/sphere and (2, 2)-tightness for the cylinder. The proof uses a new characterisation of simple (2, 1)-tight graphs and an inductive construction requiring generic rigidity preservation for 5 graph moves, including the two Henneberg moves, an edge joining move and various vertex surgery moves.