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A foundational theorem of Laman provides a counting characterisation of thefinite simple graphs whose generic bar-joint frameworks in two dimensions areinfinitesimally rigid. Recently a Laman-type characterisation was obtained forframeworks in three dimensions whose vertices are constrained to concentricspheres or to concentric cylinders. Noting that the plane and the sphere have 3independent locally tangential infinitesimal motions while the cylinder has 2,we obtain here a Laman-Henneberg theorem for frameworks on algebraic surfaceswith a 1-dimensional space of tangential motions. Such surfaces include thetorus, helicoids and surfaces of revolution. The relevant class of graphs arethe (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 ofsimple (2,1)-tight graphs and an inductive construction requiring genericrigidity preservation for 5 graph moves, including the two Henneberg moves, anedge joining move and various vertex surgery moves.

A characterisation of generically rigid frameworks on surfaces of revolution