A characterisation of generically rigid frameworks on surfaces of revolution

Anthony Nixon; John Owen; Stephen Power

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arXiv · 1210.7073

A characterisation of generically rigid frameworks on surfaces of revolution

Anthony Nixon·John Owen·Stephen Power

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Abstract

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-Henneberg 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.

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BibTeX RIS

Anthony Nixon, John Owen, Stephen Power. 2014-06-09. A characterisation of generically rigid frameworks on surfaces of revolution. https://arxiv.org/abs/1210.7073

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