A Characterization of Generically Rigid Frameworks on Surfaces of Revolution

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-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.

A constructive characterisation of circuits in the simple (2,2)-sparsity matroid

“…A characterisation of circuits in M * (2, 1) would be a step towards proving the analogue of Conjecture 5.7 for frameworks on a surface of revolution [17], such as a cone [11].…”

“…When the (k, l)-tight graph is simple, they still induce a matroid and we denote it as M * (k, l). Recursive constructions for the bases of M * (2, l) (l = 2, 1) can be found in [16,17,15]. In this paper we study circuits in M * (2,2).…”

Double-Distance Frameworks and Mixed Sparsity Graphs

“…Bar-joint frameworks restricted to surfaces (see [10,11]) as well as bar-joint frameworks in non-Euclidean spaces (see [6,7]) give examples of bar-joint frameworks in a metric space setting.…”

A constructive characterisation of circuits in the simple (2,1)‐sparse matroid

“…We expect that our characterisation will be useful for the problem of characterising the global rigidity of realisations of graphs 1 on surfaces of revolution (such as the cone). In [14], rigidity of such frameworks was, generically, shown to be equivalent to the graph being (2, 1)-tight. In [7, Conjecture 1] it was conjectured that the graphs that are generically globally rigid on the cone are those which are 2-connected and contain a spanning subgraph that is (2, 1)-tight.…”

“…In the case of simple graphs each intermediate graph in the recursive construction needs to be simple; this necessitates establishing new constructions. When = 2 constructions are known for several classes [12][13][14].…”