Rigidity of symmetric frameworks on the cylinder

Abstract

A bar-joint framework (G, p) is the combination of a finite simple graph G = (V, E) and a placement p : V → R d . The framework is rigid if the only edge-length preserving continuous deformations of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in R 3 restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.

Rigidity of symmetric frameworks in normed spaces

“…In [16] we analysed symmetric frameworks all of whose points are constrained to lie on a single surface. This situation can be modelled by linearly constrained frameworks where the linear constraints are chosen to represent the normals to the given surface.…”