Abstract
A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to frameworks that are symmetric with respect to some point group but are otherwise generic, and secondly to frameworks in Euclidean 3-space that are constrained to lie on 2-dimensional algebraic varieties. We combine these two settings and consider the rigidity of symmetric frameworks realised on such surfaces. First we establish necessary conditions for a framework to be symmetry-forced rigid for any group and any surface by setting up a symmetry-adapted rigidity matrix for such frameworks and by extending the methods in Jordan et al. (2012) to this new context. This gives rise to several new symmetry-adapted rigidity matroids on group-labelled quotient graphs. In the cases when the surface is a sphere, a cylinder or a cone we then also provide combinatorial characterisations of generic symmetry-forced rigid frameworks for a number of symmetry groups, including rotation, reflection, inversion and dihedral symmetry. The proofs of these results are based on some new Henneberg-type inductive constructions on the group-labelled quotient graphs that correspond to the bases of the matroids in question. For the remaining symmetry groups in 3-space—as well as for other types of surfaces—we provide some observations and conjectures.
Highlights
A finite simple graph embedded into Euclidean space Rd with vertices interpreted as universal joints and edges as stiff bars is known as a bar-joint framework
Using the orbit rigidity matrix, combinatorial characterisations of symmetry-forced rigid symmetry-generic frameworks have recently been established for a number of symmetry groups in the plane [10,15,16]
We extend these concepts and some of these combinatorial results to symmetric frameworks in 3D whose joints are constrained to surfaces
Summary
A finite simple graph embedded into Euclidean space Rd with vertices interpreted as universal joints and edges as stiff bars is known as a bar-joint framework. Using the orbit rigidity matrix, combinatorial characterisations of symmetry-forced rigid symmetry-generic frameworks have recently been established for a number of symmetry groups in the plane (under the assumption that the symmetry group acts freely on the framework joints) [10,15,16]. 6, 7, 8 and 9 we use the orbit-surface rigidity matrix to derive combinatorial characterisations of symmetry-forced rigid frameworks which are embedded generically with inversive or certain improper-rotational (where an improper rotation is a rotation followed by a reflection in a plane perpendicular to the rotation axis) or dihedral symmetry on the sphere, with rotational, reflective or inversive symmetry on the cylinder or with rotational, reflective, inversive or certain improper-rotational symmetry on the cone. This situation induces a symmetry-preserving motion in a framework that counts to be minimally rigid without symmetry! For this case and some others we conjecture that the necessary counts we derived here are sufficient
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