Rigid Cylindrical Frameworks with Two Coincident Points

Abstract

We develop a rigidity theory for frameworks in R 3 which have two coincident points but are otherwise generic and only infinitesimal motions which are tangential to a family of cylinders induced by the realisation are considered. We then apply our results to show that vertex splitting, under the additional assumption that the new edge is redundant, preserves the property of being generically globally rigid on families of concentric cylinders.

Infinitesimal Rigidity in Normed Planes

“…With the geometric results of the previous section in hand, we can use the combinatorics of [14] to prove the difficult sufficiency direction of our main result on coincident frameworks. We begin with the following result which can be extracted from the proof of [14,Theorem 4].…”

“…The difficulty that already arises in this context shows how necessary the genericity assumption in those papers really was. Frameworks with coincident points have been considered in the Euclidean context [10,12] and applied to global rigidity there [4], as well as for frameworks on surfaces [14].…”

“…Here we provide a detailed geometric analysis of the effect of certain graph operations on the rigidity of a coincident point framework in a normed plane. In Section 4 we combine these geometric results with combinatorial results of [14] to establish a purely combinatorial characterisation of independence in the 'coincident point normed plane rigidity matroid' and we deduce from this a delete-contract characterisation of coincident point rigidity in any strictly convex normed plane. In Section 5 we provide our other main results; these concern global rigidity.…”

“…It was shown in [14] that the edge sets of the uv-sparse subgraphs of G form the independent sets of a matroid, and when |V | ≥ 5 this matroid has rank 2|V | − 2.…”

Global rigidity of generic frameworks on the cylinder

“…We next consider the generalised vertex splitting operation. In [12,Theorem 5.2] it was proved that the standard vertex splitting operation, with the additional assumption that the new graph is rigid when we delete the bridging edge (i.e. the edge joining the two copies of the split vertex), preserves generic global rigidity on the cylinder.…”