Rigidity of Frameworks on Expanding Spheres

Abstract

A rigidity theory is developed for bar-joint frameworks in R d+1 whose vertices are constrained to lie on concentric d-spheres with independently variable radii. In particular, combinatorial characterisations are established for the rigidity of generic frameworks for d = 1 with an arbitrary number of independently variable radii, and for d = 2 with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in R 3 . Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.

Frameworks with coordinated edge motions

“…As discussed above, coordinated rigidity is an example of a rigidity theory with an enlarged set of allowed motions. Coordinated frameworks also generalise a model for frameworks on expanding spheres introduced in [16]. A number of recent results in condensed matter theory [7,9,19] show that (nearly) minimally rigid frameworks can be "tuned" to have a number of interesting geometric and material properties.…”

Global rigidity of generic frameworks on the cylinder

“…An alternative approach to this problem would be to try to extend the recursive characterisation of M * 2,2 -circuits to a recursive characterisation of graphs which are 2-connected and redundantly rigid on Y, and then use it to show that all generic realisations of these graphs have a maximum rank equilibrium stress. 4) In [21] frameworks on 'expanding' spheres were considered. Our v-free rigidity model is analogous to this for the cylinder.…”

“…Since the analogous problem when all vertices are free to move off the cylinder independently is exactly the well known 3-dimensional rigidity problem, it may be of interest to extend this further to consider the case when two or more subsets of the vertex set can move independently off the cylinder. It is not hard to adapt the examples given in [21] to show that the obvious sparsity count is not sufficient to imply minimal rigidity when there are at least three subsets moving independently, but it may be sufficient when there are at most two subsets.…”

Frameworks with Coordinated Edge Motions

“…The allowed motions are the continuous deformations in the space of allowed configurations. The motivation for studying these types of frameworks is to interpolate between rigidity in dimension d and dimension d + 1.A model for this expanding spheres setup, which is present in [16], is based on Whiteley's coning construction [23]. We first add a new vertex v 0 to G and fix it at the origin (the centre of the spheres) and then we connect v 0 to each of the vertices of G. The new edges joining v 0 with the vertices in V 0 must have fixed unit length, and the remaining new edges joining v 0 with vertices in V 1 ∪ .…”

“…. ∪ V k do not have fixed length, but all of the ones in the same class V j must have the same length.Inspired by [16] we consider frameworks in which not all of the bars are fixed-length in a more general fashion. We identify, in advance k ∈ "coordination classes" of edges which are allowed to change their length, subject to edge length differences being preserved within each coordination class.Our study of coordinated rigidity in such a general setup is also motivated by some recent results in condensed matter theory.…”