## 13 Further Explorations
### Global Rigidity, Universal Rigidity and Superstability  
Rigid frameworks may have many equivalent realisations, but what happens if the framework is unique (up to isometries)? This is the question of global rigidity, which we now consider from the projective viewpoint. Two frameworks are _equivalent_ if they have the same edge lengths. A framework \((G,p)\) in \(\mathbb{R}^{d}\) is _globally rigid_ if every equivalent framework \((G,q)\) in \(\mathbb{R}^{d}\) arises from \((G,p)\) from an isometry of \(\mathbb{R}^{d}\). A deep result of Gortler, Healy and Thurston [165] confirmed that generic realisations of a graph are either all globally rigid or none of them are. A graph is called generically globally rigid if all its generic frameworks are globally rigid. Hendrickson [166] proved two natural necessary conditions for generic global rigidity: \((d+1)\)-connectivity and redundant rigidity. Here a graph is _redundantly rigid_ if it remains rigid after any single edge is removed. These conditions are also sufficient in 2-dimensions [167] but do notcharacterise global rigidity in \(\mathbb{R}^{d}\) for any \(d\geq 3\)[168; 169]. In some special cases, such as body-bar frameworks, redundant rigidity is necessary and sufficient for generic global rigidity [128]. Global rigidity has also been considered for linearly constrained frameworks [170]. In this context natural analogues of Hendrickson's conditions hold and a natural stress matrix condition is sufficient for generic global rigidity. Moreover in 2-dimensions there is an efficient combinatorial characterisation of generic global rigidity. More general slider constraints and projective ideas should be explored. As discussed, linear constraints model sliders where the points at infinity are pinned. It would be interesting to extend these global rigidity results to different types of sliders, as was done for infinitesimal rigidity in [78]. It would also be valuable to generalise the results of [170] to higher dimensions and to allow non-generic linear constraints. We will also mention an additional concept. A framework \((G,p)\) in \(\mathbb{R}^{d}\) is _dimensionally rigid_ if there are no equivalent frameworks with a higher dimensional span. Note that dimensionally rigid frameworks can be flexible, but this notion was shown to be important in the study of global and universal rigidity by Alfakih [171]. (Universal rigidity is an extension of global rigidity where we require that all equivalent frameworks in \(\mathbb{R}^{D}\), for any \(D\geq d\), are congruent.) Global rigidity is _almost_ projectively invariant in the following senses [172; 173]. 1. Dimensional rigidity is projectively invariant [171]. 2. Transfer of metric: a graph \(G\) is generically globally rigid in \(\mathbb{R}^{d}\) if and only if it is generically globally rigid in the spherical space \(\mathbb{S}^{d}\), the hyperbolic space of dim \(d\), and the Minkowskian space of dim \(d\)[11; 69; 173]. 3. Coning: a graph \(G\) is generically globally rigid in \(\mathbb{R}^{d}\) if and only if the cone graph is generically globally rigid in \(\mathbb{R}^{d+1}\)[173]. 4. Open projective neighborhoods: if a framework \((G,p)\) is globally rigid, then within the projective images, an open neighborhood of projectively equivalent frameworks shares the global rigidity [173]. We explored the projective conditions for a generically isostatic graph to have a non-trivial infinitesimal motion. If the graph is redundantly rigid then we would anticipate that there are several polynomial conditions for there to be an infinitesimal motion. Some unusual cases arise for bipartite graphs such as \(K_{5,5}\) in \(\mathbb{R}^{3}\). For \(K_{5,5}\), the 10 points lying on a conic gives a non-trivial in-out infinitesimal motion (Theorem 7.3). This has been exploited by Connelly [168] to show that \(K_{5,5}\) realised 'near' a sphere is not globally rigid, though it is redundantly rigid in \(\mathbb{R}^{3}\) and 4-connected.