## 6 Joints at Infinity and Sliders
### Point-Hyperplane Frameworks and Projections from Spherical Frameworks
\(G\) _can be realised as an infinitesimally rigid point-hyperplane framework in_ \(\mathbb{R}^{d}\) _such that each vertex in_ \(X\) _is realised as a hyperplane and each vertex in_ \(V\setminus X\) _is realised as a point._
2. \(G\) _can be realised on the sphere_ \(\mathbb{S}^{d}\) _with each vertex in_ \(X\) _on the equator and each vertex in_ \(V\setminus X\) _is realised in the open upper hemisphere._
3. \(G\) _can be realised as an infinitesimally rigid bar-joint framework in_ \(\mathbb{R}^{d}\) _such that the points assigned to_ \(X\) _lie on a hyperplane._  
Using the results in [43] this provides the following combinatorial characterisation of graphs which can be realised as infinitesimally rigid bar-joint frameworks in the Euclidean plane with a given set of collinear points. Given a graph \(G=(V,E)\), \(X\subseteq V\) and \(A\subseteq E\), let \(\
u_{X}(A)\) denote the number of vertices of \(X\) which are incident to edges in \(A\). **Corollary 2** ([42]). : _Let \(G=(V,E)\) be a graph and \(X\subseteq V\). Then the following are equivalent:_  
1. \(G\) _can be realised as an infinitesimally rigid bar-joint framework in_ \(\mathbb{R}^{2}\) _such that the points assigned to_ \(X\) _lie on a line._
2. \(G\) _can be realised as an infinitesimally rigid point-line framework in_ \(\mathbb{R}^{2}\) _such that each vertex in_ \(X\) _is realised as a line and each vertex in_ \(V\setminus X\) _is realised as a point._
3. \(G\) _contains a spanning subgraph_ \(G^{\prime}=(V,E^{\prime})\) _such that_ \(E^{\prime}=2|V|-3\) _and, for all_ \(\emptyset\
eq A\subseteq E^{\prime}\) _and all partitions_ \(\{A_{1},\ldots,A_{s}\}\) _of_ \(A\)_,_ \[|A|\leq\sum_{i=1}^{s}(2\
u_{V\setminus X}(A_{i})+\
u_{X}(A_{i})-2)+\
u_{X}(A)-1.\]  
The combinatorial condition in (c) is more complicated than a standard vertex-edge count. However in [43], it is shown that the condition can be efficiently checked by a combination of standard rigidity algorithms and matroid union. It is also worth noting that currently there is no known recursive construction of the family of graphs satisfying (c). **Example 9**. : _We return to the sliders of the window mechanism in Figure 18. As displayed in (a) we have 6 regular vertices, one auxiliary vertex (with dotted incident edges) to hold the two collinear edges collinear, and the red line of the slider, making \(|V|=8\). We can count \(|E|=12\) with 6 regular edges, 3 auxiliary edges, and 3 edges attaching vertices to the slider'vertex'. With \(|E|=12<2\times 8-3\) the structure has a non-trivial infinitesimal motion, which is finite unless there is an additional dependence._  
_In Figure 18c we have shifted the line of the slider to pass through the vertices which are attached to the slider in the original mechanism. A careful inspection of the constraint equations above and the corresponding rigidity matrix detects that there are no occurrences of \(r_{j}\), just its derivative \(\dot{r}_{j}\). We can replace the line of the slider by any parallel line (keeping the same normal, which does occur) with no change in solution space. In general, we can choose a hyperplane to be anywhere within a parallel class determined by its normal. This holds in all dimensions and in some figures it may be convenient to place all sliders as lines through the origin._