## 6 Joints at Infinity and Sliders
### Point-Hyperplane Frameworks and Projections from Spherical Frameworks  
Let \((G,p,\ell)\) be a point-hyperplane framework in \(\mathbb{R}^{d}\). Then we may consider this framework as a point-hyperplane framework \((G,\tilde{p},\ell)\) in the affine space \(\mathbb{A}^{d}\) by taking \(\tilde{p}_{i}^{T}=(p_{i}^{T},1)\) for all \(i\in V_{P}\). So \((G,\tilde{p},\ell)\) is the point-hyperplane framework with \(G=(V_{P}\cup V_{L},E)\), \(\tilde{p}:V_{P}\to\mathbb{A}^{d}\) and \(\ell=(a,r):V_{L}\to\mathbb{S}^{d-1}\times\mathbb{R}\). Using a central projection, we may then transfer \((G,\tilde{p},\ell)\) to a spherical framework \((G,\phi\circ(\tilde{p},\ell))\) in \(\mathbb{S}^{d}_{\geq 0}\) (the upper hemisphere including the equator) by defining \(\phi(\tilde{p})=\frac{\hat{p}}{\|\hat{p}\|}\) for each \(\hat{p}_{i}\) with \(i\in V_{P}\), and by regarding each hyperplane \(\ell_{i}=(a_{i},r_{i})\) with \(i\in V_{L}\) as the point \((a_{i},0)\) on the equator of \(\mathbb{S}^{d}\). It can then be shown (as detailed in [42]) that there existsan isomorphism between the space of infinitesimal motions of \((G,\tilde{p},\ell)\) and \((G,\phi\circ(\tilde{p},\ell))\). Thus, \((G,\hat{p},\ell)\) is infinitesimally rigid if and only if \((G,\phi\circ(\hat{p},\ell))\) is infinitesimally rigid. We illustrate this transfer of infinitesimal rigidity in Figure 16. By a simple count, the framework is flexible. Since the placement of the sliders in the plane does not matter, up to normals, the motion is illustrated in Figure 16c, with two positions illustrated with the same length bar sliding along the lines. This motion illustrates the classic example of a ladder sliding along a wall and the floor. Consider the projective framework in Figure 17 with a collinear triangle (a). While it satisfies the count \(|E|=2|V|-3\), the dependence in the collinear triangle guarantees an infinitesimal motion. When the collinear triangle is on the line at infinity--three sliders with fixed angles (b)--the third angle is dependent and can be omitted. The infinitesimal motion becomes a finite motion with the interior triangle rotating while the slider lines spread and contract (c), (d). This is illustrated in the video SlidersInfinity.mov linked in Supplementary Materials. Given a bar-joint framework \((G,q)\) on the sphere \(\mathbb{S}^{d}\), we may rotate the whole framework in \(\mathbb{S}^{d}\) so that all points are moved off the equator, and then invert all points that lie on the lower hemisphere to obtain a spherical framework \((G,q^{\prime})\) that lies on the strict upper hemisphere \(\mathbb{S}^{d}_{>0}\). This framework may now be projected up (using the inverse of the map \(\phi\)) to a bar-joint framework \((G,\hat{p})\) in the affine space \(\mathbb{A}^{d}\) (or equivalently, the Euclidean space \(\mathbb{R}^{d}\)). All of these operations preserve infinitesimal rigidity. Moreover, points of \((G,q)\) lie on  
Figure 16: A point-line framework in \(\mathbb{R}^{2}\) (**a**) and the corresponding spherical bar–joint framework obtained by coning up (**b**). The points on the equator correspond to the lines of the point-line framework. Figure (**c**) has brought the sliders to the end points of the bar in (**a**) with a visible motion taking \(u_{1},u_{2}\) to \(u^{\prime}_{1},u^{\prime}_{2}\). Figure 17: A Desargues framework with a collinear triangle (**a**) must have a non-trivial infinitesimal motion. When realised with sliders (the triangle of lines with fixed angles) as in (**b**) the three angles are dependent, so one can be omitted (**c**,**d**). There are additional realisations (**c**,**d**) arising from a finite motion. a hyperplane in \(\mathbb{S}^{d}\) if and only if the corresponding points of \((G,\tilde{p})\) lie on a hyperplane in \(\mathbb{A}^{d}\). In summary, we have the following result. **Theorem 3** ([42]). : _Let \(G=(V,E)\) be a graph and \(X\subseteq V\). Then the following are equivalent:_  
1.