## 3 Projective Rigidity
### Euclidean and Spherical Spaces  
In the previous section we have seen that if all the projective points are finite, then the projective rigidity matrix is equivalent to the affine and the Euclidean rigidity matrix. We can follow the template of [71] to show that we can also transfer infinitesimal (or static) rigidity between Euclidean space and spherical space. Note first that we may interpret the affine rigidity matrix \(\overline{\mathbf{R}}(G,\bar{p})\) of the framework \((G,\bar{p})\) in \(\mathbb{A}^{d}\) as the rigidity matrix of a framework in \(\mathbb{R}^{d+1}\) that has an extra vertex pinned at the origin, which is joined to all the vertices of \(G\). To see this, simply consider the final \(|V|\) rows of \(\overline{\mathbf{R}}(G,\bar{p})\) as rows corresponding to edges from the new joint at the origin to the points \(\bar{p}_{i}\). Since the new joint at the origin is fixed, there are no additional columns for this joint in the Euclidean rigidity matrix. We may then scale the points \(\bar{p}_{i}\) so that the resulting points \(\bar{p}_{i}^{s}\) all have unit length. This gives the following matrix:\[\mathbf{R}^{\mathbf{s}}(G,\bar{p}^{\mathrm{s}})= \begin{array}{c}ij\\ 0\ \ldots\ 0\ \ \ \ (\bar{p}^{\mathrm{s}}_{j}-\bar{p}^{\mathrm{s}}_{i})\ \ 0\ \ldots\ \ 0\ \ \ (\bar{p}^{\mathrm{s}}_{i}-\bar{p}^{\mathrm{s}}_{j})\ \ 0\ \ldots\ \ 0\\ 0\ \ldots\ 0\ \ \ \ \ \bar{p}^{\mathrm{s}}_{i}\ \ 0\ \ldots\ 0\ \ 0\ \ \ 0\ \ \ldots\ 0\\ \vdots\\ 0j\ \ 0\ \ldots\ 0\ \ \ \ 0\ \ \ 0\ \ \ldots\ 0\ \ \ \ \bar{p}^{\mathrm{s}}_{j}\ \ 0\ \ldots\ 0\\ \vdots\\ \vdots\\ \end{array} \]  
\(\mathbf{R}^{\mathbf{s}}(G,\bar{p}^{\mathrm{s}})\) is the rigidity matrix for the spherical framework \((G,\bar{p}^{\mathrm{s}})\) where the row vector corresponding to \(0i\),  
\[(0\ \ldots\ 0\ \ \ \bar{p}^{\mathrm{s}}_{i}\ \ \ 0\ \ldots\ 0),\]  
represents the constraint that the joint \(\bar{p}^{\mathrm{s}}_{i}\) remains on the unit sphere \(\mathbb{S}^{d}\) (or equivalently, the velocity vector at this joint must be tangent to the sphere for any infinitesimal motion). Thus, it is clear that affine (and hence also Euclidean) rigidity and spherical rigidity are equivalent at the infinitesimal level. Alternatively, we may see this correspondence as follows. The spherical distance constraint which preserves the angle between the bars joining the origin with \(\bar{p}_{i}\) and \(\bar{p}_{j}\) (or equivalently the arc length between \(\bar{p}_{i}\) and \(\bar{p}_{j}\) along the surface of the sphere) is given by \(\bar{p}_{i}\cdot\bar{p}_{j}=c\), where \(c\) is a constant. The constraint that each point \(\bar{p}_{i}\) has distance \(1\) from the origin is given by \(\bar{p}_{i}\cdot\bar{p}_{i}=1\). By differentiating these constraints we obtain the linear system  
\[\bar{p}_{i}\cdot\dot{\bar{p}}_{j}+\bar{p}_{j}\cdot\dot{\bar{p}}_{i} = 0\] \[\bar{p}_{i}\cdot\dot{\bar{p}}_{i} = 0.\]  
The matrix corresponding to this linear system is the projective rigidity matrix \(\mathbf{\tilde{R}}(G,\bar{p})\). Moreover, the space of trivial infinitesimal motions is the space of infinitesimal rotations in \(\mathbb{R}^{d+1}\), which has dimension \(\binom{d+1}{2}\). We can make the transfer of infinitesimal motions between a framework in \(\mathbb{A}^{d}\) and the corresponding framework in \(\mathbb{S}^{d}\) explicit as follows (see [42] for details). Let \((G,\bar{p})\) be a framework in \(\mathbb{A}^{d}\), and let \(\phi:\mathbb{A}^{d}\to\mathbb{S}^{d}_{>0}\) be defined by \(\phi(\bar{p}_{i})=\frac{\bar{p}_{i}}{\|\bar{p}_{i}\|}=\bar{p}^{\mathrm{s}}_{i}\). If \(\dot{\bar{p}}_{i}=(\dot{p}_{i},0)^{T}\) is the velocity vector of an infinitesimal motion of \((G,\bar{p})\) at \(\bar{p}_{i}\), then the velocity vector of the infinitesimal motion of \((G,\bar{p}^{\mathrm{s}})\) at \(\bar{p}^{\mathrm{s}}_{i}\) is  
\[\psi_{\bar{p}_{i}}(\dot{\bar{p}}_{i})=\frac{\dot{\bar{p}}_{i}-(\dot{\bar{p}}_ {i}\cdot\dot{p}_{i})\mathbf{e}}{\|\bar{p}_{i}\|},\]  
where \(\mathbf{e}=(0,\ldots,0,1)^{T}\).