## 5 Coning and Projecting
### Coning and Projection for \(\mathbb{S}^{d}\), \(\mathbb{M}^{d}\), and \(\mathbb{H}^{d}\)  
We have given the full theory of coning in projective form. As such it is just a matter of interpretation to observe that coning will preserve all the static and infinitesimal rigidity properties of a framework in any of the projective metrics.  
For example, the projective proof can be directly reinterpreted to prove the full transfer of infinitesimal and static properties for coning from \(\mathbb{S}^{d}\) to \(\mathbb{S}^{d+1}\). As we have seen in the previous section, if finite projective points are pulled back (re-weighted) to have length \(1\), then the affine rigidity matrix becomes the spherical rigidity matrix, and the lower rows for the vertices \(i\) can be geometrically interpreted as rows for cone-rays from a fixed origin to the points \(\widehat{p}^{s}_{i}\).  
These results can be extended in a straightforward fashion to the Minkowski spaces and their corresponding'spheres' of radius \(-1\) and \(1\) (hyperbolic space and de Sitter space [71]). For Minkowski space \(M^{d}_{1}\) we can just multiply the \(r\) relevant columns for vertices in the matrix by \(-1\). The signature of the added dimension is optional, so we have coning from \(M^{d}_{r}\) to \(M^{d+1}_{r}\) and to \(M^{d+1}_{r+1}\). With this coning, applied within Minkowski space,  
Figure 12: Plane momenta (**a**) are geometrically confirmed as a non-trivial motion when the areas of triangles \(M(a)\lor c\) and \(M(c)\lor a\) are not equal and of opposite orientation (**b**). When the plane framework is coned to \(p_{0}\), the plane momenta expand to \(3D\) quadrilaterals in planes through the cone point (**c**).  
Figure 13: A \(1\)-story building is essentially a cone (**a–c**). The rigidity of the roof depends on the projection (**d**) in the vertical direction.  
we can now complete the details of taking spheres of radius \(-1\) to obtain the hyperbolic spaces and the spheres of radius \(1\) to get de Sitter space. We can also project down to an arbitrary hyperplane which does not contain the cone point. In general this will go to a lower dimensional Minkowski space. It is possible to choose a hyperplane which has a Euclidean metric so that the image is Euclidean but not Minkowskian. It is also possible to cone up from Euclidean space with the added dimension having a negative signature so the cone lives in a Minkowskian metric.  
We can cone any framework up from \(\mathbb{M}_{1}^{d}\) to \(\mathbb{M}_{1}^{d+1}\), with any cone vertex, in the same projective way as above. For this geometric dimension, we assume the added dimension has signature \(+1\) in the metric. (see Remark 2 for other possibilities.) We can also project down from a cone point in \(\mathbb{M}_{1}^{d+1}\) to a hyperplane. At one extreme, the hyperplane has only the coordinates with signature \(+1\) and we end up in the Euclidean space \(\mathbb{E}^{d}\). At the other extreme, the hyperplane contains the subspace with signature \(-1\), and we end up with \(\mathbb{M}_{1}^{d}\). For the hyperbolic metric, and the companion de Sitter metric, one may mimic the transformation from the Euclidean space to the spherical space-but within the Minkowskian metric.  
To transfer from the affine rigidity matrix to the rigidity matrix for \(\mathbb{M}_{1}^{d}\), we simply multiply the \(d\)-th column of each vertex corresponding to the points of the framework by \(-1\)[71]. All key matrix properties are unchanged, so static and infinitesimal rigidity properties are transferred from Euclidean space to \(\mathbb{M}_{1}^{d}\). Note that the full space of translations of Euclidean space transfers to translations in \(\mathbb{M}_{1}^{d}\). Projective centers of motion and momenta for vertices will also transfer. The vectors illustrated in Figure 10b are tangent to the hyperbolas (spheres of radius \(1\) and \(-1\) in the metric) and perpendicular (in the Minkowskian metric) to the vectors pointing to the central point \((0,0,0)\).