## 5 Coning and Projecting
### Coning a Framework from \(\mathbb{P}^{d}\) to \(\mathbb{P}^{d+1}\)
Note that this coning includes projective points at infinity as vertices of the original framework, so this is an expected extension to include sliders as in [42] and Section 6. We can also projectively place the cone point on the hyperplane at infinity in \(\mathbb{P}^{d+1}\), making all the cone connections from the original vertices into sliders. In this form, the equivalence of the equilibrium stresses is even more obvious: simply drop the last coordinate of any applied loads and resolving vectors. For infinitesimal motions or for projective momenta, the only change under coning is that there are more trivial motions--essentially those for the cone point. We capture this change in the following corollary. **Corollary 1**. : _Let \(G=(V,E)\). Consider a framework \((G,p)\) in projective space \(\mathbb{P}^{d}\) and the coned framework \((G^{c},(\hat{p},\hat{O}))\) in \(\mathbb{P}^{d+1}\). With the cone point fixed, the two frameworks have isomorphic spaces of infinitesimal motions._  
Proof. : Delete the last columns for the cone vertex (pinning it down in the vocabulary of the later sections) from the rigidity matrix of \((G^{c},(\hat{p},\hat{O}))\). The new matrix is obtained from the rigidity matrix for \((G,p)\) by adding \(|V|\) columns and \(|V|\) linearly independent rows. It is immediate that the kernels will be isomorphic. As a further modest corollary to this proof, we also see that _any stress_ on a framework in \(\mathbb{P}^{d+1}\) projects to a stress in \(\mathbb{P}^{d}\). This holds for the projection from any point in \(\mathbb{P}^{d+1}\), including from an existing vertex, in which case the edges through this vertex are erased from the graph. Geometrically, we can transfer the momenta in \((G,p)\) to momenta in a general cone \((G^{c},(\hat{p},p_{0}))\) by simply joining the original momenta to a multiple of the cone point \(\alpha p_{0}\): \(M(a)\) goes to \(M(a)\lor\alpha p_{0}\). Recall that, if the joints span the projective space \(\mathbb{P}^{d}\), then an infinitesimal motion is non-trivial if and only if there is a pair of joints with \(M(a)\lor c+M(c)\lor a\
eq 0\). Note that for a plane momentum \(M(a)\), \(M(a)\lor c\) is the oriented area of the triangle with the momentum \(M(a)\) as the base and \(c\) as the third vertex (Figure 12b). When the momenta are expanded towards the cone point, we still have \(M(a)\lor c\lor\alpha p_{0}+M(c)\lor a\lor\alpha p_{0}\
eq 0\), and the momenta of the cone, fixing \(p_{0}\), represents a non-trivial infinitesimal motion. Conversely, momenta for the cone framework fixing \(p_{0}\) can be intersected by a plane containing the vertices to give momenta in that subspace, which is non-trivial if and only if the original momenta represented a non-trivial motion. **Example 5**. : _Consider a \(1\)-story building with a vertical post under each joint on the (almost flat) roof [76] (see Figure 13a). This can be viewed as a cone from a point at infinity on the lines of the posts (b) which is still projectively a cone (c). This means that the rigidity of the roof (whether it is plane or not) relative to the cone point depends only on the projection (d). To make this building rigidly attached to the ground, we need to add \(3\) further braces in the walls, preventing motions around the cone vertex, which would be translations. This matches the analysis in Section 6 where these constraints to infinity become sliders. It is possible to extend this coning analysis to multi-story buildings (a stacking of cones) [76]._