## 5 Coning and Projecting
### Coning a Framework from \(\mathbb{P}^{d}\) to \(\mathbb{P}^{d+1}\)  
In the following, we consider coning in projective space. Given a framework \((G,\bar{p})\) in projective space \(\mathbb{P}^{d}\), we add a new cone vertex placed at \(\hat{O}=(0,\ldots,0,1)\) in \(\mathbb{P}^{d+1}\) to  
Figure 11: Coning and moving vertices radially in and out on the cone rays does not change infinitesimal rigidity. obtain the corresponding coned framework \((G^{c},(\hat{p},\hat{O}))\) in \(\mathbb{P}^{d+1}\). We have the following basic result. (Projective Coning) Given a projective framework \((G,\hat{p})\) in \(\mathbb{P}^{d}\), the coned framework \((G^{c},(\hat{p},\hat{O}))\) in \(\mathbb{P}^{d+1}\) has an isomorphic space of equilibrium stresses, so \((G,\hat{p})\) is statically rigid if and only if the framework \((G^{c},(\hat{p},\hat{O}))\) in \(\mathbb{P}^{d+1}\) is statically rigid. Conversely, given a cone framework \((G^{c},(\hat{p},\hat{O}))\) in \(\mathbb{P}^{d+1}\), the projection from the cone vertex into any hyperplane \(H\) gives a framework \((G,\hat{p})\) in \(H\equiv\mathbb{P}^{d}\) with an isomorphic space of self stresses, so \((G^{c},(\hat{p},\hat{O}))\) is statically rigid in \(\mathbb{P}^{d+1}\) if and only if \((G,\hat{p})\) is statically rigid in \(\mathbb{P}^{d}\). Finally, if we pull or push vertices along the rays from the cone to the original vertices, replacing \(\hat{p}_{i}\) by \(\hat{p}_{i}+\alpha\hat{O}\), \(\alpha\
eq 0\), static rigidity is preserved. Proof. : Recall that the projective rigidity matrix of \((G,\hat{p})\) is the \((|E|+|V|)\times(d+1)|V|\) matrix  
\[\mathbf{\hat{R}}(G,\hat{p})= \begin{array}{ccccccccc}&i&&&&j\\ &&&&\vdots&&&&\\ 0&\ldots&0&\bar{p}_{j}&0&\ldots&0&\bar{p}_{i}&0&\ldots&0\\ &&&&\vdots&&&&\\ 0&\ldots&0&\bar{p}_{i}&0&\ldots&0&0&\ldots&0\\ &&&&\vdots&&&&\\ \end{array} \end{array}\]  
We can fill out this matrix with the columns for an added cone vertex \(\hat{O}\) in \(\mathbb{P}^{d+1}\) connected to all existing vertices. With \(\hat{O}=(0,\ldots,0,1)\), we create the following matrix:  
\[ \begin{array}{ccccccccc}&\bar{p}_{1}&\ldots&\bar{p}_{|V|}&\hat{p}_{1}^{d+2}& \ldots&\hat{p}_{|V|}^{d+2}&\hat{O}\\ \vdots&&&&0&&0\\ \vdots&&&&\\ \{\hat{O}\}&&&&[I_{|V|}]&&\hat{p}_{i}\\ \vdots&&&&\\ \{\hat{O}\}&&&&0&&O\end{array} \]  
Now consider an equilibrium stress on the original framework. With added coefficients of \(0\) on all the added rows, it is still an equilibrium stress on the coned framework, that is, a row dependence for the extended matrix. Consider an equilibrium stress on the coned framework. 1. Looking at the columns for \(\bar{p}_{1},\ldots,\bar{p}_{|V|}\) it must be an equilibrium stress on the original framework. 2. Looking at the columns for \(\hat{p_{1}}^{d+2},\ldots,\hat{p_{|V|}}^{d+2}\) the coefficients on the bars \(\{\hat{O},\hat{p}_{i}\}\) must all be zero. 3. Looking at the residual column for \(\hat{O}\), the coefficient on the row \(\{\hat{O}\}\) must also be \(0\). So we conclude that the equilibrium stress is also an equilibrium stress of the original framework. If we pull and push vertices along the rays from the cone to the original vertices, replacing \(\hat{p}_{i}\) by \(\hat{p}_{i}+\alpha\hat{O}\), the rank of the matrix is preserved. We can also apply a projective transformation to the cone, placing \(\hat{O}\) anywhere off the original hyperplane in \(\mathbb{P}^{d+1}\) and preserving the original framework by keeping the original \(\mathbb{P}^{d}\) fixed. To complete the static rigidity statements, consider how coning changes the counts of edges and vertices. We have  
\[|E|=d|V|-\binom{d+1}{2}\]  
if and only if  
\[|E^{c}|=|E|+|V| = (d+1)|V|+(d+1)-\left[\binom{d+1}{2}+(d+1)\right]\] \[= (d+1)|V^{c}|-\binom{d+2}{2}.\]  
Thus, a framework has full rank for static rigidity in \(\mathbb{P}^{d}\) if and only if the coned framework has full rank for static rigidity in \(\mathbb{P}^{d+1}\).