## 3 Projective Rigidity
### Infinitesimal and Static Rigidity in Projective Space
We define the _motion_ or _momentum_\(M(p)\) at the point \(p\) to be \(Z^{\prime}\vee\tilde{p}:=\sum_{i=1}^{b}(Z^{\prime}_{i}\vee\tilde{p})\). Recall that if \(u\) is an infinitesimal motion of a framework \((G,p)\) in Euclidean \(d\)-space, then the velocity vectors \(u_{i}\) at the points \(p_{i}\) satisfy the linear equations in (1). This linear system takes on an even simpler form in projective space. As we have seen above, the _momentum of the point_\(p_{i}\) is given by the \(d\)-extensor \(M(p_{i})=(u_{i},-u_{i}\cdot p_{i})\), so for every edge \(ij\) of \(G\) we obtain  
\[0 = (p_{i}-p_{j})\cdot(u_{i}-u_{j}) \tag{3}\] \[= (u_{i}\cdot p_{i})-(u_{i}\cdot p_{j})-(u_{j}\cdot p_{i})+(u_{j} \cdot p_{j})\] \[= M(p_{i})\cdot\tilde{p}_{j}-M(p_{j})\cdot\tilde{p}_{i}.\]Moreover, recall that geometrically the momentum \(M(p_{i})\) at \(p_{i}\) is a weighted section of a hyperplane of \(\mathbb{R}^{d}\) containing \(p_{i}\) with normal vector \(u_{i}\). The associated projective hyperplane will be denoted \(\overline{M}(p_{i})\). An equation for this hyperplane is given by \(M(p_{i})\lor x=0\), and hence we also have  
\[M(p_{i})\vee\tilde{p}_{i}=M(p_{i})\cdot\tilde{p}_{i}=0\quad\text{ for all }i\in V. \tag{4}\]  
In the sequel, we will often use the notation \(M_{i}=M(p_{i})\). The matrix corresponding to the linear system (3) and (4) is the _projective rigidity matrix_\(\tilde{\mathbf{R}}(G,\tilde{p})\). This matrix is the \((|E|+|V|)\times(d+1)|V|\) matrix of the form  
\[\tilde{\mathbf{R}}(G,\tilde{p})= \begin{array}{ccccccccc}&i&&&&j\\ &&&&\vdots&&&&\\ 0&\ldots&0&\tilde{p}_{j}&0&\ldots&0&\tilde{p}_{i}&0&\ldots&0\\ &&&&\vdots&&&&\\ 0&\ldots&0&\tilde{p}_{i}&0&\ldots&0&0&0&\ldots&0\\ 0&\ldots&0&0&0&\ldots&0&\tilde{p}_{j}&0&\ldots&0\\ &&&&\vdots&&&&\end{array} \end{array},\]  
where the entries are considered as row vectors. By the discussion above, the dimension of the space of rigid body motions (trivial infinitesimal motions) is \(\binom{d+1}{2}\), provided the vertices span the whole space. Assuming the framework has at least \(|V|=d\) vertices, a projective framework \((G,\tilde{p})\) is infinitesimally rigid if and only if the rank of \(\tilde{\mathbf{R}}(G,\tilde{p})\) is \((d+1)|V|-\binom{d+1}{2}\). If \(|V|<d\) then, as in the Euclidean case, the framework is infinitesimally rigid if and only if \(G\) is complete and \(\tilde{p}\) is in general position (has no affine dependence). In Figure 5, the plane momenta are shown as arrows-though they are actually weighted extensors \(\lambda_{a}ac\) along the line to the center. In our companion paper [27], these vectors are equivalent to 'parallel drawings of the plane configuration'. The paper [60] offers an extended study of how to projectively construct the centers of motion of the bars of a plane framework. A simple illustration is given in Figure 5. The preprint [28] includes a number of examples that are analyzed geometrically in terms of centers of motion. These projective momenta, as the 'centers of motion of vertices', round out the projective representation of infinitesimal kinematics. Figure 5: An infinitesimal motion has equal projections on a bar (**a**). In the plane, if the vectors are turned 90 degrees, we have weighted line segments or momenta for vertices in (**b**), whose lines meet in a point, and whose ends produce a segment parallel to the bar. This point is the center of motion (**c**) for the bar. A choice of scale for the weight of the center as a projective point generates a line parallel to the bar (**d**). We present one class of frameworks amenable to this approach of centers of motion of bars for plane frameworks [28]. We will revisit the geometry of momenta in Sections 5 and 7. Consider a cycle of quadrilaterals in the plane. The relative center of motion of any two bars in the plane is the difference of the two projective centers of the bars [28], or the center of motion of the second bar, when the first one is held fixed (by subtracting its center from all other centers).