## 3 Projective Rigidity
### Infinitesimal and Static Rigidity in Projective Space  
In this section, we give a brief summary of the development of the theory of infinitesimal rigidity in projective space using Plucker coordinates and extensors. We start by describing infinitesimal rigid body motions in projective space. In the following, we will use the notation \(\tilde{p}= \begin{pmatrix}p\\ 1\end{pmatrix} \in\mathbb{R}^{d+1}\) for a point \(p\in\mathbb{R}^{d}\). Let \(p_{1},\ldots,p_{k}\) be \(k\) points that span an affine subspace \(\overline{U}\) of \(\mathbb{R}^{d}\) of dimension \((k-1)\), and let \(U\) be the \(k\)-dimensional linear subspace of \(\mathbb{R}^{d+1}\) spanned by \(\tilde{p}_{1},\ldots,\tilde{p}_{k}\). Then the Plucker coordinate vector (or \(k\)-extensor) \(P_{U}=\tilde{p}_{1}\vee\cdots\vee\tilde{p}_{k}\) determined by \(U\) (up to a scalar) is said to be the \(k\)_-extensor associated with_\(\overline{U}\). Consider an infinitesimal rotation of \(\mathbb{R}^{d}\). It has a \((d-2)\)-dimensional axis (or center) \(W\). Let \(c_{1},\ldots,c_{d-1}\) affinely span \(W\). In the projective setting, the center is a subspace of dimension \(d-1\) in \(\mathbb{P}^{d}\) spanned by the vectors \(\tilde{c}_{1},\ldots,\tilde{c}_{d-1}\). We let \(Z=\tilde{c}_{1}\vee\cdots\vee\tilde{c}_{d-1}\) be the \((d-1)\)-extensor associated with \(W\). Then for any point \(p\
otin W\), \(Z\vee\tilde{p}\) is a \(d\)-extensor associated with the hyperplane \(\operatorname{span}(W+p)\). Now, for some vector \(v\) that is normal to \(\operatorname{span}(W+p)\), \(Z\vee\tilde{p}\) can be written as \((v,-v\cdot p)\), where \(v\cdot p\) is the dot product of \(v\) and \(p\) (see [58], for example). The length of \(v\) is proportional to the distance between \(W\) and \(p\) (and to the volume of the simplex determined by \(c_{1},...,c_{d-1}\), \(p\)) so that for some constant scalar \(\alpha\), the first \(d\) entries of \(\alpha(Z\vee\tilde{p})\) represent the velocity vector of the rotation around \(W\) at the point \(p\). The vector \(Z^{\prime}=\alpha Z\) is called the _center of the rotation_. Next we describe an infinitesimal translation of \(\mathbb{R}^{d}\) in the direction of a (free) vector \(t\in\mathbb{R}^{d}\). In projective space, a translation may be thought of as a rotation around an axis at infinity, so we may mimic the description of an infinitesimal rotation given above. More precisely, the \((d-1)\)-extensor associated with the axis at infinity for the translation in the direction of \(t\) is obtained by taking the orthogonal complement \(U\) of \(\operatorname{span}(t)\), fixing a basis \(u_{1},\ldots u_{d-1}\) of \(U\), and then taking the \((d-1)\)-extensor \(Z=\hat{u}_{1}\vee\ldots\vee\hat{u}_{d-1}\), where \(\hat{u}_{i}= \begin{pmatrix}u_{i}\\ 0\end{pmatrix} \) is the point at infinity in the direction of \(u_{i}\). Now, as above, for any point \(p\) we consider the \(d\)-extensor \(Z\vee\tilde{p}\) and observe that the first \(d\) coordinates of this vector are independent of \(p\) and proportional to \(t\). So for some constant scalar \(\alpha\), we have \(Z^{\prime}\lor p=(t,-t\cdot p)\). The vector \(Z^{\prime}=\alpha Z\) is called the _center of the translation_ in the direction of \(t\). Now, an arbitrary infinitesimal rigid body motion \(M\) is the vector sum of infinitesimal rotations and translations. If \(Z^{\prime}_{i}\), \(i=1,\ldots,b\), are the corresponding centers of these infinitesimal rigid body motions, then the velocity vector assigned to \(p\) under \(M\) is given by the first \(d\) coordinates of the vector \(\sum_{i=1}^{b}(Z^{\prime}_{i}\vee\tilde{p})\). The vector \(Z^{\prime}=\sum_{i=1}^{b}Z^{\prime}_{i}\) is called the _screw center_ of \(M\). Note that the screw center can in general not be expressed as a \((d-1)\)-extensor. (As indicated by the name'screw', in \(\mathbb{R}^{3}\) it can be represented as the sum of an extensor for a rotation, and an extensor for translation along the axes of the rotation [59].)