## Part I Projective Geometry in Core Rigidity Results  
### 2 Introduction to Euclidean Rigidity Theory  
To ease transition to our desired, more thoroughly projective presentation, we begin with a brief description of the more familiar Euclidean presentation of rigidity theory. See [30; 31; 32; 33; 34], for example, for more details. A \(d\)-dimensional (_bur-joint) framework_\((G,p)\) is an ordered pair consisting of a finite, simple graph \(G=(V,E)\) and a map \(p:V\to\mathbb{R}^{d}\). We think of a framework as a set of stiff bars (corresponding to the edges of \(G\)) that are connected at their ends by joints (corresponding to the vertices of \(G\)) that allow bending in any direction of \(\mathbb{R}^{d}\). Loosely speaking, such a framework is called _rigid_ if every continuous deformation of the vertices which fixes the bar lengths arises from a congruence of \(\mathbb{R}^{d}\). Otherwise, the framework is said to be _flexible_ (Figure 1). See [30], for example, for a detailed definition. An _infinitesimal motion_\(u:V\to\mathbb{R}^{d}\) of \((G,p)\) is an assignment of velocity vectors to the joints so that the distance between any pair of joints connected by a bar is preserved at first order:  
\[(p_{i}-p_{j})\cdot(u_{i}-u_{j})=0\text{ for all }ij\in E, \tag{1}\]  
where \(p_{i}=p(i)\) and \(u_{i}=u(i)\). An infinitesimal motion is called _trivial_ if it arises as the derivative of a rigid body motion of \(\mathbb{R}^{d}\), restriced to \(p\). The dimension of the space of trivial infinitesimal motions of a framework that affinely spans \(\mathbb{R}^{d}\) is \(\binom{d+1}{2}\). This space  
Figure 1: A rigid (**a**) and a flexible (**b**) framework in the plane. The motion shown in (**c**) takes the framework in (**b**) to the framework in (**d**). is generated by \(d\) independent translations and \((\binom{d}{2})\) independent rotations. Infinitesimal motions are illustrated in Figure 2.  
A framework \((G,p)\) in \(\mathbb{R}^{d}\) is _infinitesimally rigid_ if every infinitesimal motion of \((G,p)\) is trivial, and _infinitesimally flexible_ otherwise. The _rigidity matrix_\(R(G,p)\) of \((G,p)\) is the \(|E|\times d|V|\) matrix of the system (1), where \(u\) is unknown; that is, \(R(G,p)\) is of the form:  
\[i\] \[R(G,p)=\ ij\left( \begin{array}{ccccccccc}&&&&\vdots&&&&\\ 0&\ldots&0&(p_{i}-p_{j})&0&\ldots&0&(p_{j}-p_{i})&0&\ldots&0\\ &&&&\vdots&&&&\end{array} \right),\]  
where the entries of the matrix are considered as row vectors. The space of infinitesimal motions of \((G,p)\) is the kernel of \(R(G,p)\), and if the joints of \((G,p)\) affinely span all of \(\mathbb{R}^{d}\) then \((G,p)\) is infinitesimally rigid if and only if rank \(R(G,p)=d|V|-\binom{d+1}{2}\). It is well known that an infinitesimally rigid framework is rigid [30]. The converse is also true, provided that \((G,p)\) is _regular_, that is, if rank \(R(G,p)\geq\text{rank }R(G,q)\) for all \(q\in\mathbb{R}^{d|V|}\)[30]. Note that 'almost all' realisations \((G,p)\) of a graph \(G\) are regular, in the sense that the set of configurations \(p\) for which \((G,p)\) is regular is a dense open subset of \(\mathbb{R}^{d|V|}\). This is because they are the complement space of an algebraic variety defined by the determinants of a finite number of submatrices of the rigidity matrix. We say that \((G,p)\) is _generic_ if the coordinates of \(p\) are algebraically independent over the rationals. Clearly, a generic framework is regular, and the set of generic realisations of a graph \(G\) is still a dense (but not an open) subset of \(\mathbb{R}^{d|V|}\). The rigidity matrix \(R(G,p)\) of \((G,p)\) defines the _rigidity matroid_ of \((G,p)\) on the ground set \(E\) by linear independence of the rows of \(R(G,p)\). It is easy to see that any two generic frameworks with the same underlying graph \(G\) have the same rigidity matroid [30]. This is called the _\(d\)-dimensional rigidity matroid_ of \(G\), and we will denote it by \(\mathcal{M}_{d}(G)\). See [34; 35] for background on the use of matroid theory in rigidity.