## Part I Projective Geometry in Core Rigidity Results
### 2 Introduction to Euclidean Rigidity Theory
The above is sometimes called the kinematic approach to rigidity. We now also briefly describe the dual notion of static rigidity. An _equilibrium load_\(f\) on a framework \((G,p)\) is an assignment of a vector \(f(i)\) to each point \(p(i)\) such that \(\sum_{i\in V}f(i)=0\) and  
\[\sum_{i\in V}\big{(}f(i)_{j}p(i)_{k}-f(i)_{k}p(i)_{j}\big{)}=0\]  
for all \(1\leq j<k\leq d\), where we use the notation \(x_{t}\) for the \(t\)-th coordinate of a vector \(x\). These conditions on \(f\) are equivalent to there being no net force and no net torque. If we regard an equilibrium load as a vector in \(\mathbb{R}^{d|V|}\), then the set of equilibrium loads on \((G,p)\)  
Figure 2: Velocity vectors of a trivial infinitesimal motion (**a**) and non-trivial infinitesimal motions (**b**,**c**) of frameworks in the plane. forms a \((d|V|-{d+1\choose 2})\)-dimensional subspace of \(\mathbb{R}^{|V|}\). A _stress_\(\rho\) of \((G,p)\) is an assignment of a scalar \(\rho(e)\) to each edge \(e\) of \(G\). A stress \(\rho\)_resolves_ an equilibrium load \(f\) if  
\[\sum_{j:ij\in E}\rho(ij)(p(i)-p(j))=-f(i)\quad\text{ for all }i\in V, \tag{2}\]  
in which case we say that \(f\) is resolvable by \((G,p)\). See Figure 3 for an illustration. A stress \(\omega\) that resolves the zero load is called an _equilibrium stress_ (or _self-stress_) of \((G,p)\). Note that the set of equilibrium stresses of \((G,p)\) is a subspace of \(\mathbb{R}^{|E|}\). A framework \((G,p)\) that has only the zero equilibrium stress is called _independent_ (since in this case the rigidity matrix of \((G,p)\) has linearly independent rows). Otherwise, \((G,p)\) is called _dependent_. A framework that is infinitesimally rigid and independent is called _isostatic_. A framework \((G,p)\) is _statically rigid_ if every equilibrium load is resolvable by \((G,p)\). A classical fact which can be traced back to Maxwell and which follows from linear duality is the following. **Theorem 1**. : _A framework \((G,p)\) in \(\mathbb{R}^{d}\) is infinitesimally rigid if and only if it is statically rigid._  
We sketch a proof of this result and refer the reader to [29; 36] for details. **Sketch of proof. ** Equation (2) is equivalent to \(\rho^{T}R(G,p)=-f\), and hence the space of resolvable equilibrium loads is isomorphic to the row span of the rigidity matrix \(R(G,p)\). Let \(S(p)\) and \(M(p)\) be the space of equilibrium stresses and infinitesimal motions of \((G,p)\), respectively. Then, by the rank-nullity theorem, we have  
\[|E|-\dim S(p)=d|V|-\dim M(p).\]  
If \((G,p)\) is statically rigid, then \(\dim S(p)=|E|-(d|V|-{d+1\choose 2})\), and hence, by the equation above, \(\dim M(p)={d+1\choose 2}\), which says that \((G,p)\) is infinitesimally rigid. The converse is similar. Since for a given graph \(G\), all generic realisations of \(G\) as a \(d\)-dimensional bar-joint framework share the same rigidity properties (that is, they are either all rigid or all flexible) [30], we may define a graph to be _rigid_ (_isostatic_) in \(\mathbb{R}^{d}\) if some (equivalently, any) generic realisation of \(G\) is rigid (isostatic) in \(\mathbb{R}^{d}\). If the edge set of \(G\) is dependent in the rigidity matroid \(\mathcal{M}_{d}(G)\) and the removal of any edge yields an independent set in \(\mathcal{M}_{d}(G)\), then we say that \(G\) is a _(rigidity) circuit_ in \(\mathbb{R}^{d}\). It is a major research area in rigidity theory to obtain necessary and sufficient combinatorial conditions for the rigidity of graphs that can be checked in polynomial time. Using the well known recursive graph construction moves \(0\)-extension and \(1\)-extension, also known colloquially as Henneberg moves (since they were originally studied by Henneberg [7; 37]), Pollaczek-Geiringer showed that a graph is rigid in the plane if and only if it contains a spanning subgraph \(G=(V,E)\) satisfying \(|E|=2|V|-3\) and \(|E^{\prime}|\leq 2|V^{\prime}|-3\)  
Figure 3: (**a**) An equilibrium load on a framework \((K_{3},p)\) in the plane. This load can be resolved by \((K_{3},p)\) as shown in (**b**).