## Part II Projective Theory of Connected Body Frameworks  
### 9. Body-Bar Frameworks  
By expanding vertices to be larger rigid structures or rigid bodies, we find the combinatorics of generic rigidity simplifies, and we can give full combinatorial characterisations in all dimensions with efficient algorithms and with informative projective geometric conditions for singularity [58; 125]. This setting has been a playground for developing resultswhich we can then work to extend back to bar-joint frameworks. The theory has also been the basis for conjectures and then theorems about the rigidity of molecular structures, particularly proteins [126] (Section 10.5). The full theory is projectively invariant and will be presented with projective coordinates. This projective presentation is the only efficient way to work with the algebraic representations for these structures. Take a loopless multigraph \(G=(V,E)\) where \(V=\{1,2,\ldots,|V|\}\). We define a _bodbar framework_\((G,p)\) in \(\mathbb{P}^{d}\), with each edge _ab_ represented by the weighted 2-extensor \(p_{a}p_{b}=ab=a\lor b\) (with \(a<b\)) and its Plucker coordinates in \(\mathbb{P}^{d}\). Definition 7: The rigidity matrix \(M(G,p)\) for the body-bar framework \((G,p)\) in \(\mathbb{P}^{d}\) has one row for each bar and \(\binom{d+1}{2}\) columns for each body, with the columns for \(B_{1}\) followed by those for \(B_{2}\), etc. If \((a,b)\) is a bar with endpoints \(a\) in body \(B_{i}\) and \(b\) in body \(B_{j}\), then the row corresponding to \((a,b)\) in \(M(G,p)\) has the 2-extensor ab in the \(\binom{d+1}{2}\) columns for \(B_{i}\) and \(ba=-ab\) in the \(\binom{d+1}{2}\) columns for \(B_{j}\), and \(0\) in all other columns. (Under this definition, many frameworks are equivalent. Indeed, all that matters is the 2-extensor, or line, \(ab\), not the location of the two points \(a\) and \(b\) on that line, as long as they are distinct.) A _motion_ of \((G,p)\) is an assignment of a center \(Z_{i}\) to each body \(B_{i},1\leq i\leq m\), so that the length of each bar \((a,b)\) is instantaneously preserved, that is, \(Z_{i}^{*}\cdot ab-Z_{i}^{*}\cdot ab=0\). (Recall the definition of dual extensors in Section 3.1). If we let \(Z^{*}\) be the vector \((Z_{1}^{*},Z_{2}^{*},\ldots,Z_{m}^{*})\) of length \(m\binom{d+1}{2}\) then we require that \(M(G,p)(Z^{*})^{\mathsf{T}}=0\). An equilibrium stress of a body-bar framework \((G,p)\) in \(\mathbb{P}^{d}\) is a row dependence of \(M(G,p)\). A body-bar framework \((G,p)\) in \(\mathbb{P}^{d}\) is _independent_ if the only equilibrium stress is zero on all edges. A body-bar framework \((G,p)\) in \(\mathbb{P}^{d}\) is _infinitesimally rigid_ if \(M(G,p)\) has rank \(\binom{d+1}{2}(|V|-1)\). As noted above, there are many _equivalent_ body-bar frameworks with the bars sharing the same extensors, but perhaps using different points along the same lines. Multiplying the bar-extensors by a scalar can correspond to sliding a pair of points along the line, at a different distance or just to a different weighting of the projective points. Furthermore, the bodies do not have any location. Any point can be assumed to lie on body \(B_{i}\), and the same point can be assigned to lie on another body along another line. When looking at an application, or a coming figure (e.g., Figures 42 and 45) we will depict a set of locations for the ends of the bars, but the analysis will apply to equivalent body-bar frameworks. This equivalence is also projective, so the end of a bar, or an entire bar, can 'lie at infinity' in a figure or an application of the results. Theorem 16 (Tay [58,125]): For a generic set of lines \(p\) and a body-bar framework \((G,p)\) in \(\mathbb{P}^{d}\), the following are equivalent:  
1. \((G,p)\) is infinitesimally rigid and independent as a body-bar framework;
2. \(|E|=\binom{d+1}{2}(|V|-1)\) and for all non-empty subsets of edges \(E^{\prime}\) on bodies \(V^{\prime}\), we have \(|E^{\prime}|\leq\binom{d+1}{2}(|V^{\prime}|-1)\);
3.