## Part II Projective Theory of Connected Body Frameworks
### Static Rigidity and Stresses in Body-Bar Frameworks  
We can transfer the entire theory of statics to body-bar frameworks. One approach is to replace bodies by rigid bar-joint sub-frameworks. Instead, we will directly transfer the definitions, with illustrations from dimensions 2 and 3. Recall that forces are expressed with 2-extensors, which are also used for rows of the body-bar rigidity matrix. Figure 49: (**a**) two independent intersecting lines generate a plane pencil (red lines) which are linear combinations of the original lines. (**b**) shows three independent lines which generate two plane pencils. (**c**) shows four independent lines which generate three plane pencils that share a common line. (**d**) shows five independent lines which share a common line and generate four plane pencils sharing a common line which is the center of the singular line complex. Figure 50: (**a**) shows 6 independent lines in space which generate all lines in space. (**b**) shows 3 lines through a point \(p\) which generate all lines through \(p\), such as the added red line \(l\). (**c**) shows three lines in a plane \(P\) which generate all lines in that plane, such as the added red line \(l\). (**d**) shows 6 dependent lines when the point \(p\) in (**b**) lies in the plane \(P\) in (**c**) and they each generate the common line \(l\). In the theory for most types of finite structures with a rigidity matrix, infinitesimal rigidity requires that the column rank is \(\binom{d+1}{2}(|V|-1)\) and static rigidity requires that the row rank is also \(\binom{d+1}{2}(|V|-1)\). This immediately gives the following result. Proposition 8: _A body-bar framework in \(\mathbb{P}^{d}\) is infinitesimally rigid if and only if it is statically rigid in \(\mathbb{P}^{d}\)._  
Figure 51: Given a body-bar framework with a load (**a**), we can graphically confirm it is an equilibrium load (**b**). With the two bars joining a pair of bodies, a resolution will use a line through their crossing point (**c**) and give forces along these lines (**d**) to resolve the load. **Example 33**. : _Consider the bicycle wheel in \(3\)-dimensions (Figure 52). This can be represented as a body-bar framework with two bodies and the spokes as bars. However, the material of the thin spokes will only support forces in tension. For infinitesimal rigidity, we need a proper equilibrium stress which is positive on the spokes. With this sign restriction it is a tensegrity framework (Section 12). For example, we may only need 7 spokes which span the space of bars to make the wheel rigid - which can be enough bars to sustain an equilibrium stress \(\omega\) between two bodies. If \(\omega\) is a positive equilibrium stress on the 7 bars, then any load on the wheel and axle can be resolved on 6 independent bars. Adding a sufficiently large multiple of the positive equilibrium stress \(\omega\) on the 7 bars, the load is now resolved on the spokes, with all coefficients positive. This gives a flavour of what we will explore further in Section 12._  
_This rigidity is not quite projective but it is locally projective: any small projective transformation will preserve the signs of the equilibrium stress and keep the body-bar framework statically rigid with a positive equilibrium stress on the 7 bars. We note that this approach requires only one bar beyond the minimal \(6\) for static rigidity--not the \(12\) tensegrity members that Buckminster Fuller speculated._  
A _cut set_\(S\subset E\) of a body-bar framework is a subset \(E^{\prime}\) of edges whose removal separates the graph of the framework into two or more distinct components. The following theorem is folklore among civil engineers and gives a useful property of such cut sets [29]. **Proposition 9** (Cut Set Equilibrium). : _Let \(S^{\prime}=\{(p_{i},q_{i}):1\leq i\leq k\}\subset S\) be the edges of a cut set \(S\) of a framework \((G,p)\) which are directed into a connected component of \(G-S\).