## 10 Body-Hinge Frameworks  
### Body-Hinge Basic Transfer  
For \(d=3\) the following result was conjectured in 1976 by Janos Baracs, a structural engineer leading the Structural Topology Research Group [136]. The proof of the theorem was then observed independently by Tay [125] and Whiteley [49]. In view of the essential difficulties which remain for spatial bar-joint frameworks it is a pleasant surprise that these hinge structures retain their combinatorial simplicity. Recall that a \((d-1)\)-extensor is a \(\binom{d+1}{2}\)-dimensional vector which satisfies the Plucker-relations. (A \((d-1)\)-extensor is dual to a 2-extensor.) Recall also that a _screw_ is a general \(\binom{d+1}{2}\)-dimensional vector, or a sum of \((d-1)\)-extensors. A body-hinge structure (or body-hinge framework) \((G,H)\) in \(d\)-space is a graph \(G=(V,E)\) together with a mapping \(H\) from \(E\) into the space of \((d-1)\)-extensors of projective \(d\)-space: \(H(e)=H_{e}=H_{ij}\) if \(a_{1}(e)=i\) is the initial vertex of \(e\) and \(a_{2}(e)=j\) is the final vertex of \(e\). An infinitesimal motion of a body-hinge structure is an assignment of screw centers \(S_{i}\) to each vertex \(v_{i}\) of the graph such that for every oriented edge \((i,j)\): \(S_{i}=-S_{j}=\omega_{e}H_{e}\) for some scalar \(\omega_{e}\). A body-hinge framework is infinitesimally rigid if every infinitesimal motion is trivial, with all bodies receiving the same centre. Notice that \(S_{i}=-S_{j}=\omega_{e}H_{e}\) is a set of \((\binom{d+1}{2}-1)\) equations, and a hinge is equivalent to \((\binom{d+1}{2}-1)\) bars. The body-hinge framework can only be infinitesimally rigid in \(\mathbb{P}^{d}\) if we replace each hinge by \((\binom{d+1}{2}-1)\) bars and the resulting body-bar framework contains a  
Figure 56: Three lines meeting three lines in nine points of intersection form a projectively special position (**a**) on a quadric surface which is known to be flexible. Releasing the last intersection to make the lines more generic (**b**) yields a structure that is infinitesimally rigid. Figure (**c**) counts to be infinitesimally flexible. subset with the required \(\binom{d+1}{2}(|V|-1)\) edges for body-bar rigidity. Following the body-bar analysis, this subset must partition into \(\binom{d+1}{2}(|V|-1)\) trees, such that the edges from any hinge are in distinct trees, with up to \((\binom{d+1}{2}-1)\) bars between any two bodies connected by a single hinge. What is less obvious is that if this replacement gives such a tree partition into \(\binom{d+1}{2}(|V|-1)\) trees, such that the edges between a pair of bodies are in distinct trees, with up to \((\binom{d+1}{2}-1)\) bars between any two bodies, then there is an infinitesimally rigid geometric realization with the edges between any two bodies all incident with a single line - the line of a geometric hinge [48]. This realization is found when the trees are realized along the edges of a simplex in \(\mathbb{P}^{2}\). This is illustrated in Figure 57 for \(d=3\). With the 6 trees realized along the edges of a tetrahedron, up to 5 of the trees meet a single edge (\(T_{4}\)), which becomes the line of a hinge sharing these 5 trees. These observations, with some additional details, are captured in Theorem 19. **Theorem 19** (Tay-Whiteley [48]). : _For a graph \(G\) the following are equivalent:_  
1. \(G\) _has realisations as an infinitesimally rigid body-hinge structure in_ \(\mathbb{P}^{d}\)_._
2. \(G\) _contains_ \(\binom{d+1}{2}\) _spanning trees which use any hinge edge at most_ \(\binom{d+1}{2}-1\) _times._
3.