## 10 Body-Hinge Frameworks
### Body-Hinge Motion Assignments  
This presentation is adapted and extended from the initial 3-dimensional projective analysis in [53]. The initial presentation was projective and therefore included sliders at infinity as hinges in \(\mathbb{P}^{3}\). We present the basic results in the form of a natural generalisation to all dimensions. An analogous approach also occurs in the study of cofactors for splines and we will see that results and methods transfer (Section 11).  
Given two bodies \(B_{i}\), \(B_{j}\) and a hyperplane for the hinge \(H_{ij}\), this becomes a geometric constraint on the centers of motion (codimension 2 extensors) \(S_{i}\) of the bodies:  
\[S_{i}-S_{j}=\omega_{ij}H_{ij}.\]  
The edge \(ij\) is directed, as is \(H_{ij}\). The same equation can be written as \(S_{j}-S_{i}=\omega_{ij}H_{ji}\) with \(H_{ji}=-H_{ji}\) since \(\omega_{ji}=\omega_{ij}\). Given any body-hinge framework in \(\mathbb{P}^{d}\), we can track what happens around a cycle \(C:\langle B_{1};H_{12};B_{2};\ldots:B_{k};H_{k1};B_{1}\rangle\). We observe that the hinge equations collapse the entries \(S_{i}\) so that the equation becomes the _cycle condition_\(\sum_{(ij)\in C}\omega_{ij}H_{ij}=\mathbf{0}\). We call such an assignment of scalars a _motion assignment_ of the body-hinge structure.  
Figure 57: Any 5 trees on edges of a tetrahedron are all incident on a shared hinge line.  
**Proposition 10**.: _Any infinitesimal motion of a body-hinge framework in \(\mathbb{P}^{d}\) gives a unique motion assignment. Conversely, any motion assignment of scalars which satisfies the cycle condition on all cycles gives an infinitesimal motion for the body-hinge framework, unique up to a trivial motion for an initial body._  
In \(\mathbb{P}^{3}\), starting with a spherical polyhedron which has vertices, edges and faces, and making the faces into bodies, the cycles for the motion assignment are all generated by the cycles around vertices. This is a key property of any simply-connected topological surface. In addition, in \(\mathbb{P}^{3}\), the hinges as 2-extensors are also candidates for stresses on bars and there is a transfer between motion assignments and equilibrium stresses. This connection will reappear and will be extended as a key property in Section 10.6.  
**Proposition 11** (Crapo and Whiteley [53]).: _Any infinitesimal motion of a body-hinge framework in \(\mathbb{P}^{3}\) on the faces and edges of a polyhedral manifold gives a unique motion assignment which is an equilibrium stress on the framework of vertices and edges. Conversely, given a spherical polyhedron with an equilibrium stress on the vertices and edges of the polyhedron, the scalars form a motion assignment for an infinitesimal motion of a body-hinge framework on the faces and edges of the polyhedron._  
Figure 58 shows some cycles where this correspondence transfers to give results for infinitesimal rigidity (a) and for infinitesimal (actually finite) flexibility (b).  
For more general oriented topological surfaces in 3-space, such as a torus, the topology can be realised geometrically as a polyhedral body-hinge framework in 3-space with vertices, edges as hinges, and faces as discs. A motion assignment to the hinges and faces of the polyhedron which satisfies the cycle conditions still implies an equilibrium stress in the corresponding bar-joint framework on the vertices and edges. However the converse does not hold. Given an equilibrium stress on the vertices and edges of the toroidal polyhedron, there are cycles of faces and edges on a toroidal polyhedron which do not disconnect the graph. Therefore the transferred equilibrium stress scalars may not satisfy the cycle condition for a motion assignment on such a cycle.