## 7 Pure Conditions
### Bipartite Frameworks and Quadratic Surfaces
After a projective transformation, the momenta remain tangent to the new quadric and the Euclidean velocity will be normal to the quadric. This geometric reasoning extends to all dimensions, giving a center of motion for each bar which is the polar of the bar in the quadric in the space. This polarity for momenta is a new result for projective momenta._  
_Moreover, if we apply a projective transformation to the entire configuration, to obtain other non-degenerate quadric surfaces, the momenta transfer immediately with the same projective transformation, along with the polarity. It will take some more subtle limiting arguments to transfer to degenerate quadric surfaces, in the manner of [95]._  
_In a general dimension \(d\), the momenta of ends of the bar \((a,b)\) are weighted hyperplanes, and \(M(a)\wedge M(b)\) is the weighted center of motion of the bar. This is a striking new geometric result which depends on projective geometry of polarities about quadrics and the special infinitesimal motions of frameworks on quadrics. Notice that if a bar is a diagonal of the sphere (through the center of the sphere) the momenta are parallel hyperplanes, meeting at a projective 'center' at infinity, representing a translation of the bar._  
Figure 22: A complete bipartite framework on a circle has a non-trivial infinitesimal motion moving \(a_{i}\) out along rays and \(b_{j}\) in along rays (**a**). The two velocities for any pair of points on the circle have equal projections on the line of the chord (**b**). We can summarize this example with the following new result. **Proposition 1**. : _Given a bipartite framework \((G,p)\) realizing a bipartite graph \(K_{m,n}\) in \(\mathbb{P}^{d}\) with all vertices on a quadratic surface \(Q\), the polar of the vertices \((G,p)\) in the quadric gives a multiple of the momenta of the vertices and the polar of the edges gives a multiple of the projective centers of motion of the bars for a non-trivial infinitesimal motion._  
The geometry of centers of motion, including these momenta of vertices, is rich and not well explored. However, some further examples are found in [28], where there was a focus on planar graphs and connections with projections of spherical polyhedra. This connection will also reappear in our companion paper [27], where we explore reciprocal diagrams. **Corollary 4**. : _A complete bipartite graph \(K_{m,n}\) is generically rigid in \(\mathbb{P}^{d}\) if and only if (i) \(m,n\geq d+1\); and (ii) \(m+n\geq{d+2\choose 2}\)._  
**Example 13**. : _Consider the graph \(K_{5,5}\). This graph is generically rigid in \(\mathbb{P}^{3}\). Since \(|E|=25=3|V|-5\) it also has an equilibrium stress. If we consider a realisation where all points lie on a quadric (one geometric condition) then it is infinitesimally flexible, with a larger space of equilibrium stresses. With one bipartite side of 5 points in a plane, these 5 points must lie on a plane conic and also generate an infinitesimal motion, which actually extends to a finite motion._  
**Example 14**. : _A framework realising the graph \(K_{4,5}\) plus any single bar in \(\mathbb{P}^{3}\) has a non-trivial infinitesimal motion if and only if there is a quadric surface through the nine joints which also contains the line of the added bar or if the four joints \(a_{1},a_{2},a_{3},a_{4}\) are coplanar ([95] Corollary 2.1)._  
**Example 15**. : _Consider \(K_{6,6}\) realised as a generic framework in \(\mathbb{P}^{4}\). With \(|E|=36=(4|V|-10)-2\) we immediately see that the space of non-trivial infinitesimal motions is at least 2-dimensional. Since any 12 vertices have a \(3\)-dimensional space of conics through all the vertices, there is actually a \(3\)-dimensional space of non-trivial infinitesimal motions. There must be an equilibrium stress in all generic realisations, even though these frameworks are flexible. This is an example of a circuit which is not predicted by any simple count of vertices and edges.