## 7 Pure Conditions
### Bipartite Frameworks and Quadratic Surfaces  
The family of complete bipartite graphs have fully understood rigidity properties, both generically and geometrically in all dimensions. The original theory for these graphs was developed, using statics, in [92]. An early example of \(K_{3,3}\) in the plane with conics was presented by Sang [93], and an applied 3-dimensional example of \(K_{4,6}\) with a quadric was found by row-reduction in a Master's thesis in geodesy for the bipartite graph of satellite positions and ground stations [94]. We will present the overall results as transferred to infinitesimal kinematics in [95,96]. The second widely studied class of generically rigid graphs are the simplicial manifolds, which are far from bipartite. See Section 8.4 and [97]. A key result in this direction, obtained by Fogelsanger [98], is that the graph of any triangulation of a closed 2-manifold is generically rigid in \(\mathbb{R}^{3}\). **Theorem 5** (Whiteley [95]). : _A framework realizing the bipartite graph \(K_{m,n}\) with partite sets \(A\) and \(B\)\((m,n\geq 2)\) in \(\mathbb{R}^{d}\) (for \(d>1\)) has a nontrivial infinitesimal motion if and only if either_  
1. _the joints of_ \(A\cup B\) _lie on a quadric surface,_
2. _one side (_\(A\) _or_ \(B\)_) lies on a hyperplane along with at least one joint of the other side, or_
3. _one side (_\(A\) _or_ \(B\)_) lies on a hyperplane_ \(H\) _and lies on a quadric surface within the hyperplane._  
**Corollary 3**. : _Any bipartite framework (with more than 2 joints) realised with all its joints on a quadric surface in \(\mathbb{P}^{d}\) (for \(d>1\)) will have a non-trivial infinitesimal motion._  
The essential geometric feel for Corollary 3 can be found by observing that this is true for a sphere as the quadric, and that in some sense (including through the complex  
Figure 21: The octahedron is infinitesimally flexible in 3-space if and only if four opposite faces are concurrent (**a**). For an equilibrium stress, the components of the equilibrium stress in the plane of \(a,b,c\) must lie in the plane at \(a,b,c\) and meet in a point \(p\) in the plane (**b**). The momenta for vertices \(a,b,c\) intersect the plane of triangle \(abc\) in plane momenta which meet in a point \(p\)–the center of motion of the triangle—which is on all four planes (**c**). numbers) all quadrics are projective images of the sphere. Note that for a sphere as the quadric the velocities are radial in-out of equal length. See Figure 22.  
**Example 12**. : _There is a deeper projective form shown in Figure 23. The projective momenta of the vertices on the sphere are now weighted hyperplanes tangent to the d-sphere (a), in all dimensions, with equal weights at each joint. In the plane, the construction of the center of motion of a bar as the intersection of the momenta of its ends (\(c_{ab}=M(a)\wedge M(b)\)) is also the construction of the polar point to a line of the bar through the circle in the conic polarity (Figure 23b). The weight of this center of rotation is scaled to ensure \(\beta_{ab}c_{ab}\lor a=M(a)\)._  
_With momenta, the 'in-out' motion becomes clockwise/counterclockwise tangents for the two classes of vertices (a). Following the property that the in-out velocities are of equal length, the momenta must be equal weight multiples of the polar tangent lines. In the plane, with the momenta tangent to the circle, a projective transformation of the circle will create a more general conic, with the momenta now tangent to the new conic. If we take limits of such conics, we can find the momenta for any conic. For degenerate conics (e.g., two lines meeting or parallel) there is still a non-trivial infinitesimal motion, but the momenta are more subtle [95]._  
_In \(\mathbb{R}^{3}\), the momenta will be weighted tangent planes to the sphere, and the projective center will be a line (2-extensor) which is the intersection of the two momenta planes at the ends of the bar, and also the polar of the line in the sphere.