## 7 Pure Conditions
### Small Examples  
The following two examples illustrate pure conditions as a single projective polynomial that captures when a generically isostatic graph has an equilibrium stress or equivalently a non-trivial infinitesimal motion. These, and many other examples, are explored at length in [28; 29], using both projective kinematics and projective stresses, in \(\mathbb{P}^{2}\) and \(\mathbb{P}^{3}\). **Example 10**. : _Consider the graph in Figure 20a. With 6 vertices and 9 edges, this graph is generically isostatic in \(\mathbb{P}^{2}\) (recall the 3 Tree2 partition in Figure 4a). If either of the triangles is collinear, \([a_{1}a_{2}a_{3}]=0\) or \([b_{1}b_{2}b_{3}]=0\), then there is an equilibrium stress, and these terms are factors of the pure condition. If neither triangle is collinear, then consider the remaining 3 edges \(a_{1}b_{1}\), \(a_{2}b_{2}\) and \(a_{3}b_{3}\). For simplicity, assume that \(a_{1},a_{2},a_{3}\) have \(0\) as their momenta. If there is a non-trivial infinitesimal motion, the momentum for \(b_{1}\) must be a multiple of \(a_{1}b_{1}\) and then the relative center \(c\) of this motion must lie on this line. Similarly, the relative center must lie on \(a_{2}b_{2}\), and \(a_{3}b_{3}\), so the three bars must be concurrent, and the two triangles are perspective from \(c\)[61]. This concurrence can be written, using Grassmann-Cayley algebra, as the simple polynomial equation,_  
\[[a_{1}b_{1}a_{3}][a_{2}b_{2}b_{3}]-[a_{1}b_{1}b_{3}][a_{2}b_{2}a_{3}]=0.\]  
_If we consider the condition for an equilibrium stress, with neither triangle collinear, the equilibrium stress \(\omega_{1}a_{1}b_{1}+\omega_{2}a_{2}b_{2}+\omega_{3}a_{3}b_{3}\) onto the triangle \(b_{1},b_{2},b_{3}\) requires that these three forces are concurrent, so the three bars are concurrent. We can capture all these conditions in the product of the conditions (or in the logic of the separate conditions):_  
\[[a_{1}a_{2}a_{3}][b_{1}b_{2}b_{3}]([a_{1}b_{1}a_{3}][a_{2}b_{2}b_{3}]-[a_{1}b_ {1}b_{3}][a_{2}b_{2}a_{3}])=0.\]  
_This condition for a non-trivial motion is folklore within the older rigidity community [28; 29], and we will return to it several more times in this paper. This figure is also a cycle of three quadrilaterals--the case \(n=3\) already described above in Example 2--giving the two triangles being perspective from a line (Figure 20b)._  
_The example is also intimately connected to Desargues Theorem of projective geometry, which says that the two triangles being perspective from a point, or one of the two triangles being collinear is equivalent to the two triangles being perspective from a line: corresponding edges intersect at points along a line [61]. Some classic statics textbooks for engineers include appendices which give static proofs of these types of projective geometry theorems [87]. Statics has a long history in projective geometric reasoning, including the balance of weighted points in Mobius barycentric coordinates and classical proofs of Ceva's theorem._  
**Example 11**. : _Consider the graph \(G\) of an octahedron, depicted in Figure 21, which is generically isostatic in \(\mathbb{R}^{3}\) (see also Theorem 14). A theorem of Bennett [28,29,88] shows that this has an infinitesimal motion if and only if the four alternate faces (in yellow in (a)) meet in a single point. This geometry can be expressed by a single projective polynomial which will be named the pure condition in Section 7.4 below. The polynomial that expresses the concurrence of the planes is:_  
\[[a_{1}a_{2}b_{3}b_{1}][a_{2}a_{3}b_{1}b_{2}][a_{3}a_{1}b_{2}b_{3}]+[a_{1}a_{2}b_ {3}b_{2}][a_{2}a_{3}b_{1}b_{3}][a_{3}a_{1}b_{2}b_{1}].\]  
_This requires that the octahedron must be non-convex and hence is far from the results of Cauchy for triangulated convex polyhedra (Section 8.4). It is a theorem of projective geometry that if one set of four opposite faces meet in a single point then the other four faces also meet in a single point. This theorem will follow from the analysis below.