## 7 Pure Conditions
### Computing Pure Conditions: Pinned Frameworks, and \(d\)-Directed Graphs  
The pure conditions of a graph \(G=(V,E)\) can be computed by taking a Laplace decomposition of the determinant of the associated rigidity matrix for a generic realisation squared off either (i) by adding \(\binom{d+1}{2}\) tie-down rows to remove the infinitesimal degrees of freedom [85] or (ii) by deleting \(d\)-tuples of columns to pin down certain vertices. This second option provides objects that are regularly studied in mechanical engineering [80; 106; 105]. We will summarise some of these techniques, including connections to strongly directed graphs, because these also have applications both to mechanical engineering, under the name of Assur Graphs, as well as to computing pure conditions for other rigidity-like matrices such as cofactor matrices in Section 11. We also note that many of these methods and results have analogues for body-bar frameworks (Section 9), for multivariate splines (Section 11) and for the dual concepts of liftings and parallel drawings in our companion paper [27]. We begin by adding rows to the projective matrix for a _tie-down_\(T\) that blocks all of the trivial motions, adapting [85] (recall Section 7.5). We will illustrate this process using an example in 3-space. **Example 19**. : _Consider the pinned tetrahedron in Figure 28a. There is a unique way to orient the remaining edges to result in a \(3\)-directed graph. This translates to a pure condition for the tied-down framework \((G\cup T,p)\) which is a single bracket condition \([a_{1}a_{2}a_{3}a_{4}]\). The framework is dependent if and only the four vertices are coplanar. We have_  
set of cycles [106] (see Figure 29b,c). However the strongly connected components are invariant under such reversals. In \(\mathbb{P}^{2}\), with a tie-down of size \(3\) to block the trivial motions, there will be analogous \(2\)-directed orientations of the tied down graph. However, related to Laman's theorem and its counts, the existence of a \(2\)-directed orientation of the tied down graph is necessary but not sufficient for the graph to be isostatic [106]. This connects to work in mechanical engineering on Assur graphs [106], which we return to below. **Example 20**. : _We illustrate the process with one more example, which has served as the provocation for a number of explorations, and will reappear in Section 10.6. Consider the framework in Figure 29a. With the central tetrahedron tied down we get a factor \([a_{1}a_{2}a_{3}a_{4}]\) as calculated in Example 19. The remaining edges have two \(3\)-directed orientations, differing by reversing the directed cycles in Figure 29b,c:_  
\[(b_{1},b_{2})(b_{2},b_{3})(b_{3},b_{4})(b_{4},b_{1})\text{ reversing to }(b_{1},b_{4})(b_{4},b_{3})(b_{3},b_{2})(b_{2},b_{1}).\]  
_Together these two \(3\)-directed orientations give the overall pure condition:_  
\[[a_{1}a_{2}a_{3}a_{4}]([a_{1}a_{2}b_{2}b_{1}][a_{2}a_{3}b_{3}b_{2}][a_{3}a_{4} b_{4}b_{3}][a_{4}a_{1}b_{1}b_{4}]\]  
\[-[a_{1}a_{2}b_{4}b_{1}][a_{2}a_{3}b_{1}b_{2}][a_{3}a_{4}b_{2}b_{3}][a_{4}a_{1} b_{3}b_{4}]).\]  
_It is not immediately obvious what sign should be between the two terms. The next example will give a simple calculation which clarifies this sign. If we swap which cycle of vertices the tetrahedron is attached to (Figure 29d), we have a new factor \([b_{1}b_{2}b_{3}b_{4}]\) but the remaining edges generate a factor which is, up to \(\pm 1\), the same. We will return to this'swapping' of blocks (the tetrahedra) and holes (the unfilled quadrilaterals) in Section 10.6 [107,108]._  
**Example 21**. : _Consider the general cycle of triangles around a rigid central n-gon which is already pinned (Figure 30). This was analysed in ([28] Section 3), offering an additional method worth describing. In this framework, every vertex \(a_{i}\) is attached to two grounded vertices \(b_{i-1}\) and \(b_{i}\), and will therefore have a projective momentum which is a multiple of the extensor for that triangle \(M(a_{i})=\lambda_{i}b_{i-1}b_{i}a_{i}\).