## 3 Projective Rigidity
### Plucker Coordinates and Extensors  
Consider the projective \(d\)-space \(\mathbb{P}^{d}\). Recall that a point in \(\mathbb{P}^{d}\) is represented as a vector in \(\mathbb{R}^{d+1}\), but two non-zero vectors \(p\) and \(q\) represent the same projective point if and only if \(p=\lambda q\) for some \(\lambda\
eq 0\). These \((d+1)\)-dimensional vectors are called the _homogeneous coordinates_ for the points. If the last coordinate \(p_{d+1}\) of \(p\) is non-zero, we say that \(p\) is _finite_, with \(p_{d+1}\) as _weight_. In this case, we can represent \(p\) as \((p_{1},\ldots,p_{d},1)\). If \(p_{d+1}=0\), then \(p\) is called _infinite_ (as it lies in the hyperplane at infinity) and has weight zero. Let \(U\) be a \(k\)-dimensional linear subspace of \(\mathbb{R}^{d+1}\) and let \(\{u_{1},\ldots,u_{k}\}\) be a set of basis vectors of \(U\). We let \(A(u_{1},\ldots,u_{k})\) be the \(k\times(d+1)\) matrix whose \(i\)th row is the transpose of \(u_{i}\). For a \(k\)-element subset \(\{i_{1},\ldots,i_{k}\}\) of \(\{1,\ldots,d+1\}\), the \((i_{1},\ldots,i_{k})\)-th _Plucker coordinate_ of \(U\) is defined as the determinant of the \(k\times k\) submatrix obtained from \(A(u_{1},\ldots,u_{k})\) by taking the \(i\)-th columns for \(1\leq j\leq k\) in some predetermined order. The _Plucker coordinate  
Figure 4: A graph is isostatic in \(\mathbb{R}^{2}\) if and only if the edges can be decomposed into 3 trees, exactly 2 meeting at each vertex (called 3_Tree2_ for short) Figure (**a**). The tree decomposition is _proper_ if no non-trivial subtrees of distinct trees have the same span [47]). Failure is illustrated in Figure (**b**) which is not proper—the subgraph in the circle is covered by two trees (hence has \(|E^{\prime}|=2|V^{\prime}|-2\)). vector_\(P_{U}\) of \(U\) is the \({d+1\choose k}\)-dimensional vector consisting of these \({d+1\choose k}\) Plucker coordinates of \(U\) in some predetermined order. Note that \(U\) determines \(P_{U}\) up to a scalar multiple. In the terminology used in the Grassmann-Cayley algebra, which considers Plucker coordinate vectors at the symbolic level (that is, without the specification of an order for the coordinates), the vector \(P_{U}\) is often also called a \(k\)-_extensor_ and is denoted by \(u_{1}\vee\dots\lor u_{k}\). The subspace \(U\) is also called the _support_ of \(P_{U}\). We will adopt this notation and terminology which is commonly used in rigidity theory, while keeping in mind that we always assume that the coordinates are given relative to an ordered basis. **Example 1**. : _Consider a line in \(\mathbb{R}^{3}\) given by the points \(a=(a_{1},a_{2},a_{3})\) and \(b=(b_{1},b_{2},b_{3})\). Let \(U\) be the subspace of \(\mathbb{R}^{4}\) spanned by the vectors \(\bar{a}=(a_{1},a_{2},a_{3},1)\) and \(\bar{b}=(b_{1},b_{2},b_{3},1)\). To obtain the Plucker coordinate vector of \(U\) we consider the \(2\times 4\) matrix \(A\) whose first and second row are \(\bar{a}\) and \(\bar{b}\), respectively, and take the determinants of six \(2\times 2\) submatrices of \(A\) by choosing ordered pairs of columns in the following order: \((4,1),(4,2),(4,3),(2,3),(3,1),(1,2)\). This gives_  
\[P_{U}=(b_{1}-a_{1},b_{2}-a_{2},b_{3}-a_{3},a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{ 1}b_{3},a_{1}b_{2}-a_{2}b_{1})^{T}=(d,x\times d),\]  
_where \(d=b-a\), \(x\) is any point on the line, and \(x\times d\) represents the static moment of the line with respect to the origin._  
We may form the _dual space_ of \(U\), denoted by \(U^{*}\), as follows. Consider the linear system given by the following dot products  
\[x\cdot u_{\ell}=0\qquad\ell=1,\dots k,\]  
were the variables are \(x=(x_{1},\dots,x_{d+1})\). The matrix corresponding to this system has rank \(k\) and so we let \(U^{*}\) be the \((d+1-k)\)-dimensional solution space to this system. The _dual Plucker coordinate vector_ of \(U\), \(P_{U^{*}}\), is defined to be the vector that consists of the Plucker coordinates of \(U^{*}\), which are called the _dual Plucker coordinates_ of \(U\).