## Part II Projective Theory of Connected Body Frameworks
### Projective Line Dependences and the Stewart Platform  
Consider two bodies in \(d\)-space joined by \(\binom{d+1}{2}\) bars. This is infinitesimally rigid if and only if these lines are independent in projective space. The Stewart Platform (Figure 48) illustrates how the line geometry in \(\mathbb{P}^{3}\) appears in the analysis of relative motions of two bodies [6; 131; 132]. In particular, the Stewart Platform with a zigzag pattern of shared vertices [133], as shown in Figure 48b, has independent connecting bars and is hence generically isostatic. This is also called a hexapod. A classic textbook [134], written for North American structural engineering students, chose to simplify the communication of dependencies for 6 bars connecting a body to the ground by only illustrating the singular configuration where all bars meet a single line (Figure 48c--with line \(ab\)). This choice was because the more careful complete communication would have required more knowledge of projective geometry, which the author knew but the engineering students did not. This lack of knowledge of projective geometry may still hold true, at least in North America. This singular configuration with all braces meeting a single line was at the heart of the Tay River Bridge disaster, also illustrated in [102]. Consider the \(6\times 6\) matrix formed by the coordinates of 6 lines joining two bodies in \(\mathbb{P}^{3}\):  
\[\begin{array}{c}41\quad 42\quad 43\quad 23\quad 31\quad 12\\ a_{1}\lor b_{1}\quad\left( \begin{array}{cccccc}a_{1}b_{1}\\ a_{1}b_{1}\end{array} \right)_{41}\quad(a_{1}b_{1})_{42}\quad(a_{1}b_{1  
fixed. In the language of the Theory of Screws [6; 59; 131], the bars are _null lines of the screw_. The null lines of a screw form a line complex which includes a pencil of lines through every point in space. If the screw solution is itself a line, then all the bars of the line complex intersect this line, forming a _singular line complex_, with the pencil of lines formed by joining some point to this line. More generally, some configurations of lines are linearly dependent and therefore will support stresses. Others are independent in the sense that no linear combination will be dependent. The sums of such lines will create screws, but only some independent lines will generate additional lines as sums of the generators, rather than screws. We offer a brief illustration of the larger theory, nicely presented in [131]. We offer a short summary when independent lines generate other lines as linear combinations. 6 independent lines will generate all lines, and all 2-extensors (screws) as linear combinations. When working with examples of frameworks, it is valuable to be able to detect the projective geometric dependence of bars. This is well presented in [131; 132]. We will summarize some key observations about how dependencies of lines appear in 3-space. We begin with two lines and work up to six lines (Figure 49). **Two independent lines. **  
_If two lines in 3-space are skew (not intersecting) then no other line will appear among the linear combinations. If two lines intersect, then all lines through this intersection in their plane will appear as linear combinations forming a projective pencil. This is projective, so if the lines are parallel meeting at infinity, then other coplanar and parallel lines lie in the same projective pencil._  
**Three independent lines. **  
_If three lines are mutually skew, then they will lie on a ruling of a quadratic surface. Linear combinations will be other lines on the regulus of this quadratic surface. Four lines on this regulus will be dependent. This is something engineers have been trained to watch for, at least visually._  
_If three lines lie together in a plane, but are not mutually concurrent, then all other lines in the plane will be a linear combination of these three lines, As a dual, if three lines are concurrent, but not coplanar, they are independent and a fourth line through this point is dependent (Figure 49,c).