## Part II Projective Theory of Connected Body Frameworks
### Body-Bar Projective Conditions and Centers of Motion  
White and Whiteley [58] present an analysis of the pure conditions for body-bar frameworks in \(\mathbb{P}^{d}\) with techniques of directed graphs analogous to those described in Section 7.6. In that paper, there was a focus on how the conditions relate to infinitesimal motions of bodies when an edge is deleted from a generically isostatic body-bar framework or becomes dependent due to a special projective configuration. Figure 45 illustrates a few simple special (singular) positions for body-bar frameworks in the plane [58]. More generally, White [52, 57] explores a number of applied geometric results which can be well described in the projective Grassmann-Cayley algebra. **Example 27**. : _Figure 45a shows a generic rigid body-bar framework in the plane with two bodies connected by \(3\) independent bars. The row of the rigidity matrix for a bar ab imposes the condition: \(abS_{1}-abS_{2}=ab(S_{1}-S_{2})=ab(S_{12})=0\), where \(S_{1}\) and \(S_{2}\) are the centers of motion for the bodies \(B_{1}\) and \(B_{2}\), and \(S_{12}\) is the relative center of motion of the bodies. Projectively, this says that the relative center must lie on the line ab. This also holds for the bars cd and \(ef\). There is a non-trivial infinitesimal motion if and only if there is a projective point on all three bars (\(b\)). Given such a point, we can choose an arbitrary center \(S_{1}\) and find a center \(S_{2}\) which produces this relative center (\(c\)). If the three bars meet in a point \(S_{12}\), projectively (including all bars being parallel), then there is a non-trivial infinitesimal motion, and vice versa. One vision of this is: holding body \(B_{1}\) fixed, body \(B_{2}\) can rotate about the point \(S_{12}\) which lies on all the bars._  
**Example 28**. : _Consider the plane body-bar framework in Figure 46a. The pure condition for this example has a projective construction as follows: writing \(C(G)=(a\wedge b)\vee(c\wedge d)\vee(e\wedge f)\), we see that \(C(G)=0\) precisely when the three points of intersection determined by the pairs of bars are collinear. In general, when we consider the relative centers of any three bodies in the plane with a non-trivial infinitesimal motion, we obtain:_  
\[S_{12}+S_{23}+S_{31}=[S_{1}-S_{2}]+[S_{2}-S_{3}]+[S_{3}-S_{1}]=0\]  
_which is a simple accordion collapse where reordering the brackets gives middle terms \((-S_{i}+S_{i})=0\). Since \([S_{i}-S_{i}]\) is a projective point (with a weight) for the relative centre of the two bodies, this equation says that the three relative centres must projectively add to \(0\). Projectively, three points can only add to \(0\) if they are collinear._  
_This result also illustrates a more general theorem of Arnold-Kempe [52, 57]:_ If three bodies are in motion in the plane, the relative centers of motion of the three pairs of bodies are  
Figure 45: Two plane bodies with \(3\) bars (**a**). The singular position is where the three bars meet in a point (**b**) which is the relative center of motion. There are many choices for the centers of motion for the bodies, provided that the relative center stays positioned at the projective point of intersection (**c**). (Note here that while the bodies are drawn as circles, they can take any shape and can extend arbitrarily far.) collinear in the projective plane. _Notice that this is a projective statement. Some, or even all, of the relative centers can be on the projective line at infinity corresponding to relative translations. If the bars in Figure 46a are parallel, then this is what happens to the relative centers._  
The body-bar matrices have several simplifying features compared to the bar-joint matrices. The columns of any single body effectively provide a handy tie-down. A further simplifying feature of the body-bar matrices is that an entry \(a_{i}b_{j}\) occurs in just one row (recall Definition 7), and the determinant of a matrix after a tie-down is linear in the Plucker coordinates of each bar.