## Part II Projective Theory of Connected Body Frameworks
### Coning Body-Bar Frameworks
_the graph satisfies_ \(|E|=({d+1\choose 2}-1)|R|-({d+1\choose 2})\) _and for all subgraphs with at least two rods we have_ \(|E^{\prime}|\leq({d+1\choose 2}-1)|R^{\prime}|-({d+1\choose 2})\)_;_
3. _the graph has a_ \(({d+1\choose 2})\text{Tree}({d+1\choose 2}-1)\) _partition into_ \(({d+1\choose 2})\) _disjoint trees,_ \(({d+1\choose 2})-1\) _at each rod._  
Figure 54: Two rods in space can be joined with three bars. Even formed as a clamp (**a**), this structure is flexible. With four bars (**b**) it is generically rigid: \(|E|=4=(2\times 5-6)=5|R|-6\). Three rods can be joined in several patterns with 9 bars (**c**). Tanigawa proved an extended theorem with bodies, rods and bars in 3-space [135]. The variant of body-hinge frameworks (see sections below) where hinges can connect more than two bodies was also analysed and proven by Tay and by Tanigawa. When a bar (or other rod) meets a rod at infinity, they form what Tay called a'slip joint' [125,129]. In these papers, this is modeled as a slider along the bar. These slip joints are also found in physical models in mechanical engineering. A special case of interest is where pairs of rods are _clamped_ together at shared points. For this special case of _rod and clamp frameworks_ we have necessary counts for independence and infinitesimal rigidity. However, a full combinatorial theory has not yet been developed for these structures. A rod and clamp framework in 3-space consists of 1-dimensional rods (collinear bodies), and selected pairs of rods are clamped together at common points. This may be modelled by placing three bars between each selected pair of rods so that they go through a common point. In such a framework each rod has 5 degrees of freedom and each incidence of two rods (pinned or shared vertex) removes 3 common degrees of freedom (relative translations), so we conjecture that the constraint count for independence is \(3|I^{\prime}|\leq 5|R^{\prime}|-6\) (where \(|R^{\prime}|\) is the number of rods and \(|I^{\prime}|\) is the number of incidences of pairs of rods). More generally, if we allow more than two rods to be incident with a point, then this can be modeled using incidence structures \(S=(V,R;I)\), where \(I\subseteq V\times R\), and the constraint count becomes \(3|I^{\prime}|\leq 3|V^{\prime}|+5|R^{\prime}|-6\). In the following we consider some basic examples. **Example 35**. : _Consider the example in Figure 56. In (b), we have 6 rods and 8 clamps, each on two rods. The counts are \(|R|=6\) and \(3|I|=24\). This gives \(3|I|=24=30-6=5|R|-6\). Experimentally, this is infinitesimally rigid. If we remove one rod (c) we have \(|L|=5\) and \(3|I|=18\) and hence \(3|I|=18<25-6=5|R|-6\) so the structure must be infinitesimally flexible. Adding back the last rod with two clumps is adding 5 degrees of freedom and 6 constraints. The lines of the rods can still be generic. However, if we ask for one further intersection (a), then the 6 lines must be rulings of a quadric surface by classic projective geometry. It is known that this configuration is finitely flexible. (This is actually a question from an old undergraduate Tripos exam from Cambridge in the 1870s. )_  
Figure 55: Three rods in \(\mathbb{P}^{3}\) are generically rigid with three triples of connecting bars (a), and still infinitesimally rigid with the triples each concurrent (b), even with the rods crossing effectively as clamps (c). As an extension of this example, every \(K_{m,n}\) with \(m,n\geq 3\) will be flexible, even though these will appear to be over-counted by increasing gaps between \(3|I|\) and \(5|R|-6\). In addition, if we have \(K_{4,4}\) minus one edge, then a classical theorem of projective geometry called the 16 th point theorem guarantees that the 16 th intersection must occur. The algebraic structure of these rod and clamp frameworks is not quite the same as for the previous types of frameworks. In \(3\)-dimensions, we can make generic choices for the two points \(a,b\) defining the line of a rod.