## Part III Maximal Abstract Rigidity Matroids and Multivariate Splines  
Over the last 35 years, there has been a growing recognition of the strong similarity in combinatorics, geometric techniques, and results in two distinct fields, each projectively invariant:  
1. the projective and combinatorial theory of frameworks, both bar-joint and panel-hinge in \(\mathbb{P}^{3}\); and
2. bivariate \(C^{1}_{2}\) splines for a polygonal decomposition \(\Delta\) of a disc in the plane, written \(S^{1}_{2}(\Delta)\) in approximation theory [23]. This focuses on finding a piecewise degree 2 surface (\(C_{2}\)) for each polygonal cell so that they fit together over the edges with globally continuous first derivatives (\(C^{1}\)) across the whole surface. This space of splines is sometimes studied as the row dependencies (cofactors) of a rigidity type cofactor matrix based on the edges and vertices of the cell decomposition [22; 24]. Over the years this similarity became a lens for a deeper analogy through which tools, conjectures and results were transferred between the fields and were written up both in publications and in circulating preprints [21; 24]. For example, vertex splitting was first derived as a technique for \(C^{1}_{2}\)-cofactors while proving the generic version of Cauchy's Theorem for bivariate \(C^{1}_{2}\) splines [121]. The technique was then transferred to rigidity theory [120] as a now standard basic inductive technique. A very recent result [149] presents a direct algebraic transfer between \(C^{d-2}_{d-1}\) splines on a conic in the plane and rigidity in \(\mathbb{R}^{d}\) along the moment curve \((n^{d},n^{d-1},\ldots,n)\). \begin{table}
\begin{tabular}{l l l l l} \hline \hline \multicolumn{4}{c}{**Table of Geometric Structures and Distance Constraints: Euclidean, Minkowski**} \\ \hline
**Dim.** & **Geometric Objects** & **Necessary Counts** & **Suff.** & **Sect.** \\ \hline \(d=1\) & point line (bar-joint) & \(|E|\leq|V^{\prime}|-1\) & Yes (\(=\)) & \\ \(d=1\) & line pin-rod & \(|P^{\prime}|\leq|F^{\prime}|-1\) & Yes (\(=\)) & \\ \hline \(d=2\) & point edge (bar-joint) & \(|E^{\prime}|\leq 2|V^{\prime}|-3\) & Yes (\(=\)) & [13; 14] \\ \(d=2\) & bar face (body-bar) & \(2|P|\leq 3|B|-3\) & Yes & Section 9.1 [58] \\ \(d=2\) & point face (body-pin) & \(2|P|\leq 3|B|-3\) & Yes & Section 10 \\ \(d=2\) & pinned rods & \(2|I^{\prime}|\leq 2|V^{\prime}|+3|F^{\prime}|-3\) & Yes & Section 10.7 [145] \\ \hline \(d=3\) & bar–joint & \(|E|\leq 3|V|-6\) & No & [34] \\ \(d=3\) & body-bar & \(|E|\leq 6|B|-6\) & Yes (\(=\)) & Section 9.1 [58] \\ \(d=3\) & rod and bar & \(|E|\leq 5|R|-6\) & Yes (\(=\)) & Section 9.6 [135] \\ \(d=3\) & body–hinge & \(5|H|\leq 6|B|-6\) & Yes & Section 10 [58] \\ \(d=3\) & flat-body hinge & \(5|H|\leq 6|B|-6\) & Yes & Section 10.4 [50] \\ \(d=3\) & molecular body hinge & \(5|H|\leq 6|B|-6\) & Yes & Section 10.4 [50] \\ \(d=3\) & body-pin & \(3|P|\leq 6|B|-6\) & No & \\ \(d=3\) & clamped rods & \(3|P|\leq 5|R|-6\) & No & Section 9.6 \\ \(d=3\) & edge-face (sheetworks) & \(|E|\leq 3|F|-6\) & No & Section 8.2 [114] \\ \(d=3\) & point-face (sheetworks) & \(|I|\leq 3|V|+3|F|-6\) & No & Section 8.2 [114] \\ \hline \(d>3\) & bar–joint & \(|E|\leq d|V|-(d_{+1}^{+2})\) & No & [34] \\ \(d\) & body-bar & \(|E|\leq(d_{+2}^{+2})(|B|-1)\) & Yes (\(=\)) & Section 9.2 [58] \\ \(d\) & body–hinge & \([(d_{+2}^{+2})-1]|H|\leq(d_{+1}^{+2})(|B|-1)\) & Yes & Section 10 [58] \\ \(d\) & flat-body hinge & \([(d_{+2}^{+2})-1]|H|\leq(d_{+1}^{+2})(|B|-1)\) & Yes & Section 10.4 [50] \\ \(d\) & rod and bar & \(|E|\leq(d_{+1}^{+2})-1]|R|-(d_{+1}^{+2})\) & Yes & Section 9.6 [135] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structures with distance constraints, and necessary counting conditions. Pins and clamps mean shared vertices of larger bodies. We will use the notation \(C^{r}_{d}\)-cofactors throughout this section, except when we are directly relating to the broader approximation theory literature over cell decompositions, where \(S^{r}_{d}(\Delta)\) will be used.