## 11 Multivariate Splines and Cofactor Matroids
### Smoothing Cofactors for Splines and Compatibility Conditions  
We first present the basic cofactors of bivariate splines following a pattern which strongly matches with the approach above for motion assignments for body-hinge frameworks. This connection informed some methods used in [23,150]. We initially consider the faces and edges of a planar graph realised in the plane, without crossings, and ask about the space of all surfaces which are piecewise quadratic over each face, and when two faces share an edge, they meet with continuous 1 st derivatives (common tangent planes) forming the \(S^{1}_{2}\)-_bivariate splines_. The _compatibility condition_ below describes algebraically when the two faces meet over the line \(p_{i}p_{j}\) with a continuous 1 st derivative at the shared line. It offers a basic equation central to our analysis. **Lemma 2** (Chui and Wang [150]). : _Two bivariate quadratic polynomials \(S^{j}\) and \(S^{k}\) meet with continuous 1 st derivatives over the line \(A^{jk}x+B^{jk}y+C^{jk}=0\) if and only if, for some scalar \(\beta^{jk}\) we have_  
\[S^{k}-S^{j}=\beta^{jk}(A^{jk}x+B^{jk}y+C^{jk})^{2}.\]  
With projective coordinates for the vertices \(p_{i},p_{j}\) and the variable affine point \(X=(x,y,1)\), the equation of the line from \(p_{i}\) to \(p_{j}\) is written \([p_{i}p_{j}X]\). The \(\beta^{ij}\) are termed _smoothing cofactors_. For simplicity, we assume \(j<k\), giving an orientation to each edge, and reversing the orientation, the equation \(S^{j}-S^{k}=-\beta^{kj}[p_{i}p_{j}X]^{2}\) implies that \(\beta^{kj}=-\beta^{jk}\). As before, when we have an oriented cycle of faces and edges \(C\), the compatibilty equation is \(\sum_{(ij)\in C}\beta^{jk}[p_{i}p_{j}X]^{2}\equiv 0\). This is called the _conformality condition_ when applied to a face-edge cycle around a vertex of an oriented manifold (Figure 65). This is a polynomial identity, with 6 different powers \(x^{i}y^{j}\)\(i,j\leq 2\): \(x^{2},xy,y^{2},x,y,1\), which must hold identically for these 6 powers, analogous to the 6 coordinates of the 2-extensors in \(\mathbb{P}^{3}\). We can ask about which lines are dependent or independent in the plane, with these \(C^{1}_{2}\)-cofactor conditions. **Example 40**. : _Consider Figure 65. Any three lines through a point are independent but any four lines through a point are dependent (a), with the line joining ab as the dependent \(4 th\) line at each of a and b, as we will confirm in the next section. A generic set of \(4\), \(5\), or \(6\) lines in the plane is \(C^{1}_{2}\)-cofactor independent, including a generic cycle of the form (b). See also the figures and examples in Section 11.4 for generic triangulated spheres [24; 121]. However a set with three lines meeting in each of the two points \(a,b\) is \(C^{1}_{2}\)-cofactor dependent, as the line joining the two vertices is a linear combination of each of the triples (a)._  
_If the 6 lines form the zig-zag of an octahedron (b,c) the projective condition for dependence has been identified in multiple analyses [24; 151]. As illustrated in (c) the projective condition is the plane concurrence of the lines \(a_{1}b_{1},a_{2}b_{2},a_{3}b_{3}\). We do not know what other projective conditions will make 4, 5 or 6 lines \(C^{1}_{2}\)-cofactor dependent. This is a problem for future work. It would be possible to take the determinant of the \(6\times 6\) matrix below, which must be projectively invariant, and seek the relevant projective condition._  
An extended approach for splines which is highlighted in approximation theory is the investigation of row dependencies which are polynomials of bounded degree. This moves to algebraic geometry and homology theory. There are some key results in recent papers [115; 152]. More generally, this cofactor condition extends to higher bivariate splines with piecewise degree \(d\) polynomials with continuous \(r\)th derivatives from the space of \(S^{r}_{d}\)-splines with smoothing cofactors which are polynomials of fixed degree.