Week 5

1.  Theorem (formula) for 4 disks in general configuration.

2.  Special case: Descartes' formula, and its extended version, both for tangent disks.

Define data matrix  $\mathbf D=[C_1|C_2|C_3|C_4]$  where $C_i = \begin{bmatrix}\dot x_i\\ \dot y_i\\ \beta_i\\ \gamma_i\end{bmatrix}$ 

Define the 4x4 configuration matrix $\mathbf F$ as a Gramian of inner products of pairs of disks: $$F_{ij} = \langle C_i , C_j \rangle$$  The general (matrix) formula is $$\boxed {\quad D\, f D^T \ = \ g \phantom{\Big|^2}\ }$$ where $f = F^{-1}$ (needs to be calculated from $F$ for a particular configuration), and $g=G^{-1}$, i.e., $$g = \begin{bmatrix} -1&\ 0&0&0\\ \ 0 & -1 &0&0\\ \ 0& \ 0&0 & 2 \\  \ 0&\ 0&2&0 \end{bmatrix}$$  Teaser: Draw four mutually perpendicular circles.