piece of information will play an important role in the main results of Section \[bond\_geometric\].

We recall that in the case of inversion and butterfly points the matrix $M$ is of rank $1$, since it is non zero and has rank $\leq 1$, as implied by the boundary condition $h = 0$. Hence $M$ is of the form $v w^t$ for two non zero vectors whose coordinates satisfy $$v_1^2 + v_2^2 + v_3^2 \; = \; 0 \quad \mathrm{and} \quad w_1^2 + w_2^2 +
w_3^2 \; = \; 0$$ Then we can think of $v$ and $w$ as points of the conic $C = \big\{ \alpha^2 +
\beta^2 + \gamma^2 = 0 \big\}$ in ${\mathbb{P}}^2_{{\mathbb{C}}}$. Notice that, although $v$ and $w$ are not unique, they always give the same pair of points on the conic. We would like to think of $v$ and $w$ as directions in ${\mathbb{R}}^3$, and in order to do this we provide an identification between $C$ and $S^2$. This identification is accomplished in two steps, namely first we consider an isomorphism $C \cong
{\mathbb{P}}^1_{{\mathbb{C}}}$ and then we take the bijection between ${\mathbb{P}}^1_{{\mathbb{C}}}$ and $S^2$ given by the stereographic projection[^3].\
The isomorphism $C \cong {\mathbb{P}}^1_{{\mathbb{C}}}$ is given 