under translations (this was the reason why we chose $N$ in this way). So by translations from both sides, we obtain $x = y = 0$. The value of $r$ cannot be changed by any rotation that fixes $x = y = 0$, but we still can apply a rotation of the form $\left( 
\begin{smallmatrix}
    c & d & 0 \\
    -d & c & 0 \\
    0 & 0 & 1 
\end{smallmatrix} 
\right)$, with $c^2 + d^2 = 1$, from the left. The effect on $M$ is multiplication by $(c+id)$, and we have no effect on $r$. Projectively, this is the same as leaving $M$ untouched and multiplying $r$ by $(c+id)^{-1}$. Hence we can reach the situation where $r \in {\mathbb{R}}_{>0}$. We notice that $r$ cannot be zero, otherwise we would have $N = 0$. So inversion points have the following normal forms: $$\beta \; = \; (0: \underbrace{1:i:0:i:-1:0:0:0:0}_M:
\underbrace{0:0:0}_{x}: \underbrace{0:0:0}_{y}: r),$$ with $r\in{\mathbb{R}}_{>0}$.

### Butterfly Points. {#compactification:boundary:butterfly}

A boundary point $\beta$ with $M \ne 0$ and $N = 0$ is called a *butterfly point*. The complex dimension of the set of butterfly points is $4$: as before, we can choose $v$ and $w$ on the absolute conic curve, 