to the situation with $r = 0$. Eventually, as we have already seen in Subsection \[compactification:boundary:inversion\], we can perform rotations so that we can ensure that $\gamma$ is a real positive number. So we get normal forms of the following kind $$\beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M:
\underbrace{\gamma:i\gamma:0}_{x}: \underbrace{1:i:0}_{y}: 0),$$ with $\gamma \in {\mathbb{R}}_{>0}$.

### Collinearity Points. {#compactification:boundary:collinearity}

For the last group of points $\beta$ in $B$ we have $M = 0$ and either $x = 0$, $y \ne 0$ or $x \ne 0$, $y = 0$. These points are called *collinearity points*. There are two subsets of collinearity points, one with $x = 0$ and one with $y = 0$. Both subsets have complex dimension $2$ (since there is still a free value for $r$ to choose).

By rotations, we can achieve either $x = (1:i:0)^t$ or $y = (1:i:0)^t$. Translations act transitively on $r$, so we get two normal forms, namely $$\beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M: \underbrace{1:i:0}_{x}:
\underbrace{0:0:0}_{y}:0) ,$$ $$\beta = (0: \underbrace{0:0:0:0:0:0:0:0:0}_M: \underbrace{0:0:0}_{x}:
\underbrace{1:i:0}_{y}: 0).$$

We conclude the section about the boundary of $X$ by showing that we can associate to each inversion, butterfly and similarity point a pair $(L,R)$ of elements of $S^2$, namely oriented directions in ${\mathbb{R}}^3$. This 