# Proof: Every H-Projection is product of H-Reflections

I have some problems with the proof of the statement above (sorry for a bad translation I guess).

First of all some definitions:

Definition 1 (möbius-transformation):

Let $A \in \mathbb{C}^{2 \times 2}$ with $det A=ad-bc \neq 0$ then

1. for $c \neq 0$: $$\varphi_A: \begin{cases} \mathbb{C}_{\backslash{\lbrace -d/c \rbrace}} &\to \mathbb C \\ z &\to \frac{az+b}{cz+d}=\frac 1 c \cdot (a-\frac{ad-bc}{cz+d})\end{cases}$$
2. for $c=0$: $$\varphi_A: \begin{cases} \mathbb{C} &\to \mathbb C \\ z &\to \frac{az+b}{cz+d}=\frac a d \cdot z + \frac b d\end{cases}$$ is called möbius-tranformation.

Definition 2 (H-Projection):

Let $\mathcal{B}$ the set of all H-Projections with $\beta: \mathbb H \to \mathbb H$ and

1. (typ 1): $\beta:=\frac{az+b}{cz+d}$ with $a,b,c,d \in \mathbb R$ and $\left|\begin{array} aa & b \\ c & d \end{array}\right|>0$
2. (typ 2): $\beta:=\frac{a\overline z+b}{c\overline z+d}$ with $a,b,c,d \in \mathbb R$ and $\left|\begin{array} aa & b \\ c & d \end{array}\right|<0$

Definition 3 (H-reflection):

There are two kinds of H-reflections:

1. (typ 1): Let $g$ with $Re(z)=a$ be a H-Line. Then we have with $\sigma_g(z)=-\overline z+2a$ a H-transformation.
2. (typ 2): Let $g$ with $|z-a|^2=r^2$ be a H-Line. Then we have with $\sigma_g(z)=\frac{a \overline z+(r^2-a^2)}{\overline z -a}=\frac{r^2}{\overline z -a}+a$ a H-transformation.

Theorem 1: Every H-Projection is product of H-Reflections

Proof: Given $\beta \in \mathcal B$ with $\beta=\frac{az+b}{cz+d}$ ($a,b,c,d \in \mathbb R$ and $ad-bc>0$.

1. If $c \neq 0$, we use $$z \to \frac{ad-bc}{c^2} \cdot \frac{1}{\overline z-(-d/c)}-\frac d c \to -\left( \frac{ad-bc}{c^2}\cdot \frac{1}{z+d/c}-\frac d c\right)+\frac{a-d}{c}=\beta(z)$$ In every step was used a H-reflection, so $\beta$ is the product of two reflections.

2. If $c=0$ we can use $\beta(z)=az+b$ ($a>0$). $\beta$ is the result of $$z \to \frac{1}{\overline z} \to az \to -a \overline z \to az+b$$ and because of that a product of four reflections.

Now my question: How do we get the two/four reflection equations? I tried to use Definition 3 and put $\beta$ into it, but it didn't gave me the expectet result. Which two reflections do I have to combine?

Sorry for the long questions and thanks a lot for any hints.

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