Möbius transformation that preserve distance for two pair of given points in $\mathbb{H}$.

Question

I need to prove that for a given two pair of points $(z_1,z_2)$ and $(w_1,w_2)$ in $\mathbb{H}$ (Poincaré's upper half plane), where $d_{\mathbb{H}}(z_1,z_2)=d_{\mathbb{H}}(w_1,w_2)$, there is an Möbius transformation $m \in \text{Möb}(\mathbb{H})$ (Möbius transformations that preserve $\mathbb{H}$) so that $m(z_1)=w_1$ and $m(z_2)=w_2$.

Hope that anyone can help me. :)

Sincerely Henrik B.

Answer 1

Here is a relatively painless way:

• Consider two maps from $\mathbb H$ to the unit disk: one sends $z_1$ to the center, the other does the same with $w_1$. $$f(z)=\frac{z-z_1}{z-\bar z_1},\qquad g(z)=\frac{z-w_1}{z-\bar w_1}$$
• From the equality of distances, obtain $|f(z_2)|=|g(w_2)|$.
• Let $\phi$ be a rotation of the disk that sends $f(z_2)$ to $g(w_2)$.
• The composition $g^{-1}\circ\phi\circ f$ is the map you want.