About the congruence relation on Poincaré Half-Plane model

Question

I've been studying Hyperbolic Geometry under Hilbert Axiomatization on the Poincaré Half-Plane model. The congruence relation of segments is defined as $AB \equiv CD \Leftrightarrow \exists L \in Lob(\mathbb{H}^2 \cup r_{\infty})$ such that $C=L(A)$ and $D=L(B)$. ($L$ is a Lobachewski transformation that preserves $\mathbb{H}^2$ and $r_{\infty}$).

But this notion is very abstract to me. A Lobachevski transformation can be any composition of euclidean reflections and inversions...

Specially because I need to proof that two hyperbolic lines that are parallel to another hyperbolic line r will have their images by an isometry parallel to the image of r. I know that every isometry is a Lobachevski transformation, but how can I even describe it?

Thanks!

Answer 1

Thm 1: The group of Isometries of $\mathbb{H}^2 \simeq \mathbb{Lob}$.

Thm 2: In the half-plane model, r,s lines are such that $r \| s$ at a point P if, and only if, both $r$ and $s$ share an improper point $A_{\infty}$ where occurs $\angle(r,s)=0$.

Thm 3: Lobachevski transforms preserve angle and incidence.

Let $r,s,t$ lines in the model, where $r$ and $s$ have the ideal point $A_{\infty}$, $r$ and $s$ have the ideal point $B_{\infty}$ and $r \| s$ at a point P. Let $\Psi$ be an isometry of $\mathbb{H}^2$.

By Thm 1, $\Psi$ is a Lobachevski transform. By Thm 3, $\Psi$ preserves angles and incidence. Then, by Thm 2, we have the following:

• $\Psi(P) \in \Psi(s)\cap\Psi(t) $
• $\Psi(A_{\infty}) \in \Psi(r)\cap\Psi(t) $
• $\Psi(B_{\infty}) \in \Psi(r)\cap\Psi(s) $
• $\angle(\Psi(r),\Psi(t)) = 0$
• $\angle(\Psi(r),\Psi(s)) = 0$

Then, again by Thm 2, the result follows.

$\square$