\begin{definition}[Definition:Poincaré Plane]
Let:
:$\H = \set {\tuple {x, y} \in \R^2: y > 0}$
Let $a \in \R$ be a real number.
Let:
:${}_a L := \set {\tuple {x, y} \in \H: x = a}$
Define:
:${}_A L := \set{ {}_a L: a \in \R}$
Let $c \in \R$ be a real number and $r \in \R_{>0}$ be a strictly positive real number.
Let:
:${}_c L_r := \set {\tuple {x, y} \in \H: \paren {x - c}^2 + y^2 = r^2}$
Define:
:${}_C L_R := \set { {}_c L_r: c \in \R \land r \in \R_{>0} }$
Finally let:
:$L_H = {}_A L \cup {}_C L_R$
The abstract geometry $\struct {\H, L_H}$ is called the '''Poincaré plane'''.
This is shown to be an abstract geometry in Poincaré Plane is Abstract Geometry.
\end{definition}