Definition:Poincaré Plane

Definition
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.

Also known as
The Poincaré plane is also called the hyperbolic plane.