Definition:Poincaré Plane

Definition
Let:
 * $\mathbb H = \left\{{\left({x, y}\right) \in \R^2: y > 0}\right\}$

Let $a \in \R$ be a real number.

Let:
 * ${}_a L := \left\{{\left({x, y}\right) \in \mathbb H: x = a}\right\}$

Define:
 * ${}_A L := \left\{{ {}_a L: a \in \R}\right\}$

Let $c \in \R$ be a real number and $r \in \R_{>0}$ be a strictly positive real number.

Let:
 * ${}_c L_r := \left\{{\left({x, y}\right) \in \mathbb H: \left({x - c}\right)^2 + y^2 = r^2}\right\}$

Define:
 * ${}_C L_R := \left\{{ {}_c L_r: c \in \R \land r \in \R_{>0}}\right\}$

Finally let:


 * $L_H = {}_A L \cup {}_C L_R$

The abstract geometry $\left({\mathbb H, L_H}\right)$ 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.