Definition:Poincaré Plane

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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.


Source of Name

This entry was named for Henri Poincaré.


Sources