Definition:Poincaré Plane

From ProofWiki
Jump to navigation Jump to search



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.




Also known as

The Poincaré plane is also called the hyperbolic plane.

However, that name is also given to another concept.

Hence the use of Poincaré plane, in order to reduce ambiguity.


Also see

  • Results about The Poincaré plane can be found here.


Source of Name

This entry was named for Henri Poincaré.


Sources