Definition:Poincaré Plane
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It has been suggested that this page be renamed. In particular: Definition:Poincaré Half-Plane? Is that what it's called in your own references? Not really, for example, in Einsiedler/Ward, $\H$ is called a half-plane, but at the same time, $\H$ is an example of a hyperbolic plane. I have no reference for Poincaré (half-)plane. However, a quick search in the internet suggest this should be called the Poincaré half-plane. Until we get a reference calling it the half plane from a $\mathsf{Pr} \infty \mathsf{fWiki}$ approved source, it stays as it is, perhaps with an addition to Also known as. To discuss this page in more detail, feel free to use the talk page. |
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.
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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
- Poincaré Plane is Abstract Geometry, in which the Poincaré plane shown to be an abstract geometry.
- Results about The Poincaré plane can be found here.
Source of Name
This entry was named for Henri Poincaré.
Sources
- 1991: Richard S. Millman and George D. Parker: Geometry: A Metric Approach with Models (2nd ed.) ... (previous) ... (next): $\S 2.1$