Poincaré Plane is Abstract Geometry

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Theorem

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


Proof

We will show that the axioms for an abstract geometry hold.


Axiom 1

Let $P = \left({x_1, y_1}\right)$ and $Q = \left({x_2, y_2}\right)$ be two points in $\left({\mathbb{H}, L_H}\right)$.

If $x_1 = x_2 = a$ then $P,Q \in {}_a L$.

If $x_1 \ne x_2$ then let:

$c = \dfrac {y_2^2 - y_1^2 + x_2^2 - x_1^2} {2 \left({x_2 - x_1}\right)}$
$r = \sqrt{\left({x_1^2 - c}\right) - y_1^2}$


By definition then $P, Q \in {}_c L_r$.

So any two points in $\mathbb{H}$ lie on a line in $L_H$.

$\blacksquare$


Axiom 2

For any $a \in \R$:

$\left({a, 1}\right), \left({a, 2}\right) \in {}_a L$


Also for any $c \in \R$ and $r \in \R_{>0}$:

$\displaystyle \left({c + \frac{1}{2} r, \frac{\sqrt{3} }{2} r}\right),\left({c - \frac{1}{2} r, \frac{\sqrt{3} }{2} r}\right) \in {}_c L_r$


So every line in $L_H$ has at least two points.

$\Box$


Hence $\left({\mathbb{H}, L_H}\right)$ is an abstract geometry.

$\blacksquare$


Sources