Poincaré half-plane model

Example

The Poincaré half-plane model is the upper half-plane $\mathbb{H} = \set{(x, y) \in \R^2 : y > 0}$ with the Riemannian metric $$ds^2 = \frac{dx^2 + dy^2}{y^2}.$$ Its geodesics, the lines of the geometry, are the vertical half-lines and the semicircles centred on the $x$-axis. The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$\cosh d = 1 + \frac{(x_2 - x_1)^2 + (y_2 - y_1)^2}{2 y_1 y_2}.$$

Check of the distance on the $y$-axis: the length of the segment from $(0, 1)$ to $(0, e)$ is $\int_1^e dy/y = 1$. The formula gives $\cosh d = 1 + (e - 1)^2/(2e) = (e^2 + 1)/(2e) = \cosh 1$, so $d = 1$ as well.

Check of the hyperbolic parallel axiom: take the line $x = 0$ and the point $(1, 1)$. The vertical line $x = 1$ does not meet it. The semicircle centred at $(2, 0)$ through $(1, 1)$ has radius $\sqrt 2$, so it stays in $x \ge 2 - \sqrt 2 > 0$ and does not meet it either. So $\mathbb{H}$ is a model of hyperbolic geometry.