Beltrami–Klein model

Example

The Beltrami–Klein model of the hyperbolic plane is the open unit disc $D = \set{x \in \R^2 : \abs{x} < 1}$ with the following distance. For distinct points $p, q \in D$, the line through $p$ and $q$ meets the unit circle in two points $a$ and $b$. Label them so that the order along the line is $a, p, q, b$. Then $$d(p, q) = \frac{1}{2} \ln \frac{\abs{aq} \, \abs{pb}}{\abs{ap} \, \abs{qb}},$$ where $\abs{xy}$ is the Euclidean distance, and $d(p, p) = 0$. This is the Cayley–Klein metric whose absolute is the unit circle, and it makes $D$ a metric space. The lines of the model are the open chords of the disc.

Check on a diameter: take $p = (0, 0)$ and $q = (r, 0)$ with $0 < r < 1$. Then $a = (-1, 0)$ and $b = (1, 0)$, so $\abs{aq} = 1 + r$, $\abs{pb} = 1$, $\abs{ap} = 1$ and $\abs{qb} = 1 - r$. The formula gives $d(p, q) = \frac{1}{2} \ln \frac{1 + r}{1 - r} = \operatorname{artanh} r$, which tends to $\infty$ as $r \to 1$. So the boundary circle lies at infinite distance, and the disc is a whole infinite plane.