Poincaré disk model

Description

The Poincaré disk model of the hyperbolic plane is the unit disc $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ with the Riemannian metric $$ds = \frac{2\, \abs{dz}}{1 - \abs{z}^2}.$$ The distance between two points is the infimum of the lengths of piecewise smooth curves joining them, which makes $\mathbb{D}$ a metric space. The distance from $0$ to $z$ is $$d(0, z) = \int_0^{\abs{z}} \frac{2\, dt}{1 - t^2} = \log\frac{1 + \abs{z}}{1 - \abs{z}},$$ attained along the radius, so points near the boundary circle are infinitely far from $0$. In general, $d(z, w) = 2 \operatorname{artanh} \abs{\dfrac{z - w}{1 - \bar w z}}$.