Geodesics of the Poincare disc
= Geodesics of the Poincare disc
{c}
The geodesics are diameters and arcs of circles orthogonal to the unit circle. A radial segment is length minimizing because
$$
\int\frac{2\sqrt{\dot r^{,2}+r^2\dot\theta^{,2}}}{1-r^2}\,dt
\geq2\operatorname{artanh}r,
$$
with equality for constant angle and monotone radius. Hence the distance from the origin to $z$ is $2\operatorname{artanh}|z|$.