Hyperbolic distance in the Poincare disc
= Hyperbolic distance in the Poincare disc
{c}
For $z_1,z_2$ in the <Poincare disc model>, put
$$
\rho=\left|\frac{z_2-z_1}{1-\overline{z_1}z_2}\right|.
$$
Their <hyperbolic distance> is
$$
d(z_1,z_2)=2\operatorname{artanh}\rho
=\log\frac{1+\rho}{1-\rho}.
$$