Solution
= Solution
The <Poincare disc model> has metric
$$
ds^2=\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}.
$$
Rotations about the origin are hyperbolic isometries, so a Euclidean circle centred there is a hyperbolic circle. Its radius is the radial length
$$
\boxed{R=\int_0^r\frac{2,dt}{1-t^2}
=\log\frac{1+r}{1-r}=2\operatorname{artanh}r}.
$$