= Solution
The radial segment is a <geodesic> of the <Poincare disc model>. Its <hyperbolic length> is
$$
\boxed{d_D(0,re^{i\theta})=\int_0^r\frac{2\,dt}{1-t^2}=\log\frac{1+r}{1-r}=2\operatorname{artanh}r.}
$$
One can also verify that it minimizes <length of a curve>: along any piecewise smooth curve from zero to radius $r$, the Euclidean speed is at least the absolute radial speed. Integrating $2|dr|/(1-r^2)$ is at least the absolute change in $\log((1+r)/(1-r))$, with equality on the radial segment. Thus the answer is independent of $\theta$.
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