Take the 2-minute tour ×  I am currently trying to understand
the paper by Krioukov et. al. on hyperbolic networks, but since
I do not have a background in hyperbolic geometry (or, in that
sense, in geometry at all) I struggle to understand some points
they are making.  For instance, they aim to compute the distance
$x$ between two points $(r, \theta)$ and $(r', \theta')$ in the
hyperbolic plane by using the hyperbolic law of cosines:
\begin{equation} \mathsf{cosh}(kx) = \mathsf{cosh}(kr)
\mathsf{cosh}(kr') -
\mathsf{sinh}(kr)\mathsf{sinh}(kr')\mathsf{cos}(\Delta\theta)
\end{equation}  Here, $K=-k^2<0$ refers to the curvature, and
$\Delta \theta = \pi - |\pi - |\theta - \theta'||$. I can
intuitively follow this equation since it is similar to the
regular law of cosines in the euclidean plane. However, in the
next step they approximate it by assuming that $kr, kr'$ is
large, and $\Delta \theta > 2 \sqrt{e^{-2kr}+e^{-2kr'}}$ as
follows: \begin{equation} x = r + r' + \frac{2}{k} \ln \sin
\frac{\Delta\theta}{2}