  J. Nitsche proved that an isolated singularity of a conformal hyperbolic
metric is either a conical singularity or a cusp one. We prove by developing
map that there exists a complex coordinate $z$ centered at the singularity
where the metric has the expression of either
$\displaystyle{\frac{4\alpha^2\vert z \vert^{2\alpha-2}}{(1-\vert z \vert
^{2\alpha})^2}\vert \mathrm{d} z \vert^2}$ with $\alpha>0$ or
$\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|dz|^{2}}$.
