It's well know that one can describe the metric of a surface around a regular enough point by using geodesic polar coordinates.
$ ds^2 = d\rho^2 + G(\rho,\theta )^2 d\theta^2 $
One advantage of writing the metric this way is that it has a nice taylor expansion:
$ G(\rho,\theta) = \rho - \frac{K}{6} \rho^3 + ...$
However, this is clearly not valid for a cone around its tip, since it we have now a singular point. Also, it's not valid for "rugby balls" and similar surfaces with singularities (the first image of this paper https://arxiv.org/pdf/1007.2523v1.pdf). For these cases we have something like $ G(\rho,\theta) = \beta( \rho - \frac{K}{6} \rho^3)+ ...$ Where $\beta$ is a parameter related to the "strength" of the singularity.
My question is: given a surface with an isolated singularity, can I write an geodesic polar coordinates around this point? If yes, what is the expansion of $G(\rho,\theta)$?