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The connection between geodesics on the modular surface$\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb H}$ and regular continuedfractions, established by Series, is extended to a connection between geodesicson $\Gamma\backslash {\mathbb H}$ and odd and grotesque continued fractions,where $\Gamma\cong {\Bbb Z}_3 \ast {\Bbb Z}_3$ is the index two subgroup of$\operatorname{PSL}(2,{\mathbb Z})$ generated by the order three elements$\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix} \right)$ and$\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix} \right)$, havingan ideal quadrilateral as fundamental domain. A similar connection betweengeodesics on $\Theta\backslash {\mathbb H}$ and even continued fractions isdiscussed in our framework, where $\Theta$ denotes the Theta subgroup of$\operatorname{PSL}(2,{\mathbb Z})$ generated by $\left( \begin{smallmatrix} 0& -1 \\ 1 & 0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 2\\ 0 & 1 \end{smallmatrix} \right)$.

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions