Let $T$ be an ideal triangle in the Poincare upper half-plane $\mathbb{H}$ with the point $i$ as its "circumcenter" (by which I mean that the point $i$ is the center of symmetry of this triangle and is equidistant from the three vertices under the hyperbolic metric).
There is a natural action of the group $V_T \cong \mathbb{Z}/3\mathbb{Z}$ on this triangle, by counterclockwise rotations by 120 degrees permuting the vertices (given by the restriction of the rotation fractional linear transformation on $\mathbb{H}$ to $T$). This action leaves the point $i$ fixed.
Is there a description of the quotient cone $T/V_T$ and the map $\phi : T \rightarrow T/V_T$? I am interested in explicit coordinates. Presumably this question has something to do with ``orbifolds'', although I really don't know what they are.