By a conformal structure on a polygon, I mean all the vertices of a polygon are missing (or, equivalently, it has ideal vertices), so that it can be considered as a conformal structure on the unit disk which has on the boundary unit circle marked points corresponding to the vertices of the polygon(This is just a simple application of the uniformization theorem). And there's yet another model for this that often appears in literature;
The above is a model for a 7-gon of which I took a snapshot from Baldwin. Here each half-infinite band corresponds to a neighborhood of the vertices of the 7-gon, and there exist segments perpendicular to the "edges of the polygon" which is shown as the curves bounding the polygon and also as the height of rectangular regions in the diagram above. Then the conformal structures on the given 7-gon is parametrized by $I_1$ to $I_4$, which are the height of rectangular regions. Of course, we assume that there exist perpendiculars for every pair of edges of the polygon, and you have freedom to choose which 4 disjoint perpendiculars (($n-3$) such perpendiculars in $n$-gon in general) to parametrize the conformal structures.
The claim that I want to verify is that this indeed parametrizes all the conformal structures on a given polygon, and thus that the dimension of the moduli space of conformal structures of $n$-gon is precisely $n-3$ as expected. It is not even obvious to me that this parametrization is surjective though. Any idea or suggestion of references is appreciated.
Edit. After the discussion with Moishe Cohen, it seems not true in general that this parametrization is surjective. And it seems this construction is somehow related to the Fenchel-Nielsen coordinates(though I have no proper knowledge on F-N in case of surfaces with boundary), so I would appreciate a suggestion of reference of F-N coordinates relevant to this model(i.e., disk with marked points on boundary).