It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum\_i^r\frac{a\_i-1}{a\_i})$, where $r$ is the number of cones with stabilizer of orders $a\_1,...,a\_r$ respectively. 


Now, I don't know if everyone would like to know what is the corresponding relation when $B$ has *reflector intervals* or *reflector circles*, as I'd rather... so I dare to question: 

 
Is there a generalization in this direction? 



Branched coverings over orbifolds with reflector lines
A detailed write up of what you are looking for is also available in Chapter 13 of [Thurston's notes](http://library.msri.org/nonmsri/gt3m/). Specifically, formula 13.3.4 on page 331, has the desired formula
$$\chi(O) = \chi(X\_O) -\frac{1}{2} \sum\_{i=1} ^N (1-\frac{1}{n\_i}) -\sum\_{j=1}^M(1-\frac{1}{m\_j}),$$ 
where $O$ is the orbifold, $X\_O$ is it's underlying space, there are N points fixed locally by dihedral groups of orders $2n\_1,...,2n\_N$ and $M$ points fixed locally by rotations of orders $m\_1,...,m\_M$. 


However, I believe the question addressed (at least in part) by Hurwitz in the proof of [Hurwitz's theorem](http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_on_automorphisms), since the bound the order of an automorphism group H acting on surface of genus g is 168(g − 1) if H includes orientation reversing symmetries and 84(g-1) in the orientable case. The key observation is that the Gauss-Bonnet extends to orbifolds as well. 


Computing the Euler characteristic for the quotient of $\mathbb{H}^2$ by the (full) [(2,3,7) triangle group](http://en.wikipedia.org/wiki/%282,3,7%29_triangle_group) is a good way to see this in action. 


