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Let $U\subset X$ be an open subset of a connected Riemann surface $X$. Let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $P\in U$ be the unique point such that $z(P) =0$.

Suppose I take another point $Q\in U$. I want to use $z:U\longrightarrow B(0,1)$ to construct a coordinate around $Q$.

Question. Does the following work?

Consider an open set $V$ in $U$ whose image under $z$ is a small open disc around $z(Q)$. Then we define $w: V\longrightarrow B(0,1)$ by $w(x) = z(x) - z(Q).$ Is this is a coordinate around $Q$?

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    The translation is a Möbius transformation, but you can also use a different one to map the entire unit disk holomorphically to itself in such a way that $Z(Q) \mapsto 0$. Your formula for the new coordinate is correct. Googling Möbius transformation or the automorphisms of the unit disk will turn up an explicit formula for this map if you want one.2011-10-01

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