Finding the elliptic function that maps a triangle to the unit circle

Question

This is a task that my professor give me! I am thinking about finding the function by using the Christoffel-Weierstrass transformation. But he said that it's completely correct. We need to use the elliptic function constructed by Eisenstein and Kronecker with periods! I really do not know how to find its periods here!

I have difficulty with the problem follows:

Find the elliptic function $\zeta(z)$ which maps the triangle $ABC$ (illustrated in the figure) to the circle, such that $$\begin{cases} \zeta(0)=1^+,\\ \zeta(b)=1^-,\\ \zeta(z_0)=0. \end{cases}$$ Finding the inverse function $z(\zeta)$ of that $\zeta(z)$.

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