The Weierstrass-Enneper Representation using hodographic coordinates on a minimal surface

In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass-Enneper representation. From this we derive the hodographic coordinate $\rho \in D\subset \text{CC}$ and $\sigma$ its complex conjugate which enables us to write the Weierstrass-Enneper representation in a new way.