The Weierstrass-Enneper representation using hodographic coordinates on a minimal surface (Q1422305)

In this work, the author proves the Weierstrass-Enneper representation for minimal surfaces in Euclidean 3-space from the point of view of hodographic coordinates. The author first notes that the stereographic projection of the Gauss map can itself be used as a conformal coordinate for a minimal surface, away from umbilic points of the surface, or equivalently, away from branch points of the Gauss map. This leads to a simpler form for the Weierstrass-Enneper representation away from umbilic points. This simpler form for the representation is then proven by using hodographic coordinates.