Given a parabolic coordinate system $x=\frac{1}{2}(u^2-v^2), y=uv, z=z$. I have to find the first fundamental form. There are standard techniques (just for ref. Andrew Perssley and many others). The ans is $(u^2+v^2)(du^2+dv^2)+dz^2$.
The question : Is there any way to find this from conformal mapping.