How do I convert a parametric surface $S(u,v) = (X(u,v),Y(u,v),Z(u,v))$ to a Monge representation

Question

How do I convert a parametric surface

$S(u,v) = (X(u,v),Y(u,v),Z(u,v)) $

to a Monge representation,

$ S(x,y) = (x,y,Z(x,y))$?

Answer 1

You could try solving the system

$$\begin{cases} x=X(u,v) \\y=Y(u,v) \end{cases} $$ for the functions $u=u(x,y),v=v(x,y)$. After that you can reparametrize as

$$(x,y,Z(u(x,y),v(x,y))) .$$