Consider functions of the type $f(u,v) : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ representing a conformal parametrization of a surface.
Is there a family of closed-form functions (e.g. polynomials, trigonometric) that represents a variety of surfaces while preserving conformality?
Ideally, the family of functions would be general enough to represent any conformal mapping. I tried to frame this problem as a system of PDEs based on the first fundamental form of $f$, but it is severely underdetermined, so I am not sure if that is possible. Any pointers/suggestions are appreciated.