Atlas of a given surface

Question

How do we go about finding the atlas (collection of homeomorphisms/surface patches) of any given surface, for instance, if the surface is: $\{(x,y,z) \in \mathbf{R}^{3} : x = y^2\}$?

Answer 1

Explizit, one can use

$$\eqalign{ & \left( {u,v} \right) \in {\mathbb{R}^2} \cr & {\varphi ^{ - 1}}\left( {u,v} \right) = \left( {{u^2},u,v} \right) \cr} $$

as parametriztion, and

$$\begin{gathered} \left( {x,y,z} \right) \in {\mathbb{R}^3} \hfill \\ \varphi \left( {x,y,z} \right) = \left( {y,z} \right) \hfill \\ \end{gathered}$$

as projection.

Answer 2

Hint View the surface as the graph of a particular function of $(y, z)$.