Parametrization of surfaces

In case we want to express a surface in the space, we will need to give it as a function of two variables: $φ : U = [a, b] \times [c, d] \subseteq R^{2} ⟶ S \subset R^{3}$ , so that for every pair of coordinates (let's call them $u$ , $v$ ) the only corresponding surface point is $S$ , and vice versa.

Example

A parametrization of a sphere of radius $R$ is $\begin{array}{ccc} φ : [- \frac{π}{2}, \frac{π}{2}] \times [0, 2 π] & ⟶ & R^{3} \\ [θ, α] & ⟶ & R \cdot (\cos θ \cdot \cos α, \cos θ \cdot \sin α, \sin θ) \end{array}$

Example

A parametrization of an ellipsoid of semiaxes $a$ , $b$ and $c$ is $\begin{array}{ccc} γ : [\frac{- π}{2}, \frac{π}{2}] \times [0, 2 π] & ⟶ & R^{3} \\ [θ, α] & ⟶ & R \cdot (a \cdot \cos θ \cdot \cos α, b \cdot \cos θ \cdot \sin α, c \cdot \sin θ) \end{array}$

Example

A parametrization of the graph of a function of two variables $f (u, v)$ $\begin{array}{ccc} γ : [a, b] \times [c, d] & ⟶ & R^{3} \\ [u, v] & ⟶ & (u, v, f (u, v)) \end{array}$

Example

A parametrization of the resultant surface when turning the graph of a function $f (x)$ with respect to the $z$ axes. $\begin{array}{ccc} γ : [a, b] \times [0, 2 π] & ⟶ & R^{3} \\ [x, θ] & ⟶ & (x \cdot \cos θ, x \cdot \sin θ, f (x)) \end{array}$