# 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: $\varphi: U=[a,b]\times[c,d]\subseteq \mathbb{R}^{2} \longrightarrow S \subset \mathbb{R}^{3}$, so that for every pair of coordinates (let's call them $u$, $v$) the only corresponding surface point is $S$, and vice versa.

A parametrization of a sphere of radius $R$ is $$\begin{array}{ccc} {\varphi : \big[-\frac{\pi}{2}, \frac{\pi}{2}\big] \times [0,2\pi]} & {\longrightarrow} & {\mathbb{R}^{3}} \\ {[ \theta, \alpha] }&{ \longrightarrow} & {R \cdot ( \cos \theta \cdot \cos \alpha , \cos \theta \cdot \sin \alpha , \sin \theta )} \end{array}$$

A parametrization of an ellipsoid of semiaxes $a$, $b$ and $c$ is $$\begin{array} {ccc} {\gamma: \big[\frac{-\pi}{2},\frac{\pi}{2}\big] \times [0,2\pi]} & {\longrightarrow} & {\mathbb {R} ^{3}} \\ {[\theta, \alpha]} & {\longrightarrow} & {R\cdot(a \cdot \cos \theta \cdot \cos \alpha, b \cdot \cos \theta \cdot \sin \alpha,c \cdot \sin \theta)} \end{array}$$

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

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