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# Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. Elliptic cylinder r(u, v)=[a cos v, b sin v, u]

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$\textbf{{\color{#c34632}{\underline{Representation}}}}$

Recall, a $\textbf{parametric representation}$ of a surface $S$ in space is

$r(u,v)=[x(u,v),y(u,v),z(u,v)]=x(u,v)\mathbf{i}+y(u,v)\mathbf{j}+z(u,v)\mathbf{k}$

Therefore, for our example:

$x=a\cos v, \quad \quad \quad \quad y=b\sin v \quad \quad \text{and} \quad \quad z=u$

This is parametric representation of an elliptic cylinder. The base is an elipse with semi-major axis $a$ and semi-minor axis $b$ with $u$ being the height of the cylinder. $v$ is the angle $\theta$.

Consequently,

$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1,\quad \quad z=u$

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