Math Insight

Parametrized surface examples

Example 1

Parametrize the single cone $z=\sqrt{x^2+y^2}$.

Solution: For a fixed $z$, the cross section is a circle with radius $z$. So, if $z=\spfv$, the parameterization of that circle is $x=\spfv\cos \spsv$, $y=\spfv \sin \spsv$, for $0 \le \spsv \le 2\pi$.

The parameterization of whole surface is \begin{align*} (x,y,z) = \dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spfv) \end{align*} for $0 \le \spsv \le 2\pi$, $0 \le \spfv \le \infty$.

Of course, there's nothing sacred about $\spfv$ and $\spsv$. Could also use \begin{align*} (x,y,z) = \dlsp(r,\theta) = (r \cos \theta, r \sin\theta, r). \end{align*}

Example 2

What happens if fix the radius of the circle to $x = 3 \cos \theta$, $y= 3 \sin \theta$?

Solution. The parameterization becomes \begin{align*} (x,y,z) = \dlsp(\spfv,\theta) = (3 \cos \theta, 3 \sin\theta, \spfv) \end{align*} This is a right circular cylinder of radius 3.