Parameterizing Surfaces

# Parameterizing Surfaces

We are about to look at a new type of integral known as a Surface Integral, but before we do, we will need to first learn how to parameterize a surface. As we saw on the Parametric Surfaces page, if a surface in $\mathbb{R}^3$ with the variables $x$, $y$, and $z$ is given as a function of the other two variables (i.e, $z = f(x, y)$, $y = g(x, z)$ or $x = h(y, z)$) then parameterizing this surface is very easy.

For example, consider the surface given by $x = 2y^2 + 3z^2$. Then let $y = u$ and $z = v$. Then we can let $x = f(u, v) = 2u^2 + 3v^2$ and so the parameterization $\vec{r}(u, v) = ((2u^2 + 3v^2), u, v)$ is a parameterization of this surface as shown below: Of course, many surfaces cannot be expressed in this manner. For example, suppose that we want to parameterize the surface $2x^2 + 3y^2 + z^2 = 4$. Note that we cannot express this surface as a function of two of its variables. Now you should also notice that this surface represents an ellipsoid.

One way to parameterize this surface is by rewriting this surface as $z^2 = 4 - 2x^2 - 3y^2$ and so $z = \pm \sqrt{4 - 2x^2 - 3y^2}$. Let $x = u$ and $y = v$. Then $z = \pm \sqrt{4 - 2u^2 - 3v^2}$. Thus this surface can be parameterized in two parts as:

(1)
\begin{align} \quad \vec{r}(u, v) = \left (u, v, \pm \sqrt{4 - 2u^2 - 3v^2} \right ) \end{align}

Of course, this is somewhat a messy parameterization, so let's try using spherical coordinates instead. Let $x = \sqrt{2} \sin \phi \cos \theta$, $y = \frac{2}{\sqrt{3}} \sin \phi \sin \theta$ and $z = 2 \cos \phi$. Then a nicer parameterization of our ellipse is for $0 ≤ \phi ≤ \pi$ and $0 ≤ \theta ≤ 2\pi$ is:

(2)
\begin{align} \vec{r}(\phi, \theta) = \left ( \sqrt{2} \sin \phi \cos \theta , \frac{2}{\sqrt{3}} \sin \phi \sin \theta, 2 \cos^2 \phi \right ) \end{align}

The following image is a graph of our ellipse with this parameterization. In general, for an ellipse $ax^2 + by^2 + cz^2 = d$ for $a, b, c, d > 0$ then the general parameterization of this ellipse is given by:

(3)
\begin{align} \vec{r}(\phi, \theta) = \left\{\begin{matrix} x(\phi, \theta) = \sqrt{\frac{d}{a}}\sin \phi \cos \theta\\ y(\phi, \theta) = \sqrt{\frac{d}{b}} \sin \phi \sin \theta \\ z(\phi, \theta) = \sqrt{\frac{d}{c}} \cos \phi \end{matrix}\right. \quad 0 ≤ \phi ≤ \pi \: , \: 0 ≤ \theta ≤ 2\pi \end{align}

Another type of parameterization we will look at are surfaces obtained by revolutions. Suppose that we have a continuous function $y = f(x)$ for $a ≤ x ≤ b$ and that we revolve this surface about the $x$-axis by an angle of $\alpha$ for $0 ≤ \alpha ≤ 2\pi$. Then we can parameterize the surface generated as:

(4)
\begin{align} \quad \vec{r}(x, \theta) = \left\{\begin{matrix} x(x, \theta) = x \\ y(x, \theta) = f(x) \cos \theta\\ z(x, \theta) = f(x) \sin \theta \end{matrix}\right. \quad a ≤ x ≤ b \: , \: 0 ≤ \theta ≤ \alpha \end{align}

We can also parameterize surfaces obtained by rotating a function in terms of $y$ or $z$ about the (respectively) $y$ or $z$ axis in a similar manner.

For example, consider the function $f(x) = 4 - x^2$, and suppose that we want to parameterize the surface generated by rotating $f$ about the $x$ axis for $-2 ≤ x ≤ 2$ and $0 ≤ \theta ≤ \pi$. Then the parametric equation for this surface is given by:

(5)
\begin{align} \quad \vec{r}(x, \theta) = \left\{\begin{matrix} x(x, \theta) = x \\ y(x, \theta) = (4 - x^2) \cos \theta\\ z(x, \theta) = (4 - x^2) \sin \theta \end{matrix}\right. \quad -2 ≤ x ≤ 2 \: , \: 0 ≤ \theta ≤ \pi \end{align}

The graph below is of the surface above described parametrically: 