# 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)

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)

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)

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)

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)

The graph below is of the surface above described parametrically:

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