Parameterizing Surfaces Examples

On the Parameterizing Surfaces page, we saw many different ways to parameterize various surfaces in $\mathbb{R}^3$. We will now look at some examples of parameterizing surfaces.

Example 1

Let $g(y) = \cos y$. Parameterize the surface generated by rotating $g$ about the $y$-axis $270^{\circ}$ for $-\pi ≤ y ≤ \pi$.

The curve that we are rotating, $x = \cos y$, about the $y$ axis shown below:

Let $y(y, \theta) = y$. Then we'll have that $x(y, \theta) = g(y) \cos \theta = \cos y \cos \theta$ and $z(y, \theta) = g(y) \sin \theta = \cos y \sin \theta$. The restrictions to our parameters $y$ and $\theta$ are that $-\pi ≤ y ≤ \pi$ and $0 ≤ \theta ≤ \frac{3 \pi}{2}$ and so parametrically we have that this surface is given by:

(1)

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

The following is a graph of the surface generated by our parametric equations.

Example 2

Parameterize the sphere of radius $a ≥ 0$ centered at the origin.

A sphere of radius $a ≥ 0$ is given by $x^2 + y^2 + z^2 = a^2$. This sphere can be parameterized by using spherical coordinates. Let $x = a \sin \phi \cos \theta$, $y = a \sin \phi \sin \theta$ and $z = a \cos \phi$. Then we have that the following parameterization of this sphere:

(2)

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

Example 3

Parameterize the surface $z^2 = y + 4x^2$.

We can rewrite the equation above as $y = z^2 - 4x^2$, and so we wish to parameterize the surface generated by the function $g(x, z) = z^2 - 4x^2$. Let $x = u$ an d$z = v$. Then $y = g(u, v) = v^2 - 4u^2$ and thus a parameterization for this surface is given by:

(3)

\begin{align} \quad \vec{r}(u, v) = \left\{\begin{matrix} x(u, v) = u \\ y(u, v) = v^2 - 4u^2 \\ z(u, v) = v \end{matrix}\right. \end{align}

A portion of the surface generated by the parametric equations above is shown below: