# 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)

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)

## 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)

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