27. Surface Parameterization

We have three ways to represent a curve:

• A $\large y=f(x)$ graph such as $\large y = x^2$
• An implicit curve such as the circle $\large x^2+y^2=4$
• A parametrized curve (vector valued function) such as $\large \vec{r}(t)=\langle 3t+2,t-1\rangle$

There are comparable ways to represent surfaces:

• A $\large z=f(x,y)$ graph such as $\large z = x^2+y^2$
• An implicit surface such as the sphere $\large x^2+y^2+z^2=4$
• A parameterized surface, which we will read about today!

Remember, to integrate along a curve, we needed to represent the curve parametrically. The last type of integral we will see is a surface integral, and we'll need to represent the surface parametrically in order to integrate on it.

Parametric Surfaces

The general form for a paramettric surface is:

$$\large G(u,v) = \langle x(u,v),y(u,v),z(u,v)\rangle$$

The variables $\large u$ and $\large v$ are called parameters. The idea is that as we run through all the possible values for $\large (u,v)$, we get the $\large (x,y,z)$ points on a surface. In the image below, we select a $\large (u,v)$ point on the left, and via the equations below, that point gets mapped to the surface on the right:

Parameterizations to Remember

There are many interesting surfaces that can be made in this way. We'll explore some fun ones in class. But we'll need four in particular for HW and test problems. In the general definition, we used $\large u$ and $\large v$ as the parameters, but we can use any letters and will often use letters that make sense for the surface. In general, the parameter domain will depend on the part of the surface we are interested in.

1. Sphere of radius $\large R$: $\large G(\theta,\phi) = \langle R\sin\phi\cos\theta, R\sin\phi\sin\theta,R\cos\phi\rangle$. Here, $\large R$ is a constant; there are only the two parameters $\large \theta$ and $\large \phi$.
2. Cylinder of radius $\large R$: $\large G(\theta,z) = \langle R\cos\theta, R\sin\theta,z\rangle$.
3. Cone: $\large G(\theta,t) = \langle t\cos\theta, t\sin\theta,t\rangle$. Notice that in the $\large x$ and $\large y$ coordinates, we have a circle of radius $\large t$ (so, a radius equal to the height).
4. Graph: $\large z = f(x,y)$: $\large G(x,y) = \langle x, y,f(x,y)\rangle$

Tangent and Normal Vectors

Derivatives with respect to the parameters give us two vectors that are tangent to the surface:

$$\large \begin{split} \vec{T_u}&=\frac{\partial G}{\partial u}=\left< \frac{\partial x}{\partial u},\frac{\partial y}{\partial u},\frac{\partial z}{\partial u}\right> \\\ \vec{T_v}&=\frac{\partial G}{\partial v}=\left< \frac{\partial x}{\partial v},\frac{\partial y}{\partial v},\frac{\partial z}{\partial v}\right> \\\ \end{split}$$

Remember, the cross product $\large \vec{u}\times\vec{v}$ is orthogonal to both $\large \vec{u}$ and $\large \vec{v}$. So the vector perpendicular to the surface, called the normal vector, is:

$$\large \vec{n} = \vec{T_u}\times\vec{T_v}$$

In the video, we see how to parameterize a cone and find a normal vector:

Tomorrow's class:

• Explore other parameterized surfaces
• Practice finding tangent and normal vectors for the standard surfaces

Section Outcomes:

I can:

• Find the parameterization and parameter domain for a sphere, a cone, a cylinder, or a graph.
• Find vectors tangent and normal to a surface.

Other resources

For more on parameterized surfaces, see https://openstax.org/books/calculus-volume-3/pages/6-6-surface-integrals