Parametric Surfaces

# Parametric Surfaces

Recall that in $\mathbb{R}^3$, a curve $C$ can be parameterized as $\vec{r}(t) = (x(t), y(t), z(t)) = x(t) \vec{i} + y(t) \vec{j} + z(t) \vec{k}$ where $x = x(t)$, $y = y(t)$ and $z = z(t)$ are parametric equations that define the curve. Extending this idea further, we can also parameterized a surface, however, we will need to use two parameters instead of one since curves are one dimensional objects and surfaces are two dimensional objects. Let $u$ and $v$ be these parameters. For $a ≤ u ≤ b$ and $c ≤ v ≤ d$ we can define a parametric surface with the parametric equations $x = x(u, v)$, $y = y(u, v)$, and $z = z(u, v)$ as:

(1)
\begin{align} \quad \vec{r}(u, v) = (x(u, v), y(u, v), z(u, v)) = x(u, v) \vec{i} + y(u, v) \vec{j} + z(u, v) \vec{k} \end{align}
 Definition: Let $R$ be the rectangle $[a, b] \times [c, d]$. A Parametric Surface on $R$ is a function $\vec{r}(u, v) = (x(u, v), y(u, v), z(u, v)) = x(u, v) \vec{i} + y(u, v) \vec{j} + z(u, v) \vec{k}$ defined for $a ≤ u ≤ b$ and $c ≤ v ≤ d$.

Let's now look at some examples of parametric surfaces. First, suppose that we are given a two variable function $z = f(x, y)$. This two variable function represents a surface in $\mathbb{R}^3$. We can easily parameterize this surface if we let $x = u$, $y = v$ and $z = f(u, v)$. For example, the function $f(x, y) = x^2 + y^2$ which represents a paraboloid that is parallel to the $z$ axis and opens upward can be parameterized for all $u$ and $v$ in $\mathbb{R}$ as:

(2)
\begin{align} \quad \vec{r}(u, v) = (u, v, f(u, v)) = u \vec{i} + v \vec{j} + f(u, v) \vec{k} \end{align}

Now recall that we were able to represent some very interested curves parametrically. For example, consider the surface given parametrically by $\vec{r}(u, v) = \left ( u, \frac{1}{2} v^2, uv \right )$ for $- 2 ≤ u ≤ 2$ and $-2 ≤ v ≤ 2$. The graph of this surface is given below:

We note that this surface intersects itself. This is not uncommon for parametric surfaces. Now note that when dealing with parametric curves, we could join two curves $\vec{r_1}$ and $\vec{r_2}$ at either of their endpoints to form a new curve. We can do the exact same thing with parametric surfaces to form a composite surface.

 Definition: If $S_1$ and $S_2$ are parametric surfaces that are joined at a portion of their respective boundaries, then together $S_1$ and $S_2$ form a Composite Surface.

For example, consider the parametric surfaces given by $\vec{r_1}(u, v) = (u, v, 0) = u \vec{i} + v \vec{j}$ for $0 ≤ u ≤ 3$ and $0 ≤ v ≤ 3$ and $\vec{r_2}(u, v) = (u, 0, v) = u \vec{i} + v \vec{k}$ for $0 ≤ u ≤ 3$ and $0 ≤ v ≤ 3$. We note that both of these surfaces represent planes. $\vec{r_1}$ represents a plane on the $xy$-axis and perpendicular to the $z$-axis, while [$\vec{r_2}$ represents a plane on the $xz$-plane and perpendicular to the $y$-axis. We can join these surfaces at part of their boundary, more specifically, on the line joining the origin with the point $(3, 0, 0)$ to form a composite surface and graphed below: