Parametrized surfaces extend the idea of parametrized curves to vector-valued functions of two variables. We can parametrize a curve with a function of one variable. The function $\dllp: [a,b] \to \R^3$ (confused?) maps the interval $[a,b]$ onto a curve in three dimensions. For example, $\dllp(t) = (\cos t, \sin t, t)$ parametrizes a helix or slinky. The below applet shows a helix with one loop, as we let $t$ range in $[0,2\pi]$.

*Parametrized helix.* The vector-valued function $\dllp(t)=(\cos t, \sin t, t)$ parametrizes a helix, shown in cyan. This helix is the image of the interval $[0,2\pi]$ (shown in magenta) under the mapping of $\dllp$. For each value of $t$, the cyan point represents the vector $\dllp(t)$. As you change $t$ by moving the blue point along the interval $[0,2\pi]$, the red point traces out the helix.

We can parametrize surfaces in the same way. To parametrize surfaces, we simply need a function of two variables. To illustrate the properties of parametrized surfaces, we will use the example function \begin{align*} \dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv). \end{align*} If we fix $\spfv=1$, then $\dlsp(1,\spsv)$ parametrizes the helix as before. If we change $\spfv$ to 0.5, then $\dlsp(0.5,\spsv)$ also parametrizes a helix, but this time with radius 0.5 rather than radius 1.

If we let $\spfv$ range from 0 to 1, and let $\spsv$ range from 0 to $2\pi$, then $\dlsp(\spfv,\spsv)$ traces out a whole family of helices, with radii between 0 and 1. The result is a surface called a helicoid, as shown below.

*A parametrized helicoid.* The function $\dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv)$ parametrizes a helicoid when $0 \le \spfv \le 1$ and $0 \le \spsv \le 2\pi$. You can drag the blue and green points on the sliders to change the values of $\spfv$ and $\spsv$. You cannot move the red point directly, but it moves with $\spfv$ and $\spsv$ be at $\dlsp(\spfv,\spsv)$. If you leave $\spfv$ fixed and change only
$\spsv$, then the red point traces out a helix. Changing $\spfv$ changes the radius of the helix. If you keep $\spsv$ constant and change only $\spfv$, the red point traces out a straight line.

If we let $\dlr$ be the region where $0 \le \spfv \le 1$ and $0 \le \spsv \le 2\pi$, i.e., $\dlr = [0,1] \times [0, 2\pi]$, then the function $\dlsp$ maps the region $\dlr$ onto the helicoid shown above. We sometimes write this as $\dlsp: \dlr \to \R^3$. (Note the region $\dlr$ is actually a subset of $\R^2$, so we could also say that $\dlsp: \R^2 \to \R^3$.)

The region $\dlr$ is a rectangle in $\spfv\spsv$-space. Any point $(\spfv,\spsv)$ in $\dlr$ is mapped to the point $\dlsp(\spfv,\spsv)$ on the helicoid. We can demonstrate this map more clearly by replacing the sliders that represent the variables in separate intervals $\spfv \in [0,1]$ and $\spsv \in [0, 2\pi]$ with the rectangle showing that $(\spfv, \spsv) \in [0,1] \times [0, 2\pi]$.

*A parametrized helicoid.* The function $\dlsp(\spfv,\spsv) = (\spfv\cos \spsv, \spfv\sin \spsv, \spsv)$ parametrizes a helicoid when $(\spfv,\spsv) \in \dlr$, where $\dlr$ is the rectangle $[0,1] \times [0, 2\pi]$. The region $\dlr$ is shown as the green rectangle floating above the helicoid. You can drag the green point in $\dlr$ to specify both $\spfv$ and $\spsv$. If, for example, you drag the green point along the bottom of the rectangle, you change $\spfv$ while leaving $\spsv=0$. Similarly, if you drag the green point along the right side of the rectangle, you change $\spsv$ while leaving $\spfv=1$. You cannot directly move the red point on the surface as it is moves with $\spfv$ and $\spsv$ to be at the point $\dlsp(\spfv,\spsv)$.

In general, the region $\dlr$ does not have to be a rectangle. But in many examples, we let $\dlr$ be a rectangle to make the calculations simpler.

You can see some examples here.