Math Insight

An introduction to parametrized surfaces

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 $[0,2\pi]$.

Applet: Parametrized helix

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Parametrized helix

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.

More information about applet.

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.

Applet: A parametrized helicoid

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: A parametrized helicoid

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.

More information about applet.

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 $\dlsp: \R^2 \to \R^3$.)

Rectangular domain to be mapped into helicoid

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 $(\spfv, \spsv) \in [0,1] \times [0, 2\pi]$.

Applet: A parametrized helicoid

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: A parametrized helicoid

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 $\dlsp(\spfv,\spsv)$.

More information about applet.

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.