Parametrized surfaces

By now, you are too familiar with parameterized curves. If we have a function c : [a, b]$\to$R³, then c(t) parametrizes a curve in three dimensions as t ranges from a to b. We have frequently used c(t) = (cos t, sin t, t) as a parametrization of a helix or slinky. For example, here is one loop of the helix, with 0$\le$t$\le$2$\pi$.

We can parametrize surfaces in the same way. To parametrize surfaces, we simply need a function of two variables (rather than a function of one variable that we needed for parametrized curves). To illustrate the properties of parametrized surfaces, we will use the example function

$$\Phi$$

If we fix u = 1, then $\Phi$(1, v) parametrizes the helix as before. If we change u to 0.5, then $\Phi$(0.5, v) also parametrizes a helix, but this time with radius 0.5 rather than radius 1.

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

Notice that if you drag the blue slider to change v, then the red point traces out a helix. You can drag the green slider to change u and thus the radius of the helix. If you keep v constant and change only u, the red point traces out a straight line.

If we let D be the region where 0$\le$u$\le$1 and 0$\le$v$\le$2$\pi$, then the function $\Phi$ maps the region D onto the helicoid shown above. We sometimes write this as $\Phi$ : D$\to$R³. (Note the region D is actually a subset of R², so we could also say that $\Phi$ : R²$\to$R³.)

The region D is a rectangle in uv-space. Any point (u, v) in D is mapped to the point $\Phi$(u, v) on the helicoid. We can demonstrate this map more clearly by replacing the above blue slider for v and green slider for u with a green point that you can drag in the rectangle D to specify both u and v. In the below demo, the region D is shown floating above the helicoid. Notice that if you drag the green point along the bottom of the rectangle, you change u while leaving v = 0. Similarly, if you drag the green point along the right side of the rectangle, you change v while leaving u = 1. (Note: you cannot drag the red point on the surface.)

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

You can see some examples here.

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