Parametrized surfaces

By now, you are too familiar with parameterized curves. If we have a function c : [a,b] → 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 a CVT of one loop of the helix, with 0 ≤ t ≤ 2π.

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

Φ(u,v) = (u cos v,u sin v,v).
If we fix u = 1, then Φ(1,v) parametrizes the helix as before. If we change u to 0.5, then Φ(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 2π, then Φ(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 ≤ u ≤ 1 and 0 ≤ v ≤ 2π, then the function Φ maps the region D onto the helicoid shown above. We sometimes write this as Φ : D → R3. (Note the region D is actually a subset of R2, so we could also say that Φ : R2 → R3.)

PIC

The region D is a rectangle in uv-space. Any point (u,v) in D is mapped to the point Φ(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.