Parametrized Surfaces: The Torus

In multivariable calculus, we often use transformations to map a two-dimensional region onto some other region in Euclidean space. A change of variables with double integrals involves a transformation of a two-dimensional region onto another two-dimensional region. These regions are really just "surfaces" with a constant z value of 0.

If we allow the value of z to vary, then our transformations can send a two-dimensional region onto a true three-dimensional surface. One such example is the torus. I won't give you the parametric equations for the torus here---that's part of your homework!!---but the example below will show you geometrically what the mapping does.

On the left you can see the rectangle

R = [0, 2π] x [0, 2π]

in uv-space. This rectangle is mapped onto the torus shown on the right; to help you see "through" the torus, I've rendered it as a wire-drawing, but of course it's really a surface.

One small square in the domain is highlighted in red; its image under the transformation is also highlighted on the right. To help you keep track of where the image rectangle is, one side is colored red, while the other side is gold. You can click on the rectangle in the domain and drag it around to see what happens to its image.

Here are some things for you to think about: what are the grid lines of this parametrized surface? Both u and v range from 0 to 2&pi, which certainly suggests rotations and/or circles are involved. What kind of rotation do you see if you keep u constant but change v? What about the other way around?

To move the rectangle in uv-space, move the cursor over the red rectangle until a square is highlighted; then click and drag the mouse. If you need to figure out your perspective, you can click on the torus and drag the picture to rotate it. Things can quickly get out of hand; press the "Home" key on your keyboard to reset the image.

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