Parametric Surfaces

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Parametric Surfaces

Based on “Parametric Surfaces” in section 5.6 of the textbook.

Key Ideas

Parametric equations for a surface have 2 variables:

The “parameter domain” is the set of values that can be used for u and v.

A “regular” parametric surface is one that really does describe a surface, as opposed to a line or line segment or a point. Regularity is defined by the requirement that

Cross products show up in regularity, among other places; the determinant mnemonic for cross product is useful.

Plotting

Mathematica’s ParametricPlot3D can plot surfaces as well as curves: give it a vector of 2-variable functions, and ranges of values for both variables.

For example, you can download a Mathematica notebook with the textbook’s cylinder (r(u,v) = ⟨ cos u, sin u, v ⟩) and torus (r(u,v) = ⟨ (2+cos u) cos v, (2+cos u) sin v, sin u ⟩).

Try some examples of your own (invent your own parametric surface and see what it looks like, or try modifying some of the ones in the book, e.g., give them elliptical rather than circular cross sections, etc.)

Tangents and Normals

Not in the part of the book you’ve read so far, but you can figure it out…

How would you find a vector tangent to a (regular) parametric surface?

Imagine holding one parameter constant while the other varies. Then you basically have a single-variable parametric function tracing a curve (which happens to lie in the surface). The tangent to that curve, i.e., its derivative, must also be tangent to the surface. So each partial derivative of a parametric surface is a tangent to that surface:

How would you find a normal vector to a (smooth) parametric surface? The normal must be perpendicular to all tangents, so calculate it as the cross product of the two tangent vectors from above.

At least, this works if the surface is regular. But if the surface is non-regular, i.e., it degenerates into a line or point, then the tangents are parallel or zero, and have a cross product of 0. This starts to make sense of the rule that regular surfaces have non-zero cross products.

As an example, see if you can find a parametric function for normals to the torus from the textbook.

Start by finding the partial derivatives…

And then take the cross product of those derivatives:

Next

Surface areas of parametric surfaces.

Please read “Surface Area of a Parametric Surface” in section 5.6 of the textbook.

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