Parametric Surfaces

How do you turn a one-dimensional vector-valued function r(t) into two-dimensional surface? Rhetorically, that's like asking how to turn f(x) into f(x,y)--you just add another variable. But instead of making the vector function r(t,z), the conventional form is r(u,v).

For instance, take <cos(u), sin(u), v>, where u ranges from 0 to 2pi and v ranges from 0 to 3. The x- and y-components dictate that in the xy-plane, you have a circle. And what does that circle do? It gets extruded upwards to 3. The end result is a cylinder with radius 1 and height 3.

What is the purpose of having another way to describe a surface? The primary reason is because the function is easier to interpret.

In the 2D plane, we're familiar with:

as a circle of radius 3. But in 3-space, that becomes a cylinder. But what's to distinguish one from another? In parametric, however, this difference is clear.