How do we parameterize surfaces?

Today was like a review of the lesson we learned back in September/October. Parameterizations are given and you have to find how the x, y, and z vector-valued functions, which are in terms of u and v, relate in order to draw the surface. Doesn't this sound like finding traces and level curves?

The concept with gridlines was also introduced. First, the region that the surface covers is drawn. Then horizontal (y= __) and vertical lines (x= __) are drawn within the region (like a grid) and the equation of the level curve is found after plugging in the horizontal and vertical line values. After putting together several level curves, the curve can be seen and sketched.

Here's a cool site I found that kind of explains how artists apply surface parameterization to create their art: http://www.thegnomonworkshop.com/tutorials/paramaterization/parameterization.html.