One way to imagine the idea of parameterizing a surface is that it gives us a way to put the parameter domain (which is actually in \(\mathbb{R}^2\)) into \( \mathbb{R}^3 \) and stretch it to become the surface. In class, you might have looked at the parameterization \(\vec{r}(u, v) = \langle u, v, u^2 + v^2 + 1 \rangle\) with parameter domain \(u^2 + v^2 < 16\) (a disk); here's one way you could visualize it:

Here are a couple more parameterizations of the same surface.