Why do parametric equations with two free variables create surfaces?

[coolbeans777]

Hey guys, how come when you have a parametric equation with two free variables it creates a surface, but when you have a parametric equation with one free variable it draws out a line? I sort of get it intuitively, one dimension is just a line, two dimensions is a plane, so I guess this sort of corresponds to this effect, but is there a more rigorous explanation? I didn't know which section to put this in, sorry if it's not the right one.

 

[HallsofIvy]

Hey guys, how come when you have a parametric equation with two free variables it creates a surface, but when you have a parametric equation with one free variable it draws out a line? I sort of get it intuitively, one dimension is just a line, two dimensions is a plane, so I guess this sort of corresponds to this effect, but is there a more rigorous explanation? I didn't know which section to put this in, sorry if it's not the right one.

That's pretty much it. A geometric set has "dimension n" means that we can determine specific points in the set using n free variables or parameters.

 

[Bacle2]

In a less natural/immediate way than HallsofIvy's comment, is the fact that curves

(reasonably-nice ones , at least ), are 1-dimensional manifolds, and surfaces are

2-D manifolds.