Question about parametrization and number of free variables

  • #1

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
HallsofIvy
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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.
 
  • #3
Bacle2
Science Advisor
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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.
 

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