Question about parametrization and number of free variables

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SUMMARY

Parametric equations with one free variable represent a line, while those with two free variables represent a surface. This distinction arises from the geometric concept that a set with "dimension n" can be defined using n free variables. Specifically, curves are classified as 1-dimensional manifolds, whereas surfaces are classified as 2-dimensional manifolds. This understanding aligns with the intuitive notion that one dimension corresponds to a line and two dimensions correspond to a plane.

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  • Basic concepts of curves and surfaces
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coolbeans777
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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.
 
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coolbeans777 said:
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.
 
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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