Is it possible to parametrize a function analytically?

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Hi, I want to know if there's a general method of changing variables of a function and reduce the original number of variables (which is what parametrization is, right?).

For example one parametrization for the function whose graphic is a circunference is x = Rcos(t) and y = Rsin(t). But is there any method to find that parametrization algorithmically?
 
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That one comes from phytagaros.

R^2=x^2+y^2
since sin^2(t)+cos^2(t) = 1

To make it parametirized you also need a limitation (or parameter), t=[0,2*pi[

i.e. if you have a inifinite line y=2x for all x, you can parameterize it by saying x=[0,3]
now you have a line with finite length.
 

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