General method of parameterization

[DivergentSpectrum]

is there a general formula/algorithm to parameterize an equation?

for example if i have a function
f(x,y)=0
then can i always change it to the form x(t),y(t)?

or if i had g(x,y,z)=0

then can i change it to something like x(u,v),y(u,v),z(u,v)

i know of examples, like if i have x^2+y^2=1 then i know its cos(t),sin(t) but there's a lot of other functions out there.

I guess what I am asking, is if there is always a parameterization in terms of elementary functions, and if not, what kinds of numeric algorithms are available?

 

[Orodruin]

This depends on the properties of the function. The general answer is no.

 

[DivergentSpectrum]

so is there an algorithm i could use in computer programming that takes f(x,y)=0 for example and returns an ordered set of points?

 

[Orodruin]

This is again going to depend on your function. Perhaps the only functions you will encounter are nice enough. A computer code will never give you the full set of points simply because of memory limitations. You might want a finite set of points that give you a reasonable approximation? The contour plotting function of softwares like MATLAB or mathematica should be able to do that for you.