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In this and this Math.SE questions askers wanted to parametrize their equations.

It seems to me that one has to, without the algorithm, figure out a symbolic trick and then symbolically manipulate the expressions to get the proper parametrization.

For example: $x^{2/3} + y^{2/3} = 1 $ and $z^3 = x^2$.

One could pick $x = \sin^3{t}$, $y = \cos^3{t}$ and $z = \cos^2{t}$.

That's a trick. Still, this kind of parametrization might not be pleasant for constructing osculating planes and similar.

So, is there a simple algorithm (something one can do easily using pen and paper) for parametrizing any two-three variable equation?

Community
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If there was an easy algorithm, then you could "easily" parametrize $ax^5+bx^4+cx^3+dx^2+ex+f=y$, $y=0$ giving an elementary solution to a quintic equation, which we know is impossible. Parametrizing equations still means you need to solve them, and some equations are hard or impossible to solve by elementary means.

Alex R.
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