If I'm given a curve as an implicit function, is there some algorithm for obtaining a parametrization of the curve?
For example, how would I parameterize
$\frac{2x}{y} - \frac{3y}{x} = 8$
I know I might use $\cosh$ and $\sinh$, but I'm wondering if there's a surefire way to parameterize a curve, when you have a formula for it like this.
I can't isolate y, clearly, but you can re-arrange it to
$ 2x^2 - 3y^2 - 8xy = 0 $
I'm hoping there's a way to find a trig rep for an arbitrary equation that I don't know about.
Where do you go from here?
Is it just trial and error / recognizing the form and choosing trig functions?