Back in the day, I was absolutely enthralled by the study of plane curves and their properties (I have Lockwood and Zwikker to thank). I learned early on that for the purposes of generating plots on a computer (and for that matter deducing equations of "derived curves" and determining other special properties), one should try to find a representation in parametric equations for your plane curve.
As I recall, in dealing with algebraic curves represented by an implicit Cartesian equation, I knew of only three tricks to derive parametric equations from an implicit equation (listed in decreasing order of effectiveness; I note that I did all these investigations even before I knew computer algebra systems existed):
1: Convert to polar coordinates to express in the form $r=r(\theta)$; the parametric equations are then
2: The $y=mx$ "trick" (I never did get to learn the formal name for this technique); to use the implicit equation for the folium of Descartes as an example:
and then by solving for x and using the relation $y=mx$ again,
(I remember this worked especially well for curves whose (only?) singular points are at the origin, but not very well for other curves; can anybody explain why?)
3: Randomly replacing x or y with any of the six trigonometric functions (maybe multiplied by a convenient constant), and hope that I can easily solve for the other variable. For instance, I managed to derive the parametric equation for the bicorn and the Dürer conchoid in this way.
Probably the only other thing I learned way after I had moved on to other things was that elliptic curves can for instance be represented as parametric equations involving the Weierstrass ℘ function or the elliptic exponential, but this is apparently limited to elliptic curves only.
Now for my question: did I miss any other useful (general?) methods for turning an implicit Cartesian equation for an algebraic curve into parametric equations?
I didn't want to ask a separate question, so: are there systematic methods for parametrizing a plane algebraic curve in terms of (Jacobi or Weierstrass) elliptic functions? For instance, we find here that the Fermat cubic $x^3+y^3=a^3$ can be parametrized in terms of Weierstrass functions, in addition to the elliptic curve example I gave previously. I've also encountered in my readings that the Cartesian ovals can also be parametrized with Weierstrass functions, but I have been unable to find an explicit construction of the parametric equations.