21
$\begingroup$

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

$\begin{align*}x&=r(\theta)\cos\,\theta\\y&=r(\theta)\sin\,\theta\end{align*}$

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:

$x^3+y^3=3xy$

$x^3+(mx)^3=3x(mx)$

and then by solving for x and using the relation $y=mx$ again,

$\begin{align*}x&=\frac{3m}{1+m^3}\\y&=\frac{3m^2}{1+m^3}\end{align*}$

(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?


Addendum, 8/7/2011

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.

  • 1
    A cursory Google search gives http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBoQFjAA&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.132.9886%26rep%3Drep1%26type%3Dpdf&ei=UdB1TMKmDYi-sQPEx_igDQ&usg=AFQjCNGEMATh_i2HVQCFrLH6OjU-zhMF_Q .2010-08-26
  • 0
    Qiaochu: I was investigating them way before I found out about things like *Mathematica* (would you believe I was fooling around with pen plotters?), so I probably could not have come up with using Gröbner bases to derive parametric equations. :D A CAS should of course have no trouble manipulating them however.2010-08-26
  • 1
    Some software very related (and interesting, I think): http://library.wolfram.com/infocenter/MathSource/727/2014-10-19

1 Answers 1