Folium of Descartes and Parametric Equations

Date: 11/23/98 at 01:12:20
From: Benjamin Goh
Subject: Plotting an implicit function (e.g. y^3 + x^3 = 3xy)

I have a mathematics dictionary at home, and I've come across a strange 
looking graph, called the folium of Descartes, with an equation of 
y^3 + x^3 = 3xy.

After thinking about it, I realized that I can't simply create a table 
of x and y values to plot this curve, since I cannot make either y or 
x the subject of the equation. Many graph programs also are unable to 
plot this graph (Graphmatica, Equation Grapher. . .)

I've found one way to plot such a graph, and that is testing every 
point on an x-y plane to see if it roughly fits the equation, but this 
method is rather slow. Is there another, better way to plot such 
implicit graphs? They are much more interesting than standard ones.  
I've been experimenting, and one of my favourites is 
tan(x^3) + log(sin(y^2)) = 3sin(x)cos(y). It has a very interesting 
pattern, but it's quite hard to plot.

Date: 11/23/98 at 08:57:01
From: Doctor Jerry
Subject: Re: Plotting an implicit function (e.g. y^3 + x^3 = 3xy)

Hi Benjamin,

Some equations can be described with parametric equations. It happens 
that the folium of Descartes can be so described.

The equation x^3 + y^3 = 3*a*x*y is equivalent to:

   x = 3*a*t/(1+t^3)
   y = 3*a*t^2/(1+t^3)

You can plot this by taking several values of t, calculating the x and 
y values, and plotting (x,y). I've plotted above equations using the 
Mathematica command:

   ParametricPlot[{3*t/(1+t^3),3*t^2/(1+t^3)},{t,-100,100}, 
   PlotRange->{-1.5,2}]

to get a rough sketch:

    

The following the link, when available, will provide you with more 
information:

   http://mathworld.wolfram.com/FoliumofDescartes.html   

This link is from the CRC Encylopedia of Mathematics, by Eric 
Weisstein. He includes a sketch of the curve and says in part:

   [It is] a plane curve proposed by Descartes to challenge Fermat's  
   extremum-finding techniques. ... The curve has a discontinuity at 
   t = -1. The left wing is generated as t runs from -1 to 0, the loop 
   as t runs from 0 to infinity, and the right wing as t runs from 
   -infinity to -1. 

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/