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### Folium of Descartes and Parametric Equations

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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.
```

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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/
```
Associated Topics:
College Euclidean Geometry
High School Equations, Graphs, Translations
High School Euclidean/Plane Geometry

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