Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2000 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["ParaPlot`"]
(*MathSource Item:0207-986*)

Needs["PlaneCurveFormulas`"](*0208-482*)

Needs["PlaneCurveGenerator`"](*0208-314*)

Needs["Graphics`ImplicitPlot`"]
(*MathSource Item:0200-529*)

$HistoryLength = 10;

Folium of Descartes

?FoliumOfDescartes

FoliumOfDescartes[][t]

ImplicitPlot[x^3  +  y^3  ==  3 x*y, {x, -6, 4}]

make the curve colored

Show@
      Graphics[{Hue[ .7], Thickness[ .009], 
          First@ImplicitPlot[x^3  +  y^3  ==  3 x*y, {x, -6, 4}]}, 
        AspectRatio -> Automatic, 
        PlotRange -> {{-1, 1} 3, {-1, 1} 3}, 
        Axes -> True]

Plot the parametric formula defined in FoliumOfDescartes[]

ParaPlot[Evaluate@FoliumOfDescartes[][t],  {t, -2, 2}, 
      AspectRatio -> Automatic]

ParaPlot[{3*t, 3  t^2}/(1 + t^3),  {t, -9, 9}, 
      PlotParameterValue -> True, 
      AspectRatio -> Automatic]

?Rotate

Rotate[FoliumOfDescartes[][t], -2*Pi/8*3]

Simplify@%