Folia of Descartes section plots

Question

First of all, good day and thanks for taking the time to read this question. The problem I'm having, is how i can plot just sections of the Folia of Descartes graph, which is build from the equation:

$x^3 + y^3 - 3xy=0$

I'm trying to do the representations that Morris Tenenbaum and Harry Pollard do, in their Ordinary differential equation book, page 17.

So far i can do the plot of the Folia of Descartes using:

ContourPlot[x^3 + y^3 - 3 x y == 0, {x, -3, 3}, {y, -3, 3}, Axes -> True, Frame -> False]

And parametric plot:

ParametricPlot[{(3 m)/(1 + m^3), (3 m^2)/(1 + m^3)}, {m, -20, 80},PlotRange -> {-3, 3}]

Now i know that the option RegionFunction of the Function ParametricPlot could help for this, however i have failed in getting the sections.

I'm looking for any pointers or ideas on how to do the sections of the folia of descartes.

PS: My English isn't that good so I hope i explain my self good enough while making this question. Thanks again and have a good day.

Answer 1

As seen on Wikipedia, the folium of Descartes has a polar representation: $$ r={\frac {3a\sin \theta \cos \theta }{\sin ^{3}\theta +\cos ^{3}\theta }}. $$

With this representation it's pretty easy to pick out relevant segments and to visualize them with PolarPlot.

f[min_, max_] := PolarPlot[
  (3 Sin[t] Cos[t])/(Sin[t]^3 + Cos[t]^3), {t, min, max},
  PlotRange -> 2.5, ImageSize -> 300,
  PlotStyle -> Directive[Thickness[0.01], Black],
  Epilog -> {HalfLine[{2^(2/3), 0}, {0, 1}], HalfLine[{0, 0}, {1, 1}]}
  ]

Row[{
  f[0, Pi/4],
  f[Pi/4 - 0.15, Pi/2]
  }]

Row[{f[Pi/2, 2 Pi/3], f[5 Pi/6, Pi]}]

You can combine images with Show. You could also extend f to take several intervals of t.