Converting a ParametricPlot parametric expression to a ContourPlot cartesian equation

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Using ParametricPlot I can plot a lemniscate expressed in parametric coordinates:

  ParametricPlot[1/(1 + Sin[t]^2) {Cos[t], Cos[t] Sin[t]}, {t, 0, 2 [Pi]}]

I want to find using Mathematica the equivalent cartesian expression and plot it using ContourPlot that I know to be:

ContourPlot[(x^2 + y^2)^2 == (x^2 \[Minus] y^2), {x, -1, 1}, {y,-1,1}]

Looking up among the MMA functions I wondered if CoordinateTransformData or TransformedField could help me but none of them has the appropriate coordinate transformation :"Parametric" -> "Cartesian" which had me baffled.

How can this be done ?

asked 29 secs ago

wojtek

27

Converting a ParametricPlot parametric expression to a ContourPlot cartesian equation