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lemniscate like bulbs from an Joukowski transform of an ellipse

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  • Subject: [mg102405] lemniscate like bulbs from an Joukowski transform of an ellipse
  • From: Roger Bagula <roger.bagula at>
  • Date: Sat, 8 Aug 2009 04:38:09 -0400 (EDT)

The Joukowski transform is a fluid dynamic/ mechanics transform often
with the aerodynamics of generalized cylinders ( wings).
Using this transform I get a three part figure with two lemniscate
type bulbs.
Clear[x, y, z, z1, x1, y1, r]
(* diaxial ellipse*)
x = Cos[t];
y = Cos[t + 2*Pi/3];
(*Rotation matrix to get major ellipse  axis on the x axis *)
r = {{Cos[2*Pi/3 + Pi/12], Sin[2*Pi/3 + Pi/12]}, {-Sin[2*Pi/3 + Pi/
12], \
Cos[2*Pi/3 + Pi/12]}};
g1 = ParametricPlot[r.{x, y}, {t, -Pi, Pi}, AspectRatio -> Automatic]
{x1, y1} = r.{x, y};
z = x1 + I*y1;
(* Joukowski transform without flow*)
f[t_] := ComplexExpand[(z + 1/z)/2]
(*angular division of the cosed figure into four parts*)
g2top = ParametricPlot[{Re[f[t]], Im[f[t]]}, {t, -2*Pi/3 + Pi/
    12, -Pi/12}, PlotRange -> All, AspectRatio -> Automatic]
g2bottom = ParametricPlot[{Re[f[t]], Im[f[t]]}, {t, 5*Pi/12, 2*Pi/3 +
  12}, PlotRange -> All, AspectRatio -> Automatic]
(* lemniscate type parts*)
g2bulb1 = ParametricPlot[{Re[f[t]], Im[f[t]]}, {t, -Pi/12, 5*Pi/12},
     PlotRange -> All, AspectRatio -> Automatic]
g2bulb2 = ParametricPlot[{Re[f[t]], Im[f[t]]}, {t, 2*Pi/3 + 3*Pi/12,
2*Pi/3 + \
3*Pi/12 + 6*Pi/12}, PlotRange -> All, AspectRatio -> Automatic]
Show[{g1, g2top, g2bottom}]
Show[{g1, g2bulb1, g2bulb2}, PlotRange -> All]

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