Re: lemniscate like bulbs from an Joukowski transform of an ellips
- To: mathgroup at smc.vnet.net
- Subject: [mg102880] Re: lemniscate like bulbs from an Joukowski transform of an ellips
- From: "rlbagulatftn" <rlb at tftn.net>
- Date: Tue, 1 Sep 2009 03:52:27 -0400 (EDT)
Two ways to get the complex inverse function:
f(z)=2*z/(z^2+1)=1/((z+1/z)/2)
The second way ( maybe should have been first?):
Clear[x, y, t]
(*diaxial elipse*)
x = Cos[t]
y = Cos[t + 2*Pi/3]
ParametricPlot[{x, y}, {t, -Pi, Pi}, AspectRatio -> Automatic]
(*Rotate -45 degrees*)
r = {{Cos[-Pi/4], Sin[-Pi/4]}, {-Sin[-Pi/4], Cos[-Pi/4]}}
{x1, y1} = r.{x, y}
ParametricPlot[{x1, y1}, {t, -Pi, Pi}, AspectRatio -> Automatic]
z = x1 + I*y1
(* Invert the Joukowski transform without circulation*)
f[t_] = FullSimplify[1/((z + 1/z)/2)]
ParametricPlot[{Re[f[t]], Im[f[t]]}, {t, -Pi, Pi},
AspectRatio -> Automatic, PlotRange -> All]
The first way I got this was doing what my Differential Geometry text
calls a Cartan matrix
( not the Cartan matrix that we use in Lie algebras at all! Which is
closely related to curvature.):
(* SO(2) type matric transform*)
m[t_] = Cos[t]*{{1, 0}, {0, 1}} + Cos[t + 2*Pi/3]*{{0, 1}, {-1, 0}}
(* Diaxial ellipse*)
ParametricPlot[{Cos[t], Cos[t +
2*Pi/3]}, {t, -Pi, Pi}, AspectRatio -> Automatic]
(* the second kind of Cartan Matrix*)
c[t_] = D[m[t], {t, 1}].Inverse[m[t]]
r = {{Cos[Pi/4], Sin[Pi/4]}, {-Sin[Pi/4], Cos[Pi/4]}}
(* Cartan matrix time diaxial ellipse and then negative 45 degree rotation*=
)
ParametricPlot[r.(c[
t].{Cos[t], Cos[t + 2*Pi/3]}), {t, -Pi, Pi}, AspectRatio -> Automatic]
Although they aren't exactly the same this result suggests that
the aerodynamic/ fluid mechanic Jowkouski transform
can be duplicated by some form of matrix transform.
That makes it an interesting problem to find the second kind
of Cartan matrix that is "equivalent"
to a Jowkouski transform.
Roger Bagula
--
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.g=
eocities.com/rlbagulatftn/Index.html
alternative email: rlbagula at ...