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Re: solving for a second function
*To*: mathgroup at smc.vnet.net
*Subject*: [mg102482] Re: solving for a second function
*From*: Roger Bagula <roger.bagula at gmail.com>
*Date*: Tue, 11 Aug 2009 04:02:39 -0400 (EDT)
*References*: <h5bk2l$hj3$1@smc.vnet.net>
There were several nice replies( the best by David Park).
A solution was found and it had a different series expansion.
The IFS work on kissing figures had one from a Lemniscate
from the C.R.C. Mathemaics tables (very old edition 1960's) that was
developed.
When plotted against the Lemniscate from the complex solutions
the result is strange. They don't appear to be the same curve at all?
Mathematica:
Clear[x, y, x1, y1, a, b, a1, b1]
{a, b} = {Sqrt[Abs[(((2*x)^2 - (2*y)^2)/((2*y)^2 + (
2*x)^2))]]*2*
x*y/(x^2 + y^2) , Sqrt[Abs[(((2*x)^2 - (2*y)^2)/((
2*y)^2 + (2*x)^2))]]*(x^2 - y^2)/(y^2 + x^2)} /. y -
> 1;
g0 = ParametricPlot[{{a, b}, {a, -b}}, {x, -1,
1}, AspectRatio -> Automatic, PlotRange ->
All, Axes -> False, PlotStyle -> Hue[.8]]
x = Re[1 + 1/(Cosh[r/2] + I*Sinh[r/2])];
y = Im[1 + 1/(Cosh[r/2] + I*Sinh[r/2])];
g1 = ParametricPlot[{{y, x - 1}, {y, -x + 1}}, {
r, -5*Pi, 5*Pi}, AspectRatio -> Automatic, Axes -> False,
PlotStyle ->
Hue[.6], PlotRange -> All]
Show[{g0, g1}]
The problem here is are these the same family of curves like ellipses
and circles,
or is the classical Bernoulli Lemniscate a different curve altogether
than
this complex one? Maybe the 2d for the IFS was developed wrongly?
Roger Bagula
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