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Problem with circles in complex plane
*To*: mathgroup at smc.vnet.net
*Subject*: [mg61095] Problem with circles in complex plane
*From*: Daniele Lupo <danwolf80_no_spam_ at libero.it>
*Date*: Mon, 10 Oct 2005 02:39:56 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Hi to all the Mathematica's Masters.
I've a problem working with circles in the Complex Plane.
I hope that someone of you can help me.
I've these values of centers and rays
CF = G/(1 + Q);
RF = (1/(1 + Q))*Sqrt[Q^2 + Q*(1 - Abs[G]^2)];
Cg = (g*Conjugate[P])/(1 - Abs[P]^2*(1 - g));
Rg = (Sqrt[1 - g]*(1 - Abs[P]^2))/(1 - Abs[P]^2*(1 - g));
I want to find the g parameter, to find a circle of the family (Cg,Rg)
that's tangent to the first circle.
G, P are complexes, with absolute value less than 1, while Q is a positive
real number.
To do this, I use this formula:
tangentray = Abs[CF - Cg] - RF
to find the ray of the tangent circle that depends of parameter. Then I use
sol = Solve[tangentray == Rg, g]
to find the g value that make these two rays equals. In this way I should
to find the value of g that identifies the tangent circle, right? I think
yes, but maybe I'm wrong... Probably it's so, but I don't know why.
The problem is that Mathematica returns me a list of four values, that are
complexes, whike I know that there MUST be a real value of g for which this
tangent circle exist. You can see this example:
rul = {G -> -0.4608904699810983 + 0.11491290040984217*I, Q -> 0.3, P ->
-0.8363463602974097 + 0.16256926406081632*I}
Show[
Graphics[
{
(* Family of Circles depending on g *)
Red, Circle[{Re[Cg /. rul], Im[Cg /. rul]}, Rg /. rul] /. {{g -> .8},
{
g -> 0.9}, {g -> 0.94}},
(* Fixed Circle *)
Blue, Circle[{Re[CF /. rul], Im[CF /. rul]}, RF /. rul]
}
], AspectRatio -> Automatic]
In this case you can see that, for a value of g between 0.9 and 0.94 there
must be a tangent circle.
Then, I find the g value
sol = Solve[ray == Rg, g]
and, with the same values, I obtain
sol /. rul
that returns
{{g -> -0.19317892523624003 + 0.14313227060687864*
I}, {g -> 0.9584729373799866 -
0.44878073249387535*I},
{g -> 0.22609724512769427 - 0.07178705469686383*
I}, {g -> -0.36357640440949757 +
0.07475060907387349*I}}
As you can see, noone of g values is real, around 0.9.
So, what's wrong in things that I do? How can I find the correct value og g
parameter?
Thanks for your answers
Daniele
PS: I'm using Mathematica 5.1 on WinXP Home
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