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MathGroup Archive 2005

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Re: Problem with circles in complex plane

  • To: mathgroup at smc.vnet.net
  • Subject: [mg61310] Re: Problem with circles in complex plane
  • From: Daniele Lupo <danwolf80_no_spam_ at libero.it>
  • Date: Fri, 14 Oct 2005 22:22:45 -0400 (EDT)
  • References: <did2ob$qd2$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Il Mon, 10 Oct 2005 06:46:35 +0000 (UTC), Daniele Lupo ha scritto:

> Hi to all the Mathematica's Masters.
> 
> I've a problem working with circles in the Complex Plane.

[CUT]

> 
> PS: I'm using Mathematica 5.1 on WinXP Home

I've resolved my problem.

I've used the equation

Abs[CF-Cg]==RF+Rg

But I've not used original values. I've tried to simplify equations, to
create a more simple expression in terms of LeafCount; after this, I've
squared it, to toggle Abs and obtain an algebraic equation:

simplified = Sqrt[(a - (b*g)/(1 - c*(1 - g)))^2 + 
   (d + (e*g)/(1 - c*(1 - g)))^2] == 
 f + (Sqrt[1 - g]*h)/(1 - c*(1 - g))


where

{a = Re[Ã0]/(1 + N), b = Re[S11], 
  c = Abs[S11]^2, d = Im[Ã0]/(1 + N), e = Im[S11], 
  f = Sqrt[N^2 + N*(1 - Abs[Ã0]^2)]/(1 + N), 
  h = 1 - Abs[S11]^2}

Then I've resolved simplified

sol = Solve[simplified, g]

And then, with Rule, I've replaced letters with their values, in this way:

sol /. {a -> Re[Ã0]/(1 + N), b -> Re[S11], 
   c -> Re[S11]^2 + Im[S11]^2, d -> Im[Ã0]/(1 + N), 
   e -> Im[S11], 
   f -> Sqrt[N^2 + N*(1 - Im[Ã0]^2 - Re[Ã0]^2)]/
     (1 + N), h -> 1 - Im[S11]^2 - Re[S11]^2}

I obtain four solutions: I've replaced numerical values, and I've choosen
solution that had given me what I want (fourth solution of the list).

I've obtained a very long expression, that my computer can't simplify
(after 28 hours, I've aborted Simplify command), but I've tried to simplify
all list... next days I'll try to simplify only the right solution.

Sorry if I've replied only today, but at my home near university I've not
an Internet connection.

Have a nice day

Daniele


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