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Re: Problem with circles in complex plane
*To*: mathgroup at smc.vnet.net
*Subject*: [mg61141] Re: [mg61095] Problem with circles in complex plane
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Tue, 11 Oct 2005 03:21:20 -0400 (EDT)
*References*: <200510100639.CAA26851@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Daniele Lupo wrote:
> 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
If you split your complex coordinates into pairs of reals the
computation works fine.
cen1 = 1/(1+q) * {reg,img};
rad1 = 1/(1+q)*Sqrt[q^2+q*(1-(reg^2+img^2))];
cen2 = g/(1-(rep^2+imp^2)*(1-g)) * {rep,-imp};
rad2 = Sqrt[1-g]*(1-(rep^2+imp^2))/(1-(rep^2+imp^2)*(1-g));
rul = {reg -> -0.4608904699810983, img-> 0.11491290040984217, q -> 0.3,
rep -> -0.8363463602974097, imp -> 0.16256926406081632};
norm[{a_,b_}] := Sqrt[a^2+b^2]
In[7]:= NSolve[(norm[cen1-cen2] == rad1+rad2) /. rul, g]
Out[7]= {{g -> 0.91362}}
Daniel Lichtblau
Wolfram Research
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