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

**References**:**Problem with circles in complex plane***From:*Daniele Lupo <danwolf80_no_spam_@libero.it>