Re: Circle equation problem
- To: mathgroup at smc.vnet.net
- Subject: [mg61553] Re: Circle equation problem
- From: "Valeri Astanoff" <astanoff at yahoo.fr>
- Date: Sat, 22 Oct 2005 00:35:52 -0400 (EDT)
- References: <dja2c3$fig$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Daniele,
This is the way I would do it, but I hope
someone will post a simpler way :
In[1]:=eq=Abs[p] == Abs[(G - X + I*Y)/(-1 + G*(X + I*Y))];
In[2]:=e1=List@@eq ;
In[3]:=e2=ComplexExpand[e1,{p,G}]
Out[3]=
{Sqrt[Im[p]^2 + Re[p]^2], Sqrt[(Y + Im[G])^2 + (-X + Re[G])^2]/
Sqrt[(-1 - Y*Im[G] + X*Re[G])^2 + (X*Im[G] + Y*Re[G])^2]}
In[4]:=e3=e2 ^2;
In[5]:=e4=e3[[1]]-e3[[2]];
In[6]:=e5=Numerator[e4//Together]
Out[6]=
-X^2 - Y^2 - 2*Y*Im[G] - Im[G]^2 + Im[p]^2 + [...]
In[7]:=cir=(X-XCenter)^2+(Y-YCenter)^2-ray^2 ;
In[8]:=ce=CoefficientList[e5,{X,Y}]//Flatten;
In[9]:=cc=CoefficientList[cir,{X,Y}]//Flatten;
In[10]:=Solve[Thread[ce == cc],{XCenter,YCenter,ray}]//FullSimplify
Out[10]=
{{ray -> -Sqrt[-4 + Abs[p]^2 + 2*G*Conjugate[G]],
XCenter -> (-1 + Abs[p]^2)*Re[G],
YCenter -> (-(-1 + Abs[p]^2))*Im[G]},
{ray -> Sqrt[-4 + Abs[p]^2 + 2*G*Conjugate[G]],
XCenter -> (-1 + Abs[p]^2)*Re[G],
YCenter -> (-(-1 + Abs[p]^2))*Im[G]}}
Of course, only the second solution is to be kept.
hth
v.a.