# Circle in the complex plane

1. Jan 24, 2009

### f95toli

I have a set of experimental data that I am trying to fit to an equation; but I have what I believe is a a silly problem (something I would probably have solved in 5 minutes 10 years ago....)
Anyway, the data set consists of a complex impedance as a function of a normalized frequency (the parameter x below).
Now, if I plot my data in the complex plane (i.e. Im(f) vs. Re(f) ) I get a nice circle which I can easily fit using a standard algorithm; from this fit I get the coordinates of the centre of the circle (C, a complex number) and its radius (R)

$|z-C|=R^2$.

Now to my problem. The equation I am trying to fit to is

$\frac{1+a_2xi}{a_1+a_2xi}$

which describes a circle in the complex plane when plotted as a function of the parameter x (but the points are not equidistant).
The goal is to determine $a_1, a_2$

How do I rewrite this equation in a form where I can use my known values (R and C) to determine $a_1, a_2$?

I am really stuck, I realize that the equation looks like a bilinear transformation but I am not even sure if that helps or not(?)