Explicit calculation of the center of a circle, image of a circle by a Möbius transformation

Question

It's a warm up calculation I decided to carry out while reading "PCT,Spin and statistics, and all that" by Streater and Wightmann. However I do not find what they have.

p.79 within the proof of Thm 2-14 p.77 (the calculation has not much to do with the proof, at least at this point. But if you are reading the book, notice that in the figure 2-7 p.79 they consider a function of z and $|u|\neq 1$ while just above, it was a function of u on the unit circle...): let's consider the following Möbius transformation

$$ T: z \mapsto \frac{u+z}{1+uz}\ ,\quad |u|\neq 1$$ (otherwise the unit circle is mapped to $\mathbb{R}$, as can be seen by a calculation analogous to the following)

The unit circle is mapped to another circle, whose center I wish to find. I recall that the inverse of a Möbius transformation (in particular, such maps are invertible...) $$ S: z \mapsto \frac{az+b}{cz+d}\ ,\ a,b,c,d \in \mathbb{C} \quad \text{is}\quad S^{-1}: z \mapsto \frac{dz-b}{-cz+a}$$ so in our case (as can also be checked directly) $$ T^{-1}: w \mapsto \frac{w-u}{1-uw}$$ Let's now write the condition for $z$ to be on the unit circle and see what conditions its image $w:=T(z)$ will then satisfy: $$ |z|²= 1 \quad \Leftrightarrow \quad |T^{-1}(w)|^2 =1 \quad \Leftrightarrow \quad \left(\frac{w-u}{1-uw}\right) \overline{\left(\frac{w-u}{1-uw}\right)}=1$$ $$ \Leftrightarrow \quad |w|^2 - 2\, \mathop{Re}(w\overline{u}) + |u|^2 = 1 - 2\, \mathop{Re}(w u) + |uw|^2$$ $$ \Leftrightarrow \quad (1-|u|²)|w|^2 - 2\, \mathop{Re}(w(\overline{u}-u)) + |u|^2 -1 = 0$$ $$ \Leftrightarrow \quad |w|^2 - 2\, \mathop{Re}\left(w\ \frac{2\,i \mathop{Im}(u)}{1-|u|²}\right) -1 = 0$$ Identifying with the equation of a circle of center $c\in \mathbb{C}$ and radius $r \in \mathbb{R}_+$: $$ |w-c|^2=r² \quad \Leftrightarrow \quad |w|^2 - 2\, \mathop{Re}(w \overline{c}) +|c|^2 - r² = 0$$ one obtains $$ c=- \frac{2\,i \mathop{Im}(u)}{1-|u|²} \quad \text{and}\quad r= \sqrt{1 + |c|²}$$

However, in the book it seems that they find $$ c= \frac{4 \left[ u(1+|u|²)- (1+|u|²) \mathop{Re}(u) \right]}{\left[ (1+|u|²)(1+u²) - 4 u \mathop{Re}(u) \right]}$$

So if a benevolent mind double checks the present calculation (or does something of its own), I'll be happy to discuss the result.

Answer 1

I almost agree with your answer, except you made a mistake in the sign: $\bar u-u=-2i\operatorname{Im}u$, not $2i\operatorname{Im}u$, so $$ c=\frac{2i\operatorname{Im}u}{1-\lvert u\rvert^2}. $$

Here is an alternative method.

Since $u\neq\pm 1$, the fixed points $z=T(z)$ is easily seen to be $z=\pm 1$. Hence $T$ maps the unit circle to a circle containing $\pm 1$, so the centre has to lie on the imaginary axis. Moreover, we compute derivative $$ T'(z)=\frac{1-u^2}{(1+uz)^2},\quad T'(\pm1)=\frac{1\mp u}{1\pm u} $$ So the unit circle (with normal direction $1$ at $\pm 1$) is mapped to a circle whose normal direction at $\pm 1$ is given by $\frac{1\mp u}{1\pm u}$. Now $$ \frac{1\mp u}{1\pm u}=\frac{(1\mp u)(1\pm\bar u)}{(1\pm u)(1\pm\bar u)}= \frac{(1-\lvert u\rvert^2)\mp (u-\bar u)}{\lvert 1\pm u\rvert^2} $$ So both normals intersect the imaginary axis at $$ c=\frac{(u-\bar u)}{1-\lvert u\rvert^2}. $$

Answer 2

$-1/u$ is mapped to $\infty$, therefore its conjugate point wrt the unit circle, which is $-\overline u$, is mapped to the center of the image circle.

Answer 3

As shown in this answer, given the LFT $\frac{z+u}{uz+1}$ and the circle with radius $1$ and center $0$, we find the antipodal points $$ 0\pm\frac{0+1/u}{|0+1/u|}\cdot1=\pm\frac{|u|}{u} $$ These points get mapped to antipodal points in the image of $\frac{z+u}{uz+1}$: $$ \frac{u+\frac{|u|}{u}}{1+|u|}\qquad\frac{u-\frac{|u|}{u}}{1-|u|} $$ Therefore, the center is $$\newcommand{\Im}{\operatorname{Im}} \frac12\left(\frac{u+\frac{|u|}{u}}{1+|u|}+\frac{u-\frac{|u|}{u}}{1-|u|}\right) =\frac{2i\Im(u)}{1-|u|^2} $$ and the radius is $$ \frac12\left|\frac{u+\frac{|u|}{u}}{1+|u|}-\frac{u-\frac{|u|}{u}}{1-|u|}\right| =\left|\frac{1-u^2}{1-|u|^2}\right| $$