Lesson on Symmetric Points, Part II
Moebius transformations are homographic.
1may11\\
\textit{ $\C$  2010, Prof. George K. Francis, Mathematics Department,
University of Illinois}

\begin{document}
\maketitle

\section{Introduction}
In the previous lesson we saw the versatility of our new equation for a circline in
the plane,
$\rho z\bar{z} + \mu \bar{z} + \bar{\mu}z + \sigma =0, \rho \ge 0, z\bar{z} \gt \rho\sigma.$
Recall that for  $\rho =0$ this describes a straight line, and otherwise it describes the circle
with
\begin{itemize}
\item  $circ(\rho,\mu,\sigma)$ denoting the circle with
\item Center at $c := -\frac{\mu}}{\rho}$
\item Radius at $r := \sqrt{z\bar{z}-\frac{\sigma}{\rho}}.$
\end{itemize}

Note that we can also write the equation as
$\rho |z|^2 + 2 \mathfrak{Re}( \bar{\mu} z) + \sigma =0$
when that is more suitable. Recalling that
$\mathfrak{Re}( \bar{\mu} z) = \mu \cdot z$, the dot-product of vectors we
might write
$\rho |z|^2 + 2 \mu \cdot z + \sigma =0.$
But while many properties you are familiar with from the vector calclus
carries over, the next is remarkable

Question 1.
Show that $\mu \cdot a z = \mu \bar{a} \cdot z$,
where $u\cdot v = \mathfrak{u\bar{v}}i = \mathfrac{\bar{u}v}.$

In this sublesson, we shall verify that Moebius transformations preserve circlines using
the anatomy lesson where we reduced a MT to a compositions of simpler transformations and
follow the same scheme of substitution as above. In each case, we replace the $z$ in the
equation above by $w$. When it is a
\begin{itemize}
\item Translation: $w = z+b$
\item Rotation:    $w = \beta z, \beta=e^{i \theta}$
\item Dilation: $w = tz, t > 0$
\item Reciprocal: $w = \frac{1}{z}$
\end{itemize}

We argue as follows, suppose $w$ satisfied the equation
$\rho_1 |w|^2 + 2 \mu_1 \cdot w + \sigma_1 =0$ it the circline
$circ(\rho_1, \mu_1, \sigma_1)$. We use subscripts for typographical reasons.
And suppose, for instance, $w=az$ where $a$ is some non-zero complex number.
(This would be the composition of a rotation and dilation.)
Substitute and manipulate to obtain
$(\rho |a|^2) |z|^2 + 2 \mu \cdot az + \sigma =0$, the equation
of a different circline where. Here is the first step.
$\rho_1 |a|^2 |z|^2 + 2 \mu_1 \bar{a} \cdot z + \sigma_1 =0 .$
This is the equation of $circ(\rho_1 |a|^2,\mu_1 \bar{a}, \sigma_1)$.
More algebra (do it in you Journal) yields that
$\rho_1 = \rho/|a|^2, \mu_1 = \mu / \bar{a}, \sigma_1 = \sigma$.
From this you can calculate that the center and radius affected by
the transformation $w=az$ as expected:
$c_1 = ac , r_1 = |a|r.$

Note that the equations of circlines all have  RHS=$0$, so we always
multiply through by a non-zero complex number without changing the
locus of points described by the equation. So the parameters
$\rho_1 = \rho$, $\mu_1 = a\mu$, and $\sigma_1 = \sigma |a|^2$ will
simplify the arithemtic to find the new center and radius.

Question 2.
Find the equation of the circline $circ(\rho, \mu, \sigma)$ under a translation, rotation and dilation. You can do the calculations on scratch paper.

Question 3.
Find the equation of the circline $circ(\rho, \mu, \sigma)$ under the
Moebius transformation $w = \frac{1}{z}$ and compare this algebraic inversion
to the geometric inversion we performed in the previous lesson.