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.