Sample Quiz Questions on Moebius Transformations

\begin{document}
\maketitle
\textbf{ $\C$  2011, Prof. George K. Francis, Mathematics Department, University of Illinois}

\section{Introduction}
Here we collect some typical problems to familiarize the reader with the
way questions are posed and answers are expected. [Items in brackets refer
to section in Hvidsten's tex.]

\begin{itemize}

\item \textbf{Problem 1:  }
Complex numbers and (plane) vectors provide complementary
descriptions of the Euclidean plane once we identify $z$
with the vector $(\Re(z), \Im(z))$, and conversely, the
vector $(x,y)$ with the complex number $x+iy$. [Hvisdsten 3.5]

\item (1a) Show that the dot product of two vectors equivalent to
the complex numbers  $z=x+iy$ and $w=u+iv$ can be written as
$\Re(z\bar{w})$ in complex form.
\item (1b) Use (1a) to show that two lines through the origin,
$\ell_{0z}$ and $\ell_{0w}$ are perpendicular if and only
if $w=tiz$ for some real number $t$.
\item (1c) Verify (1b) another way by expressing $ti$ in polar form
and interpreting $w=tiz$ as a rotation of the plane.

\item \textbf{Problem 2:  }
Recall that points $z$ and $z^*$ are defined to be \textit{ symmetric}
with respect to the unit circle if they lie on the same ray from the
origin, and their distances from the origin are reciprocal. (Because
reciprocal and inverse are English synonyms, the transformation of
the plane that takes each point of the extended plane to its symmetric
point is also called an inversion.)

\item (2a) Construct and label a figure that defines the inversion $z^*$ of
a point $z$ in the unit circle.

\item (2b) Suppose $z$ is inside the unit disk, and $z^*$ is outside. Complete
your figure in (2a) to a Thales right triangle, with the tangent from $z^*$
to the circle forming its long side.

\item (2c)
From the definition (above) calculate that
$z^* = \frac{z}{|z|^2} = \frac{1}{\bar{z}}$. (Supply the missing steps.)

\item \textbf{Problem 3: }
A theorem [Hvidsten 8.8]  says that $z,z^*$ are symmetric with respect to a
circline through 3 points $a,b,c$  if and only if
$CR(z^*,a,b,c) = \overline{CR(z,a,b,c)}$.

\item (3a) Apply this theorem to show that for every $z\in \mathbb{C}$ the
pair $z, -\bar{z}$ is symmetric relative to the imaginary axis in the
complex plane.  Hint: Find three convenient(!) points $a,b,c$ on the
imaginary axis and solve the defining equation.
Don't forget that $\infty$ is a legitimate point in the extended complex plane.

\item (3b) Draw a figure that illustrates that $z^*$ in (3a) is a Euclidean
reflection of $z$ in the y-axis.

\item \textbf{Problem 4:}
A theorem [Hvidsten 8.5]  says that every MT $w=f(z)$ and any four points
$a,b,c,d$ in the plane, $CR(a,b,c,d) = CR(f(a),f(b),f(c),f(d))$. (We say
that the cross ratio is invariant under Moebius transformations.)

\item (4a)  Use this theorem to solve the equation
\begin{eqnarray*}
CR(z, \frac{1}{1- ri},\frac{1}{1- si},\frac{1}{1- ti}) = CR(w,r,s,t) \\
\end{eqnarray*}
for  $w = f(z)$, but do this \textbf{without} computing any cross
ratios or solving messy equations.
Hint: Transform the LHS  by a succession
of similarities (define it) and multiplicative inverses (reciprocals),
until it looks like the RHS. Then read off the expression in the first
position to deduce $w$ as a function of $z$, $w=f(z)$.

\item (4b) What sort of function is the $w=f(z)$ found in (4a)?
What are some of its geometric properties?
Note: You do need not have solved (4a) to answer (4b).

\item (4c) Show that the three points
$n=\frac{1}{1- ri}, p=\frac{1}{1- si}, q=\frac{1}{1- ti}$ lie on
the circle symmetric
with the x-axis, and through the origin and the unit
point (i.e. the circle centered
at 1/2 and of radius 1/2.) Hint: The points
$\{1+ti | t \in \mathbb{R} \}$ lie on
the vertical line $x=1$. Why? Points $q, 1+ti$  are symmetric with respect
to the unit circle. Why? So the inversion of the line in the unit circle
is $circ(n,p,q)$. Why? And that is the predicted circle. Why?

\item (4d) Draw a persuasive figure for (4c), illustrating the geometric
relation between the six points. Note: You need not have proved
the conclusions in (4c) to use them here!

\end{itemize}

\end{document}

\end{itemize}

\end{document}

\end{document}