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}