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\newtheorem{ex}[thm]{Exercise}

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height7pt \kern7pt \vrule width.4pt} \hrule height.4pt}}}$}

\def\R{\mathbb R}
\def\Z{\mathbb Z}
\def\CP{\mathbb {CP}}
\def\H{\mathbb H}
\def\E{\mathbb E}
\def\S{\mathbb S}
\def\F{\mathcal F}
\def\D{\mathcal D}
\def\C{\mathbb C}
\def\Q{\mathbb Q}
\def\A{\mathbb A}
\def\til{\widetilde}
\def\M{\mathcal M}
\def\O{\mathcal O}
\def\N{\mathbb N}
\def\RP{\mathbb{RP}}
\def\T{\mathcal T}
\def\U{\mathcal U}
\def\V{\mathcal V}
\def\G{\mathcal G}
\def\Aut{\text{Aut}}
\def\Inn{\text{Inn}}
\def\Out{\text{Out}}
\def\Isom{\text{Isom}}
\def\Sym{\text{Sym}}
\def\Aff{\text{Aff}}
\def\til{\widetilde}
\def\1{{\bf 1}}
\def\id{\text{id}}

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\title{Homework 2 --- the Euclidean plane}

\author{Danny Calegari}
\address{Department of Mathematics \\ Harvard \\ Cambridge, MA 02138}
\email{dannyc@math.harvard.edu}

\maketitle

This homework is due October 6th at the start of class. Recall that $\Aff(\E^2)$ denotes
the group of {\em affine transformations} --- i.e. transformations preserving straight lines and
incidence properties --- of $\E^2$, $\Sym(\E^2)$ denotes the group of
{\em similarities} --- i.e. transformations preserving angles and ratios of lengths --- 
and $\Isom(\E^2)$ denotes the group of {\em isometries} --- i.e. transformations preserving
angles and lengths. For a group $X(\E^2)$, the subgroup of orientation--preserving
elements is denoted $X^+(\E^2)$.

\begin{prob}\label{preserve_line}
Let $H$ denote the subspace of $\E^2$ consisting of pairs of points $(x,y)$ with
$y\ge 0$. Let $G$ be the subgroup of $\Sym^+(\E^2)$ which stabilizes $H$; that is, which
takes $H \to H$ in a $1$--$1$ and invertible fashion. Show that $G$ is isomorphic to
$\Aff^+(\E^1)$. In this way identify the ``space of (oriented) lines'' in $\E^2$ with
the coset space $\Sym^+(\E^2)/\Aff^+(\E^1)$.
\end{prob}

\begin{prob}
Using the fact that $G$ is a group, prove the following geometric theorem: let
$C_1,C_2,C_3$ be three circles of different radii in $\E^2$ with disjoint interiors, and
for each pair $C_i,C_j$ let $l_{ij}^k$ with $k=1,2,3,4$ be the $4$ lines which are
tangent to both $C_i$ and $C_j$. These intersect in $6$ points, but only $2$ of these
points are on the line joining the centers of $C_i$ and $C_j$. Call these two special
points $p_{ij}^1$ and $p_{ij}^2$. In this way we get $6$ points,
$p_{12}^1,p_{12}^2,p_{13}^1,p_{13}^2,p_{23}^1,p_{23}^2$. Call this collection of points $P$.
Show that there are $4$ special lines, each of which intersects $P$ in $3$ points.
\end{prob}

\begin{prob}
Identifying the group of translations with $\C,+$ and the group of similarities fixing a point
$p$ with $\C^*,\times$, show that exponentiation (i.e. $z \to e^z$) 
defines a homomorphism $\C,+ \to \C^*,\times$. What is the kernel of this homomorphism? Show
that it is onto. Use this to show that there are subgroups of $\C^*,\times$ which are isomorphic
to the group $\Z \times \Z$. What does the orbit of a point $q \in \E^2$ (with
$p \ne q$) look like under the
action of such a $\Z \times \Z$ in $\C^*$?
\end{prob}

\begin{prob}
Fix some point $p \in \E^2$.
Show that every element of $\Isom^+(\E^2)$ can be written as a product $t_{x_0,y_0}r_\theta$
where $r_\theta$ denotes the rotation through angle $\theta$ about the point
$p \in \E^2$, and $t_{x_0,y_0}$ denotes translation through the vector $(x_0,y_0)$.

Think of $\E^2$ as the plane $z=1$ in $\R^3$. Using this identification, show that
there is an isomorphism between $\Isom^+(\E^2)$ and the group of $3 \times 3$ matrices of the
form
$$\left[\begin{matrix}
\cos(\theta) & \sin(\theta) & x_0 \\
-\sin(\theta) & \cos(\theta) & y_0 \\
0 & 0 & 1 \\
\end{matrix}\right]$$
\end{prob}

\begin{prob}
Show that there is a short exact sequence (that is, the image of one homomorphism is the kernel
of the next)
$$1 \stackrel{i} \longrightarrow \R^2 \stackrel{j} \longrightarrow \Aff(\E^2) \stackrel{\phi}
\longrightarrow GL(2,\R) \stackrel{\psi}\longrightarrow 1$$
where the homomorphisms $i,j$ are inclusions, and $\phi,\psi$ are quotients.

Show that this short exact sequence {\em splits}; that is, there is a homomorphism
$$s:GL(2,\R) \to \Aff(\E^2)$$ such that $\phi \circ s = \text{id}$, and therefore that
$\Aff(\E^2)$ is a semi--direct product of $GL(2,\R)$ and $\R^2$. What is the induced homomorphism
$GL(2,\R) \to \Aut(\R^2)$?
\end{prob}

\begin{prob}
Show that every translation can be written as a product of $2$ reflections. Show that
every rotation can be written as a product of $2$ reflections.
\end{prob}

\begin{prob}
Show that any two nonzero translations are conjugate in $\Aff(\E^2)$. Think of
$\Sym(\E^2)$ as a subgroup of $\Aff(\E^2)$, and therefore think of the rotations in
$\Sym(\E^2)$ as a particular subset of elements of $\Aff(\E^2)$. When are two rotations
conjugate in $\Aff(\E^2)$? (Hint: the trace of a $2 \times 2$ matrix is invariant under
conjugation by an element of $GL(2,\R)$.)
\end{prob}

\begin{prob}[Hjelmslev's theorem]
A {\em glide reflection} is the composition of two isometries: a reflection in a line $l$
and a translation in a direction parallel with the line $l$. $l$ is said to be the {\em axis}
of the glide reflection.

Suppose $i \in \Isom(\E^2)$ is orientation--reversing. Show that $i$ is either a reflection
or a glide reflection. For any point $x$, show that the midpoint of $xi(x)$ lies on
the ``axis'' of $i$ (either the axis of reflection or the axis of the glide reflection).
Let $pq$ and $p'q'$ be two line segments of equal length. Show there is a unique
orientation--reversing transformation of $\E^2$ to itself taking $p \to p'$ and $q \to q'$.
Deduce that for any pair of line segments $pq,p'q'$ of equal length, and points
$r \in pq,r' \in p'q'$ with $|pr|=|p'r'|$ and $|rq| = |r'q'|$, the midpoints of the
segments $pp',qq',rr'$ are collinear (they might be coincident).
\end{prob}

\begin{prob}[Hard]
The function $f_n:z \to (1 + \frac z n)^n$ is a function from $\C$ to itself which is
generically $n \to 1$. Let $S^1_n$ denote the circle of radius $n$ in $\C$ with center
$-n$. Let $\alpha_n$ be the rotation with center $-n$ through an angle of $2\pi/n$. 
Identifying $\C$ with $\E^2$, show
that the sequence $\alpha_n$ of isometries {\em has a well--defined limit}. That is, show
for each point $p$ that the sequence of points $\lim_{n \to \infty} \alpha_n(p)$ 
limits to some point $p_\infty$. Moreover,
show that the transformation $p \to p_\infty$ is an isometry $\alpha_\infty$ of $\E^2$.
What is $\alpha_\infty$? What does this have to do with the convergence
$$\lim_{n \to \infty} \Bigl(1 + \frac z n\Bigr)^n \to e^z$$
\end{prob}
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