## 311 – HWs 6 and 7

April 15, 2015

HW 6 is due Thursday, April 16 and HW 7 is due Tuesday, April 21, both at the beginning of lecture.

HW6

Work in hyperbolic geometry.

1. Let $\ell$ and $m$ be two parallel lines admitting a common perpendicular: There are points $P\in\ell$ and $Q\in m$ with $\overleftrightarrow{PQ}$ perpendicular to both $\ell$ and $m$. Suppose that $A,B$ are other points in $\ell$ with $P*A*B$, that is, $A$ is between $P$ and $B$. Let $C$ be the foot of the perpendicular from $A$ to $m$, and let $D$ be the foot of the perpendicular from $B$ to $m$.

Show that $PQ. That is, $\ell$ and $m$ drift apart away from their common perpendicular.

(Note that $\Box PACQ$ and $\Box PBDQ$ are Lambert quadrilaterals, and therefore $PQ and $PQ. The problem is to show that $AC.)

As an extra credit problem, show that for any number $r>0$ we can find $B$ (on either side of $P$) such that $BD>r$, that is, $\ell$ and $m$ not just drift apart but they do so unboundedly.

2. Now suppose instead that $\ell$ and $m$ are critical (or limiting) parallel lines, that is, they are parallel, and if $Q\in m$ and $P\in\ell$ is the foot of the perpendicular from $Q$ to $\ell$, then on one of the two sides determined by the line $\overleftrightarrow{PQ}$, any line through $Q$ that forms with $\overleftrightarrow{PQ}$ a smaller angle than $m$ does, cuts $\ell$ at some point.

On the same side as just described, suppose that $C,D$ are points on $m$ with $Q*C*D$, that is, $C$ is between $Q$ and $D$. Let $A$ be the foot of the perpendicular from $C$ to $\ell$, and let $B$ be the foot of the perpendicular from $D$ to $\ell$.

Show that $PQ>AC>BD$. That is, $\ell$ and $m$ approach each other in the appropriate direction.

As an extra credit problem, show that for any $r>0$ we can choose $D$ so that $BD. That is, $\ell$ and $m$ are asymptotically close to one another. Do they drift away unboundedly in the other direction?

HW 7

Show that the critical function $\kappa$ is continuous. Recall that $\kappa:(0,\infty)\to(0,\pi/2)$ measures the critical angle, that is, $\kappa(x)=\theta$ iff there are critical parallel lines $\ell$ and $m$ and a point $Q\in m$ such that if $P$ is the foot of the perpendicular from $Q$ to $\ell$, and $PQ=x$, then $m$ and $\overleftrightarrow{PQ}$ make an angle of measure $\theta$ in the appropriate direction.

(In lecture we verified that $\kappa$ is strictly decreasing. This means that the only possible discontinuities of $\kappa$ are jump discontinuities. We also verified that $\kappa(x)$ approaches $0$ as $x\to\infty$, and approaches $\pi/2$ as $x\to0$. It follows that to show that $\kappa$ has no jump discontinuities, it suffices to verify that it takes all values between $0$ and $\pi/2$, that is, one needs to prove that for any $\theta\in(0,\pi/2)$ there is an $x>0$ such that $\kappa(x)=\theta$.)