311 – HWs 6 and 7

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<AC<BD. 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<AC and PQ<BD. The problem is to show that AC<BD.)

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<r. 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.)

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