Homework 2

Homework 2
Due Thursday, February 21. Be sure to start it now so you can ask questions. If it turns out to be too long or too difficult I will shorten it. Joint work is allowed - even encouraged.

1. Let $P$ and $Q$ be two points in the unit disk. Find a Euclidean construction for the (hyperbolic/pseudo) line joining them. That is, find a straightedge-and-compass construction for the circle through those points that meets the unit circle at a right angle. (This is not easy. After trying it yourself (or with a classmate) feel free to try to find a solution on the web. If you do, you should write it out in your own words (and pictures) and tell me where and how you found it.)
2. Calculate the (hyperbolic/pseudo) distance from the center of the unit disk to a point at (Euclidean) distance $r < 1$ from the center. Hint. Your argument should lead to an integral that you can evaluate using what you learned in Math 141. If you don't remember what you need you should be able to look it up.
3. Use your answer to the previous problem to show that lines through the center are infinitely long - in other words, that the distance from the center approaches $\infty$ as $r \rightarrow 1$.
4. Find a place in Chapter 1 where the authors refer you to some external source for mathematics (or history) you're really curious about. Look it up and report on what you found.
5. Research problem: what do pseudocircles look like in the Poincare model? (A circle is the set of points at a fixed distance from a given point. But distances are distorted in the Poincare model, so a pseudocircle won't look like a circle unless it happens to be centered at the origin.) Note: I didn't know the answer to this problem when I asked it. I started my answer by drawing some good pictures … then I made a conjecture and lazily confirmed it with a web search.
6. Optional hard problem: calculate the distance between two points. This may be easier if the points are equidistant from the center. In general, this problem is probably easier in polar coordinates. You can assume (from symmetry) that one of the points is on the x axis, so you need to find the pseudoline through $(r, 0)$ and $(s, \theta )$ and then its length.