Chapter VIII: Frogs and their tongues (hyperbolic geodesics). (Image courtesy of Talia Blum. Used with permission.)
My vague impression is that most undergraduate geometry teaching is centered either on Euclidean/spherical/hyperbolic geometries, or on curved surfaces. As you’ve realized by now, this course is not of either kind, but we have time for a bit of each subject.
There are various ways of drawing the hyperbolic plane in the ordinary Euclidean one; obviously none of them works perfectly. Here we stick with the half-plane model, which is what you are most likely to see where hyperbolic geometry intersects with other parts of mathematics such as number theory. The downside is that we have to omit regular hyperbolic polygons, since those don’t look particularly symmetric in the half-plane picture. I’m not a regular user of hyperbolic geometry, so I breathe a sigh of relief when we reach its symmetries, which fit into a more general scheme of group actions. Students seem to have the opposite reaction: they often find it hard to use such transformations creatively to simplify computations. If their school teachers have spent long hours discussing Euclidean transformations, to get them into the required Felix-Klein-ian mindset, those seeds may have got lost in the thicket. (As I write this, I am ruefully looking out onto my front yard, hence the metaphor.) But really, any blame rests with the class design, as this chapter seems to be lacking “je ne sais quoi” to give it a clearer intellectual direction.
I have found C. Series (with S. Maloni and K. Farooq), Hyperbolic geometry (Lecture notes for the University of Warwick course MA 448), 2010, a helpful reference, especially for the various formulae involved. Of course, there are many other textbooks on hyperbolic geometry, usually aiming much higher than we do here.
Lecture 34: The Hyperbolic Plane
