I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (planar, spherical, and hyperbolic). Rice University has changed a lot since this course began being taught (many, many years ago); we now have very few students who want to be high school teachers, and in general the level of our students is such that most of our math majors perceive the course to be beneath them.
My assignment is to redesign the course. I have almost complete freedom except that I cannot require any prerequisites beyond multivariable calculus and ODE's.
Question : What textbook should I use?
Here are my thoughts about what I am looking for.
As I said, I cannot require any prerequisites beyond multivariable calculus and ODE's. However, our undergraduate students are very strong (based on test scores and high school grades, they are pretty similar to the students at eg Cornell or Northwestern). So I want a book that has plenty of meat in it.
It should contain a mixture of proofs and computation, but plenty of proofs.
There are no topics that I am required to cover, though of course it has to be geometric (in particular, this course is not a prerequisite for anything else).
I find axiomatic treatments of geometry boring.
I don't want to develop any machinery unless it has an immediate payoff. However, I am not at all adverse to developing some tools from scratch as long as they lead to something cool.
I want there to be lots of good problems.
Does anyone have any suggestions?