Textbook for undergraduate course in geometry 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?
 A: *

*Introduction to Geometry by Coxeter.

*Elementary Geometry From An Advanced Viewpoint by Moise.

*Geometry: Euclid and Beyond by Hartshorne.

A: I would recommend Continuous Symmetry : From Euclid to Klein by Barker and Howe.  I took the course as an undergraduate and enjoyed it very much.  The first chapter gives an axiomatic treatment of geometry, and is meant to be a short part of the course.  The rest of the book is a transformational approach to geometry, introducing isometries and similarities.  Felix Klein's Erlanger Programm is the guiding principle for the course. 
A: Here are few ideas:
1) I like the idea of a course about polytopes. Few books but some are excellent: "Lecture on polytopes" by Ziegler or "Convex polytopes" by Grunbaum are the obvious choices.
2)  A course about curves and surfaces + an introduction to manifolds should satisfy 1-6 without troubles. "Differential geometry of curves and surfaces" by Do Carmo is a very good book; there are plenty of excellent books about manifolds. 
3) A basic course on algebraic varieties require the use of algebra and differential calculus and gives example of spaces with pathological spaces (i.e. non Hausdorff and/or with singularities)
4) I guessed you want a more modern geometry course but without leaving the view of the formation of high school teachers. Michèle Audin wrote a very good book about affine, projective, curves and surfaces. It is aimed to future (French) high school teachers. I guess the title is "Geometry" (it is "Géométrie" in the French version).
I don't know the curriculum of a typical American student so I hope my suggestions are still pertinent (especially the point 3).
A: How about John McCleary's book ``Geometry from a differentiable viewpoint''. I've always thought that would be a nice basis for a course like this. 
A: Last semester I taught (at Colby College) a geometry course based on two books: Bonahon's "Low dimensional geometry" and Schwartz's "Mostly Surfaces". Both are relatively inexpensive as far as textbooks go, so I could require both from the students. The students really enjoyed reading both books simultaneously as the authors have very different styles but some overlap of content. The students would certainly need to know some linear algebra in addition to multivariable calculus. The course was challenging, but reasonably successful at helping students develop some "geometric imagination" and proof-writing skills. 
A: Have you seen the book "Geometry by Its History" by Ostermann/Wanner? It looks anything but boring. I liked it very much!  It has plenty of interesting mathematics.  Maybe you record the lectures and put them on net ...
A: I like Euclidean and Non-Euclidean Geometries: Development and History by Marvin J. Greenberg.  I will warn you: it is certainly an axiomatic treatment.  However, I really enjoyed the way that the book develops it.  For example, the distinction between the axioms of a geometry and theorems you can prove about them, versus the models of geometry and their various properties, is clearly drawn.  I dare say that, despite how advanced your undergraduates feel, they will learn a lot about the axiomatic method from this book.  I recommend that you give it a look; even if it is not the primary textbook for the course, you can use it as a convenient source of motivation, problems, examples, and history.  (There is a lot of history in this book, and many exercises.)
A: I wonder whether Igor Pak's "Lectures on Discrete and Polyhedral Geometry" might be appropriate as a textbook for an undergraduate geometry course. This is still in preliminary form, available on his website.  In the introduction he describes a selection of topics from the book that could be used for a basic undergraduate course. There seem to be lots of exercises, and at a quick glance a lot of the topics look quite interesting.  This whole subject is way outside my expertise, however, so I have no idea if the book would make a good basis for a course like the one you'll be teaching.
A: Concerning the fourth item in your list, I believe that axiomatic approach is not only boring, but (and it is more important) it is almost useless for further mathematical courses.
In my view the best geometry you can teach your first year undergraduates is the one based on modern treatment of linear algebra. The syllabus might look like this (it is based on the course I've taken in the recent years):


*

*The language of vector spaces and linear transformations
(bases, determinants, dual spaces)

*Euclidean structure (Gram matrices, orthogonal bases, orthogonal projections, any orthogonal operator is a composition of reflections in hyperplanes, orthogonal operator acts as a rotation in two-dimensional subspaces)

*Affine geometry and topology (norms, metrics, topology; convex sets, supporting halfspaces; polytopes as intersections of halfspaces)

*Projective spaces (homogeneous coordinates, atlases on projective space, Veronese embedding, projective transformations, duality of points and hyperplanes)

*Projective and affine quadrics and conics(rank, kernel; tangent space to a quadric; polar transformations; pencils of quadrics)
Textbooks with this kind of geometry are:


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*Michele Audin - Geometry

*Elmer Rees - Notes on geometry

*Gruenberg, Weir - Linear geometry

*Jean Gallier - Geometric Methods and Applications

*Mark Steinberger - A course in low-dimensional geometry

*Tarrida - Affine maps, Euclidean motions and Quadrics

*Dieudonne - Linear algebra and geometry

*Berger - Geometry

*Vinberg - A course in algebra, chapter "affine and projective spaces"


Such a course would give your students better understanding of the geometric nature of linear algebra (personally I think that the material one learns in a linear algebra course should be called "linear geometry"), it would show how modern mathematics simplifies classic material such as euclidean geometry and it would provide strong geometric basis for courses like algebraic geometry and topology (where familiarity with projective spaces helps a lot).
I suggest reading the preface to Dieudonne's book where he elaborates on these issues.
A: 

*Jean Frenkel – Géométrie pour l'élève professeur

*Claude Tisseron - Géométries Afﬁnes, Projectives, et Euclidiennes

*Eduardo Casas-Alvero - Analytic Projective Geometry

*Ernst Snapper, Robert J. Troyer - Metric Affine Geometry

A: I'm a a fan off
  Episodes in 19th and 20th Century Geometry [Ross Honsberger]
It may have an old-fashioned outlook, but you would have to have simply amazing undergraduate students for none of them to find this material a challenge.
I would supplement this with perhaps some chapters from Miles Reid's Undergraduate Algebraic Geometry (especially his treatment of the 27 lines on a cubic surface)
and perhaps Alain Connes' little paper on Morley's Theorem.
I realize that most people prefer more foundation/methodological approaches (comparisons of various axiomatic systems).  My bias: I want undergraduates to see more phenomena.  But for that other sort of course, I like the
 Strasbourg Master Class on Geometry.     
