Textbook for undergraduate course in geometry

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.

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):

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

2. 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)

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

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

5. 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:

1. Michele Audin - Geometry
2. Elmer Rees - Notes on geometry
3. Gruenberg, Weir - Linear geometry
4. Jean Gallier - Geometric Methods and Applications
5. Mark Steinberger - A course in low-dimensional geometry
6. Tarrida - Affine maps, Euclidean motions and Quadrics
7. Dieudonne - Linear algebra and geometry
8. Berger - Geometry
9. 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.

• Introduction to Geometry by Coxeter.
• Elementary Geometry From An Advanced Viewpoint by Moise.
• Geometry: Euclid and Beyond by Hartshorne.