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Could you recommend a rich, clear, and complete theory book on euclidean, affine and projective spaces (i.e., "geometry"); and an interesting exercise book full of non-trivial problems and exercises?

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Here are some Textbooks with this kind of geometry:

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 (I, II)

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

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  • $\begingroup$ Thank you for your answer. However, I personally don't like any of these. Do you have any other suggestions? $\endgroup$ – Dal Mar 11 '15 at 12:34
  • $\begingroup$ Care to elaborate on why you don't like any of these? Thanks. (I understand you wrote this a long time ago. Maybe you've changed your mind?) $\endgroup$ – Pedro Henrique A. Oliveira Jul 27 '18 at 17:09
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What a coincidence! I am studying the same. I am following different books, each with a distinctive flavor. Although I am not sure what would constitute a complete book, here are my picks:

1) I started out with Michele Audin's Book entitled "Geometry". Her emphasis is on linear algebra. It is a 'theorem-proof-theorem-proof' type of a book. So, I felt that book was dry. This is not a criticism of her book, but it is about my reading style. I like historical motivations. It has beautiful exercises. She makes you derive most of the classical theorems.

2) Felix Klein's 'Elementary mathematics from an advanced standpoint-Geometry' is a REAL treat to read. It starts from completely elementary considerations and builds a lot of theory with nice stories and physical intuition. He further lays foundations for 'theory of invariants' towards the end. Be warned, you will not find exercises in this book.

3) Another great book, I believe, is 'Geometry' by Brannan,Esplen,Gray. It gives you both synthetic and analytic flavors of geometry. They explain things out in clear detail. The illustrations and presentations of the topics are awesome. This is the book I go to if I want everything spelt out in detail.

Finally, I am sure all the books by Coxeter do not need any introduction.

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