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I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like to get a better geometric intuition and understanding of the problems that led from one geometry to another (the motivating ideas) before getting deep into purely algebraic theories and forgetting the geometry behind.

I have heard of the following books, though I am not sure which one would fulfill my purpose the best:

  1. Geometry by Brannan - It covers many geometries, though from the commentaries, it seems to be very basic.
  2. Geometry: Euclid and Beyond by Hartshorne - It is written by Hartshorne, who is a famous algebraic geometer, but the book does not seem to cover projective geometry. (Perhaps because Hartshorne has also written a book on the subject alone.)
  3. Geometry: A Comprehensive Course by Pedoe - It contains an introductory chapter on algebraic geometry, but doesn't cover Poincaré's upper half-plane model of hyperbolic geometry.
  4. Geometries by Sossinsky - It covers many different geometries and seems to use a strong algebraic approach, but I am not sure if it is very thorough in projective geometry based on the number of pages on the subject.

Of course, if you have any other recommendations, please tell me! Thank you!

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