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A concise reference for the basics of the subject,
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This review is from: Projective Geometry (Paperback)
This book takes the axiomatic approach to the build-up of projective geometry, which has its roots in the work of von Staudt. It's an alternative to the coordinate or "analytic" approach which is found in many older texts e.g. Todd . However, coordinates are covered toward the end of the book. I found the treatment of polarities particularly useful: these are developed first, independently of conics, which are then shown to result from polarities. Also illuminating is the treatment of affine and Euclidean geometry, which are introduced as specialisations of projective geometry. By singling out a plane at infinity (so you can define parallels) you get affine geometry; then by specifying within this a particular polarity you get a definition of right angles, and from that the Euclidean angle and distance measures. In my own (doubtless naive) view you can take these "geometries" as different levels of description of the same underlying "stuff", i.e. flat three dimensional space.
This line of development makes for a very concise presentation. For example, in the purely projective view there is just one kind of conic: ellipses, hyperbolas and so on are only distinguished when the line at infinity is specified. Which said, it might not be the easiest read for newbies. For that I'd recommend Olive Whicher . But a very solid and useful reference and definitely worth a place on the shelf.
 Todd J., Projective and Analytical Geometry; London: Pitman 1954
 Whicher O., Projective Geometry: Creative Polarities in Space and Time; London: Rudolf Steiner Press 1971