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I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum.

For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's not very connected to what we study but I could be wrong as I've only began recently.

Recently, in my "intro to proofs" course we studied plane isometries from a classical approach, and found it piqued my interest. I assume it has a modern treatment somewhere and generalizations?

Thus, I'm looking for literature involving:

  • Analytic Geometry (formal treatment of $n=2$ and $n=3$ would suffice). Atleast containing derivations of conics, quadratic surfaces, vector treatment of plane/space geometry. (to my understanding it's sometimes studied in highschool, perhaps there's a treatment an undergrad would enjoy?).
  • Transformations(*) and their derivations in euclidean space such as dilations, rotations, reflections, shear etc from coordinates point of view. Their derivations using matrices or complex numbers (as mentioned in wiki articles).

(*) Linear Algebra books which contains these applications in geometry are acceptable. I have browsed the table of contents of several reccomended introductory books on linear algebra but they don't seem to cover the geometric perspective. Correct me if I'm wrong.

Edit: One book is out of reach, the other seems above my current level, so I've changed the requested literature slightly.

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  • $\begingroup$ From your second bullet, it seems you should pick up any book on linear algebra. I'm pretty sure there are numerous lists of books on linear algebra on this site already. $\endgroup$ – rschwieb Jan 10 '14 at 15:32
  • $\begingroup$ The transformations I'm refering to are not necessarily linear, and I haven't seen these subjects in the tables of contents of some LA books. could you be more specific? $\endgroup$ – user7610 Jan 10 '14 at 15:38
  • $\begingroup$ OK, I understand :) But you do realize that dilations, rotations, reflections and shear are all linear transformations? Maybe you are more interested than you previously thought. $\endgroup$ – rschwieb Jan 10 '14 at 15:47
  • $\begingroup$ I'm indeed interested, but to my knowledge (haven't took LA yet) T(0)=0 for all Linear transformations, which is not always true in the ones mentioned here. $\endgroup$ – user7610 Jan 10 '14 at 15:52
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    $\begingroup$ OK, I take it you are defining them about arbitrary points in space. This means that they are the composition of a linear transformation and a translation, so while not strictly linear themselves they're still well within the realm of linear algebra. You might look up affine transformation article to see how matrices are used to encode these linear-plus-translation transformations. $\endgroup$ – rschwieb Jan 10 '14 at 15:57
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The book you are looking for is Emil Artin's Geometric Algebra. The book starts by showing how to derive the coordinate description of plane geometry from the usual axiomatization in terms of points and lines. The notions of translations and dilations are fundamental to this derivation. The second part of the book discusses some of the other transformations in the context of projective geometry.

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    $\begingroup$ This is an awesome book that I love, but I would definitely not have appreciated it if I were at the level the OP is now :) Still, a good book is a good book, and I hope @yaniv tries it out sometime or another. $\endgroup$ – rschwieb Jan 10 '14 at 15:50
  • $\begingroup$ what are the prerequisites? $\endgroup$ – user7610 Jan 10 '14 at 16:05
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    $\begingroup$ @Yaniv.Fish In theory you could do this without prerequisites if you enjoyed reading it. Otherwise, I'd advice a basic linear algebra book. $\endgroup$ – rschwieb Jan 10 '14 at 16:09
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Maybe this book will be of use. I've seen it in quite a few U.S. university libraries, but I don't know how easily available it is otherwise.

Karol Borsuk, Multidimensional Analytic Geometry, Monografie Matematyczne #50, PWN - Polish Scientific Publishers, 1969, 443 pages. [Translation to English by Wanda Halina Spalinska.]

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