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.
Best Answer
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.