In the 1880s, over fifty years after the discovery of the hyperbolic plane, Poincare pointed out that this plane provides a very useful context for describing the properties of the solutions of an important class of differential equations. Topics include Euclidean rigid motions, inversions, Euclidea
This is the only book I've ever seen that makes the hyperbolic plane seem so natural and accessible. It takes ordinary Euclidean plane geometry as its basis, which allows the book to cover a lot of material in a very satisfying manner, without requiring advanced background like group theory or differential geometry. (In fact, the only background required is basically exposure to proofs, and occasionally a little calculus or linear algebra.) It's also extremely well-written, and the problems are well thought out. A great text for college juniors or seniors (or even advanced sophomores), and also great for self-study.
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