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Affine, projective, spherical, M bius, and hyperbolic geometries are covered by this textbook. Special emphasis is given on the groups of transformations in the spirit of the Erlangen program of Felix Klein as transformation groups not only provide a theoretical basis for the geometries, but also allow to solve concrete problems, which are an organic part of the book.

By establishing direct connections and by looking at the same notion or problem from different points of view the author gives a harmonic overview of the geometry. Starting with classical topics such as conics and quadrics, duality and polarity, cross-ratio, spherical and hyperbolic trigonometry he leads to less classical ones like compositions of reflections, volumes of spherical and hyperbolic tetrahedra, billiards, de Sitter geometry.

As only the knowledge of linear algebra, complex numbers and basic group theory is required this book is accessible to second year undergraduate students. However, graduate students in algebraic geometry, differential geometry, or geometric topology will also find this basic inventory of what every geometer should know a helpful hand.