Introduction to Classical Geometries

This book follows Klein's proposal for studying geometry by looking at the symmetries (or rigid motions) of the space in question. This approach introduces in turn all the classical geometries: Euclidean, affine, elliptic, projective and hyperbolic. Numerous illustrations are included to reinforce understanding. Once plane geometry is well understood, it is much easier to go into higher dimensions. Therefore, the main focus of this text is on the mathematically rich two-dimensional case, although some aspects of 3- or $n$-dimensional geometries are included. The book appeals to, and helps to further develop the geometric intuition of the reader. Some basic notions of algebra and analysis are also used to convey more complete understanding of various concepts and results. Concepts of geometry are presented in a very simple way, so that they become easily accessible: the only pre-requisites are calculus, linear algebra and basic analytic geometry.