Introduction to Classical Geometries, Sachbücher von José Seade

This book develops the geometric intuition of the reader by examining 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. The main focus is on the mathematically rich two-dimensional case, although some aspects of 3- or $n$-dimensional geometries are included. Basic notions of algebra and analysis are used to convey better 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. Geometry is one of the oldest branches of mathematics, nearly as old as human culture.Itsbeautyhasalwaysfascinatedmathematicians,amongothers.Inwriting this book we had the purpose of sharing with readers the pleasure derived from studying geometry,as well as giving a tasteof its importance, its deep connections with other branches of mathematics and the highly diverse viewpoints that may be taken by someone entering this ?eld. We also want to propose a speci?c way to introduce concepts that have arisen from the heyday of the Greek school of geometry to the present day. We workwithcoordinatemodels,sincethis facilitatestheuseofalgebraicandanalytic results, and we follow the viewpoint proposed by Felix Klein in the 19th century, of studying geometry via groups of symmetries of the space in question. We intend this book to be both an introduction to the subject addressed to undergraduate students in mathematics and physics, and a useful text-book for mathematicians and scientists in general who want to learn the basics of classical geometry: Euclidean, a?ne, elliptic, hyperbolic and projective geometry. These are all presented in a uni?ed way and the essential content of this book may be covered in a single semester, though a longer period of study would allow the student to grasp and assimilate better the material in it. 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.