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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.