Projective geometry is concerned with the properties of figures that are invariant by projecting and taking sections. It is considered one of the most beautiful parts of geometry and plays a central role because its specializations cover the whole of the affine, Euclidean and nonEuclidean geometries. The natural extension of projective geometry is projective algebraic geometry, a rich and active field of research. The results and techniques of projective geometry are intensively used in computer vision. This book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclidean geometries. It covers central topics such as linear varieties, cross ratio, duality, projective transformations, quadrics and their classificationsprojective, affine and metricas well as the more advanced and less usual spaces of quadrics, rational normal curves, line complexes and the classifications of collineations, pencils of quadrics and correlations. Two appendices are devoted to the projective foundations of perspective and to the projective models of plane nonEuclidean geometries. The book uses modern language, is based on linear algebra, and provides complete proofs. Exercises are proposed at the end of each chapter; many of them are beautiful classical results. The material in this book is suitable for courses on projective geometry for undergraduate students, with a working knowledge of a standard first course on linear algebra. The text is a valuable guide to graduate students and researchers working in areas using or related to projective geometry, such as algebraic geometry and computer vision, and to anyone looking for an advanced view of geometry as a whole. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Undergraduate and graduate students and research mathematicians interested in projective geometry. Reviews "... [A]n excellent reference source for people interested in the subject."  MAA Reviews Table of Contents  Projective spaces and linear varieties
 Projective coordinates and cross ratio
 Affine geometry
 Duality
 Projective transformations
 Quadric hypersurfaces
 Classification and properties of quadrics
 Further properties of quadrics
 Projective spaces of quadrics
 Metric geometry of quadrics
 Three projective classifications
 Appendix A. Perspective (for artists)
 Appendix B. Models of nonEuclidean geometries
 Bibliography
 Symbols
 Index
