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Analytic Projective Geometry
Eduardo Casas-Alvero, Universitat de Barcelona, Spain
A publication of the European Mathematical Society.
EMS Textbooks in Mathematics
2014; 636 pp; hardcover
Volume: 15
ISBN-10: 3-03719-138-4
ISBN-13: 978-3-03719-138-5
List Price: US$78
Member Price: US$62.40
Order Code: EMSTEXT/15
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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 non-Euclidean 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 classifications--projective, affine and metric--as 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 non-Euclidean 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.


Undergraduate and graduate students and research mathematicians interested in projective geometry.


"... [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 non-Euclidean geometries
  • Bibliography
  • Symbols
  • Index
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