Projective geometry is the geometry of vision, and this book introduces students to this beautiful subject from an analytic perspective, emphasising its close relationship with linear algebra and the central role of symmetry. Starting with elementary and familiar geometry over real numbers, readers will soon build upon that knowledge via geometric pathways and journey on to deep and interesting corners of the subject. Through a projective approach to geometry, readers will discover connections between seemingly distant (and ancient) results in Euclidean geometry. By mixing recent results from the past 100 years with the history of the field, this text is one of the most comprehensive surveys of the subject and an invaluable reference for undergraduate and beginning graduate students learning classic geometry, as well as young researchers in computer graphics. Students will also appreciate the worked examples and diagrams throughout.
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'This book provides a lively and lovely perspective on real projective spaces, combining art, history, groups and elegant proofs.' William M. Kantor 'This book is a celebration of the projective viewpoint of geometry. It gradually introduces the reader to the subject, and the arguments are presented in a way that highlights the power of projective thinking in geometry. The reader surprisingly discovers not only that Euclidean and related theorems can be realized as derivatives of projective results, but there are also unnoticed connections between results from ancient times. The treatise also contains a large number of exercises and is dotted with worked examples, which help the reader to appreciate and deeply understand the arguments they refer to. In my opinion this is a book that will definitely change the way we look at the Euclidean and projective analytic geometry.' Alessandro Siciliano, Universitą degli Studi della Basilicata
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This text introduces students to projective geometry from an analytic perspective, mixing recent results with the history of the field.
Preface; Part I. The Real Projective Plane:
1. Fundamental aspects of the real projective plane;
2. Collineations;
3. Polarities and conics;
4. Cross-ratio;
5. The group of the conic;
6. Involution;
7. Affine plane geometry viewed projectively;
8. Euclidean plane geometry viewed projectively;
9. Transformation geometry: Klein's point of view;
10. The power of projective thinking;
11. From perspective to projective;
12. Remarks on the history of projective geometry; Part II. Two Real Projective 3-Space:
13. Fundamental aspects of real projective space;
14. Triangles and tetrahedra;
15. Reguli and quadrics;
16. Line geometry;
17. Projections;
18. A glance at inversive geometry; Part III. Higher Dimensions:
19. Generalising to higher dimensions;
20. The Klein quadric and Veronese surface; Appendix: Group actions; References; Index.
John Bamberg is Associate Professor of Mathematics at the University of Western Australia, where he previously obtained his Ph.D. under the auspices of Cheryl Praeger and Tim Penttila. His research interests include finite and projective geometry, group theory, and algebraic combinatorics. He was a Marie Skodowska-Curie fellow at Ghent University from 2006 to 2009, and a future fellow at the Australian Research Council from 2012 to 2016. Tim Penttila is an Australian mathematician whose research interests include geometry, group theory, and combinatorics. He was an academic at the University of Western Australia for twenty years, and a professor at Colorado State University for ten years.
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