Projective geometry is one of the most fundamental and at the same time most beautiful branches of geometry. It can be considered the common foundation of many other geometric disciplines like Euclidean geometry, hyperbolic and elliptic geometry or even relativistic space-time geometry. This book offers a comprehensive introduction to this fascinating field and its applications. In particular, it explains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplay between geometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians, as well as for computer scientists and physicists. The book is based on the author’s experience in implementing geometric software and includes hundreds of high-quality illustrations.

Review

“The author covers most of the traditional topics in real projective geometry, and extends the concepts through complex projective geometry. He provides the reader with concise proofs, clear and insightful presentations, and a coherent development of the topic. Richter-Gebert presents algebraic, visual, and diagrammatic approaches to unify the subject, while his fresh writing style make for a very readable text. … Additionally, the student and teacher will enjoy the quotations introducing each section. … Summing Up: Highly recommended. Upper-division undergraduates and faculty.” (R. L. Pour, Choice, Vol. 49 (5), January, 2012)

“This lovely book offers an introduction to the key ideas of projective geometry … . The book is written in a leisurely way that will make it accessible to undergraduate students with a basic knowledge of algebra. … There is no doubt that the author went to great lengths to write clearly and to make his book as user friendly as possible, as the well chosen illustrations, several per page, many in colour, testify.” (S. C. Coutinho, SIGACT News, January, 2014)

“The author, known for the co-development of the interactive geometry software Cinderella, here presents a thorough introduction to real and complex, mostly plane geometry. … Because of the detailed explanations … and the very readable style the book can be recommended warmly even to undergraduates as well as to computer scientists and physicists.” (G. Teschl, Monatshefte für Mathematik, Vol. 166 (2), May, 2012)

“The author describes his work as ‘a guided tour through real and complex geometry’, and these words explain perfectly the aim of the book. … the prerequisites for reading the book are just the basics of linear algebra (in terms of coordinates), so that the book seems accessible to a wide audience ranging from mathematicians to computer scientists and physicists.” (Hans Havlicek, Mathematical Reviews, Issue 2012 e)

“There is much to admire in this book, which is clearly a labor of love on the part of the author … . considerable care has been taken to make the book as accessible as possible … but at the same time the book also contains discussion of a considerable amount of material that I have never before seen in any other text, thereby making it useful as a potential reference as well as teaching tool.” (Mark Hunacek, The Mathematical Association of America, August, 2011)

“The author of this very well written and detailed book is an expert in projective geometry, especially in computational projective geometry … . apart from students this book is accessible to mathematicians as well as computer scientists and physicists. The author presents the rich interplay of geometric structures and their algebraic counterparts … .” (Rolf Riesinger, Zentralblatt MATH, Vol. 1214, 2011)

Table of Contents

1 Pappos's Theorem: Nine Proofs and Three Variations

Part I Projective Geometry

2 Projective Planes

3 Homogeneous Coordinates

4 Lines and Cross-Ratios

5 Calculating with Points on Lines

6 Determinants

7 More on Bracket Algebra

Part II Working and Playing with Geometry

8 Quadrilateral Sets and Liftings

9 Conics and Their Duals

10 Conics and Perspectivity

11 Calculating with Conics

12 Projective d-space

13 Diagram Techniques

14 Working with diagrams

15 Configurations, Theorems, and Bracket Expressions

Part Ill Measurements

16 Complex Numbers: A Primer

17 The Complex Projective Line

18 Euclidean Geometry

19 Euclidean Structures from a Projective Perspective

20 Cayley-Klein Geomet ries

21 Measurements and Transformations

22 Cayley-Klein Geometries at Work

23 Circles and Cycles

24 Non-Euclidean Geometry: A Historical Interlude

25 Hyperbolic Geometry

26 Selected Topics in Hyperbolic Geometry

27 What We Did Not Touch

About the Author

Jürgen Richter-Gebert is full Professor for Geometry and Visualization at Technical University Munich. Research in combinatorial, computational and dynamic geometry, automated geometric theorem proving and visualization. Author of the dynamic geometry program Cinderella and of the interactive visualization portal Mathe-Vital. Cinderella was awarded the Multimedia Innovation Award, the European Academic Software Award and the Deutsche Bildungssoftwarepreis. Mathe-Vital won the renowned MedidaPrix in 2008.