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.
Author(s): Jürgen Richter-Gebert (auth.)
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2011
Language: English
Pages: 571
Tags: Geometry; Algebra; Algorithms; General Algebraic Systems; Visualization; Convex and Discrete Geometry
Front Matter....Pages i-xxii
Pappos’s Theorem: Nine Proofs and Three Variations....Pages 3-31
Front Matter....Pages 33-33
Projective Planes....Pages 35-46
Homogeneous Coordinates....Pages 47-66
Lines and Cross-Ratios....Pages 67-78
Calculating with Points on Lines....Pages 79-92
Determinants....Pages 93-107
More on Bracket Algebra....Pages 109-123
Front Matter....Pages 125-127
Quadrilateral Sets and Liftings....Pages 129-143
Conics and Their Duals....Pages 145-166
Conics and Perspectivity....Pages 167-187
Calculating with Conics....Pages 189-207
Projective d -space....Pages 209-225
Diagram Techniques....Pages 227-246
Working with diagrams....Pages 247-267
Configurations, Theorems, and Bracket Expressions....Pages 269-292
Front Matter....Pages 293-296
Complex Numbers: A Primer....Pages 297-309
The Complex Projective Line....Pages 311-327
Euclidean Geometry....Pages 329-347
Euclidean Structures from a Projective Perspective....Pages 349-373
Cayley-Klein Geometries....Pages 375-398
Front Matter....Pages 293-296
Measurements and Transformations....Pages 399-422
Cayley-Klein Geometries at Work....Pages 423-442
Circles and Cycles....Pages 443-464
Non-Euclidean Geometry: A Historical Interlude....Pages 465-481
Hyperbolic Geometry....Pages 483-503
Selected Topics in Hyperbolic Geometry....Pages 505-523
What We Did Not Touch....Pages 525-555
Back Matter....Pages 557-571
