Perspectives on Projective Geometry : A Guided Tour Through Real and Complex Geometry

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. 1 Pappos's Theorem: Nine Proofs And Three Variations -- 2 Projective Planes -- 3 Homogeneous Coordinates -- 4 Lines And Cross-ratios -- 5 Calculating With Points On Lines -- 6 Determinants -- 7 More On Bracket Algebra -- 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 -- 16 Complex Numbers: A Primer -- 17 The Complex Projective Line -- 18 Euclidean Geometry -- 19 Euclidean Structures From A Projective Perspective -- 20 Cayley-klein Geometries -- 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 -- References -- Index. By Jürgen Richter-gebert. Includes Bibliographical References (p. 557-562) And Index. 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