Perspectives on Projective Geometry
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- Roderick York
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1 Perspectives on Projective Geometry A Guided Tour Through Real and Complex Geometry Bearbeitet von Jürgen Richter-Gebert 1. Auflage Buch. xxii, 571 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Weitere Fachgebiete > Mathematik > Geometrie > Algebraische Geometrie Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
2 Contents 1 Pappos s Theorem: Nine Proofs and Three Variations Pappos s Theorem and Projective Geometry Euclidean Versions of Pappos s Theorem Projective Proofs of Pappos s Theorem Conics More Conics Complex Numbers and Circles Finally Part I Projective Geometry 2 Projective Planes Drawings and Perspectives The Axioms The Smallest Projective Plane Homogeneous Coordinates A Spatial Point of View The Real Projective Plane with Homogeneous Coordinates Joins and Meets Parallelism Duality Projective Transformations Finite Projective Planes Lines and Cross-Ratios Coordinates on a Line The Real Projective Line Cross-Ratios (a First Encounter) Elementary Properties of the Cross-Ratio xvii
3 xviii Contents 5 Calculating with Points on Lines Harmonic Points Projective Scales From Geometry to Real Numbers The Fundamental Theorem A Note on Other Fields Von Staudt s Original Constructions Pappos s Theorem Determinants A Determinantal Point of View A Few Useful Formulas Plücker s μ Invariant Properties Grassmann-Plücker relations More on Bracket Algebra From Points to Determinants and Back A Glimpse of Invariant Theory Projectively Invariant Functions The Bracket Algebra Part II Working and Playing with Geometry 8 Quadrilateral Sets and Liftings Points on a Line Quadrilateral Sets Symmetry and Generalizations of Quadrilateral Sets Quadrilateral Sets and von Staudt Slope Conditions Involutions and Quadrilateral Sets Conics and Their Duals The Equation of a Conic Polars and Tangents Dual Quadratic Forms How Conics Transform Degenerate Conics Primal-Dual Pairs Conics and Perspectivity Conic through Five Points Conics and Cross-Ratios Perspective Generation of Conics Transformations and Conics
4 Contents xix 10.5 Hesse s Übertragungsprinzip Pascal s and Brianchon s Theorems Harmonic points on a conic Calculating with Conics Splitting a Degenerate Conic The Necessity of If Operations Intersecting a Conic and a Line Intersecting Two Conics The Role of Complex Numbers One Tangent and Four Points Projective d-space Elements at Infinity Homogeneous Coordinates and Transformations Points and Planes in 3-Space Lines in 3-Space Joins and Meets: A Universal System And How to Use It Diagram Techniques From Points, Lines, and Matrices to Tensors A Few Fine Points Tensor Diagrams How Transformations Work The δ-tensor ε-tensors The ε-δ Rule Transforming ε-tensors Invariants of Line and Point Configurations Working with diagrams The Simplest Property: A Trace Condition Pascal s Theorem Closed ε-cycles Conics, Quadratic Forms, and Tangents Diagrams in RP The ε-δ-rule in Rank Co- and Contravariant Lines in Rank Tensors versus Plücker Coordinates Configurations, Theorems, and Bracket Expressions Desargues s Theorem Binomial Proofs Chains and Cycles of Cross-Ratios Ceva and Menelaus
5 xx Contents 15.5 Gluing Ceva and Menelaus Configurations Furthermore Part III Measurements 16 Complex Numbers: A Primer Historical Background The Fundamental Theorem Geometry of Complex Numbers Euler s Formula Complex Conjugation The Complex Projective Line CP Testing Geometric Properties Projective Transformations Inversions and Möbius Reflections Grassmann-Plücker relations Intersection Angles Stereographic Projection Euclidean Geometry The points I and J Cocircularity The Robustness of the Cross-Ratio Transformations Translating Theorems More Geometric Properties Laguerre s Formula Distances Euclidean Structures from a Projective Perspective Mirror Images Angle Bisectors Center of a Circle Constructing the Foci of a Conic Constructing a Conic by Foci Triangle Theorems Hybrid Thinking Cayley-Klein Geometries I and J Revisited Measurements in Cayley-Klein Geometries Nondegenerate Measurements along a Line Degenerate Measurements along a Line
6 Contents xxi 20.5 A Planar Cayley-Klein Geometry A Census of Cayley-Klein Geometries Coarser and Finer Classifications Measurements and Transformations Measurements vs. Oriented Measurements Transformations Getting Rid of X and Y Comparing Measurements Reflections and Pole/Polar Pairs From Reflections to Rotations Cayley-Klein Geometries at Work Orthogonality Constructive versus Implicit Representations Commonalities and Differences Midpoints and Angle Bisectors Trigonometry Circles and Cycles Circles via Distances Relation to the Fundamental Conic Centers at Infinity Organizing Principles Cycles in Galilean Geometry Non-Euclidean Geometry: A Historical Interlude The Inner Geometry of a Space Euclid s Postulates Gauss, Bolyai, and Lobachevsky Beltrami and Klein The Beltrami-Klein Model Poincaré Hyperbolic Geometry The Staging Ground Hyperbolic Transformations Angles and Boundaries The Poincaré Disk CP 1 Transformations and the Poincaré Disk Angles and Distances in the Poincaré Disk Selected Topics in Hyperbolic Geometry Circles and Cycles in the Poincaré Disk Area and Angle Defect Thales and Pythagoras
7 xxii Contents 26.4 Constructing Regular n-gons Symmetry Groups What We Did Not Touch Algebraic Projective Geometry Projective Geometry and Discrete Mathematics Projective Geometry and Quantum Theory Dynamic Projective Geometry References Index
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