Coverart for item
The Resource Perspectives on projective geometry : a guided tour through real and complex geometry, by Jürgen Richter-Gebert

Perspectives on projective geometry : a guided tour through real and complex geometry, by Jürgen Richter-Gebert

Label
Perspectives on projective geometry : a guided tour through real and complex geometry
Title
Perspectives on projective geometry
Title remainder
a guided tour through real and complex geometry
Statement of responsibility
by Jürgen Richter-Gebert
Creator
Subject
Language
eng
Cataloging source
BTCTA
http://library.link/vocab/creatorDate
1963-
http://library.link/vocab/creatorName
Richter-Gebert, Jürgen
Illustrations
illustrations
Index
index present
Literary form
non fiction
Nature of contents
bibliography
http://library.link/vocab/subjectName
  • Algebra
  • Algorithms
  • Discrete groups
  • Geometry
  • Mathematics
  • Visualization
  • Geometry, Projective
Label
Perspectives on projective geometry : a guided tour through real and complex geometry, by Jürgen Richter-Gebert
Instantiates
Publication
Bibliography note
Includes bibliographical references and index
Contents
  • Projective Proofs of Pappos's Theorem
  • Points on a Line
  • 8.2.
  • Quadrilateral Sets
  • 8.3.
  • Symmetry and Generalizations of Quadrilateral Sets
  • 8.4.
  • Quadrilateral Sets and von Staudt
  • 8.5.
  • Slope Conditions
  • 8.6.
  • 1.4.
  • Involutions and Quadrilateral Sets
  • 9.
  • Conics and Their Duals
  • 9.1.
  • The Equation of a Conic
  • 9.2.
  • Polars and Tangents
  • 9.3.
  • Dual Quadratic Forms
  • 9.4.
  • Conics
  • How Conics Transform
  • 9.5.
  • Degenerate Conics
  • 9.6.
  • Primal-Dual Pairs
  • 10.
  • Conics and Perspectivity
  • 10.1.
  • Conic through Five Points
  • 10.2.
  • 1.5.
  • Conics and Cross-Ratios
  • 10.3.
  • Perspective Generation of Conics
  • 10.4.
  • Transformations and Conics
  • 10.5.
  • Hesse's "Ubertragungsprinzip"
  • 10.6.
  • Pascal's and Brianchon's Theorems
  • 10.7.
  • More Conics
  • Harmonic points on a conic
  • 11.
  • Calculating with Conics
  • 11.1.
  • Splitting a Degenerate Conic
  • 11.2.
  • The Necessity of "If" Operations
  • 11.3.
  • Intersecting a Conic and a Line
  • 11.4.
  • 1.6.
  • Intersecting Two Conics
  • 11.5.
  • The Role of Complex Numbers
  • 11.6.
  • One Tangent and Four Points
  • 12.
  • Projective d-space
  • 12.1.
  • Elements at Infinity
  • 12.2.
  • Complex Numbers and Circles
  • Homogeneous Coordinates and Transformations
  • 12.3.
  • Points and Planes in 3-Space
  • 12.4.
  • Lines in 3-Space
  • 12.5.
  • Joins and Meets: A Universal System
  • 12.6.
  • ... And How to Use It
  • 13.
  • 1.7.
  • Diagram Techniques
  • 13.1.
  • From Points, Lines, and Matrices to Tensors
  • 13.2.
  • A Few Fine Points
  • 13.3.
  • Tensor Diagrams
  • 13.4.
  • How Transformations Work
  • 13.5.
  • Finally
  • The δ-tensor
  • 13.6.
  • ε-Tensors
  • 13.7.
  • The ε-δ Rule
  • 13.8.
  • Transforming ε-Tensors
  • 13.9.
  • Invariants of Line and Point Configurations
  • 14.
  • pt. I
  • Working with diagrams
  • 14.1.
  • The Simplest Property: A Trace Condition
  • 14.2.
  • Pascal's Theorem
  • 14.3.
  • Closed ε-Cycles
  • 14.4.
  • Conics, Quadratic Forms, and Tangents
  • 14.5.
  • Machine generated contents note:
  • Projective Geometry
  • Diagrams in RP3
  • 14.6.
  • The ε-δ-rule in Rank 4
  • 14.7.
  • Co- and Contravariant Lines in Rank 4
  • 14.8.
  • Tensors versus Plucker Coordinates
  • 15.
  • Configurations, Theorems, and Bracket Expressions
  • 15.1.
  • 2.
  • Desargues's Theorem
  • 15.2.
  • Binomial Proofs
  • 15.3.
  • Chains and Cycles of Cross-Ratios
  • 15.4.
  • Ceva and Menelaus
  • 15.5.
  • Gluing Ceva and Menelaus Configurations
  • 15.6.
  • Projective Planes
  • Furthermore
  • pt. III
  • Measurements
  • 16.
  • Complex Numbers: A Primer
  • 16.1.
  • Historical Background
  • 16.2.
  • The Fundamental Theorem
  • 16.3.
  • 2.1.
  • Geometry of Complex Numbers
  • 16.4.
  • Euler's Formula
  • 16.5.
  • Complex Conjugation
  • 17.
  • The Complex Projective Line
  • 17.1.
  • CP1
  • 17.2.
  • Drawings and Perspectives
  • Testing Geometric Properties
  • 17.3.
  • Projective Transformations
  • 17.4.
  • Inversions and Mobius Reflections
  • 17.5.
  • Grassmann-Plucker relations
  • 17.6.
  • Intersection Angles
  • 17.7.
  • 2.2.
  • Stereographic Projection
  • 18.
  • Euclidean Geometry
  • 18.1.
  • The points I and J
  • 18.2.
  • Cocircularity
  • 18.3.
  • The Robustness of the Cross-Ratio
  • 18.4.
  • The Axioms
  • Transformations
  • 18.5.
  • Translating Theorems
  • 18.6.
  • More Geometric Properties
  • 18.7.
  • Laguerre's Formula
  • 18.8.
  • Distances
  • 19.
  • 2.3.
  • Euclidean Structures from a Projective Perspective
  • 19.1.
  • Mirror Images
  • 19.2.
  • Angle Bisectors
  • 19.3.
  • Center of a Circle
  • 19.4.
  • Constructing the Foci of a Conic
  • 19.5.
  • The Smallest Projective Plane
  • Constructing a Conic by Foci
  • 19.6.
  • Triangle Theorems
  • 19.7.
  • Hybrid Thinking
  • 20.
  • Cayley-Klein Geometries
  • 20.1.
  • I and J Revisited
  • 20.2.
  • 3.
  • Measurements in Cayley-Klein Geometries
  • 20.3.
  • Nondegenerate Measurements along a Line
  • 20.4.
  • Degenerate Measurements along a Line
  • 20.5.
  • A Planar Cayley-Klein Geometry
  • 20.6.
  • A Census of Cayley-Klein Geometries
  • 20.7.
  • 1.
  • Homogeneous Coordinates
  • Coarser and Finer Classifications
  • 21.
  • Measurements and Transformations
  • 21.1.
  • Measurements vs. Oriented Measurements
  • 21.2.
  • Transformations
  • 21.3.
  • Getting Rid of X and Y
  • 21.4.
  • 3.1.
  • Comparing Measurements
  • 21.5.
  • Reflections and Pole/Polar Pairs
  • 21.6.
  • From Reflections to Rotations
  • 22.
  • Cayley-Klein Geometries at Work
  • 22.1.
  • Orthogonality
  • 22.2.
  • A Spatial Point of View
  • Constructive versus Implicit Representations
  • 22.3.
  • Commonalities and Differences
  • 22.4.
  • Midpoints and Angle Bisectors
  • 22.5.
  • Trigonometry
  • 23.
  • Circles and Cycles
  • 23.1.
  • 3.2.
  • Circles via Distances
  • 23.2.
  • Relation to the Fundamental Conic
  • 23.3.
  • Centers at Infinity
  • 23.4.
  • Organizing Principles
  • 23.5.
  • Cycles in Galilean Geometry
  • 24.
  • The Real Projective Plane with Homogeneous Coordinates
  • Non-Euclidean Geometry: A Historical Interlude
  • 24.1.
  • The Inner Geometry of a Space
  • 24.2.
  • Euclid's Postulates
  • 24.3.
  • Gauss, Bolyai, and Lobachevsky
  • 24.4.
  • Beltrami and Klein
  • 24.5.
  • 3.3.
  • The Beltrami-Klein Model
  • 24.6.
  • Poincare
  • 25.
  • Hyperbolic Geometry
  • 25.1.
  • The Staging Ground
  • 25.2.
  • Hyperbolic Transformations
  • 25.3.
  • Joins and Meets
  • Angles and Boundaries
  • 25.4.
  • The Poincare Disk
  • 25.5.
  • CP1 Transformations and the Poincare Disk
  • 25.6.
  • Angles and Distances in the Poincare Disk
  • 26.
  • Selected Topics in Hyperbolic Geometry
  • 26.1.
  • 3.4.
  • Circles and Cycles in the Poincare Disk
  • 26.2.
  • Area and Angle Defect
  • 26.3.
  • Thales and Pythagoras
  • 26.4.
  • Constructing Regular n-Gons
  • 26.5.
  • Symmetry Groups
  • 27.
  • Parallelism
  • What We Did Not Touch
  • 27.1.
  • Algebraic Projective Geometry
  • 27.2.
  • Projective Geometry and Discrete Mathematics
  • 27.3.
  • Projective Geometry and Quantum Theory
  • 27.4.
  • Dynamic Projective Geometry
  • 3.5.
  • Pappos's Theorem: Nine Proofs and Three Variations
  • Duality
  • 3.6.
  • Projective Transformations
  • 3.7.
  • Finite Projective Planes
  • 4.
  • Lines and Cross-Ratios
  • 4.1.
  • Coordinates on a Line
  • 4.2.
  • 1.1.
  • The Real Projective Line
  • 4.3.
  • Cross-Ratios (a First Encounter)
  • 4.4.
  • Elementary Properties of the Cross-Ratio
  • 5.
  • Calculating with Points on Lines
  • 5.1.
  • Harmonic Points
  • 5.2.
  • Pappos's Theorem and Projective Geometry
  • Projective Scales
  • 5.3.
  • From Geometry to Real Numbers
  • 5.4.
  • The Fundamental Theorem
  • 5.5.
  • A Note on Other Fields
  • 5.6.
  • Von Staudt's Original Constructions
  • 5.7.
  • 1.2.
  • Pappos's Theorem
  • 6.
  • Determinants
  • 6.1.
  • A "Determinantal" Point of View
  • 6.2.
  • A Few Useful Formulas
  • 6.3.
  • Plucker's μ
  • 6.4.
  • Euclidean Versions of Pappos's Theorem
  • Invariant Properties
  • 6.5.
  • Grassmann-Plucker relations
  • 7.
  • More on Bracket Algebra
  • 7.1.
  • From Points to Determinants
  • 7.2.
  • ... and Back
  • 7.3.
  • 1.3.
  • A Glimpse of Invariant Theory
  • 7.4.
  • Projectively Invariant Functions
  • 7.5.
  • The Bracket Algebra
  • pt. II
  • Working and Playing with Geometry
  • 8.
  • Quadrilateral Sets and Liftings
  • 8.1.
Control code
ocn690089198
Dimensions
25 cm
Extent
xxii, 571 p.
Isbn
9783642172854
Other physical details
ill.
System control number
(OCoLC)690089198
Label
Perspectives on projective geometry : a guided tour through real and complex geometry, by Jürgen Richter-Gebert
Publication
Bibliography note
Includes bibliographical references and index
Contents
  • Projective Proofs of Pappos's Theorem
  • Points on a Line
  • 8.2.
  • Quadrilateral Sets
  • 8.3.
  • Symmetry and Generalizations of Quadrilateral Sets
  • 8.4.
  • Quadrilateral Sets and von Staudt
  • 8.5.
  • Slope Conditions
  • 8.6.
  • 1.4.
  • Involutions and Quadrilateral Sets
  • 9.
  • Conics and Their Duals
  • 9.1.
  • The Equation of a Conic
  • 9.2.
  • Polars and Tangents
  • 9.3.
  • Dual Quadratic Forms
  • 9.4.
  • Conics
  • How Conics Transform
  • 9.5.
  • Degenerate Conics
  • 9.6.
  • Primal-Dual Pairs
  • 10.
  • Conics and Perspectivity
  • 10.1.
  • Conic through Five Points
  • 10.2.
  • 1.5.
  • Conics and Cross-Ratios
  • 10.3.
  • Perspective Generation of Conics
  • 10.4.
  • Transformations and Conics
  • 10.5.
  • Hesse's "Ubertragungsprinzip"
  • 10.6.
  • Pascal's and Brianchon's Theorems
  • 10.7.
  • More Conics
  • Harmonic points on a conic
  • 11.
  • Calculating with Conics
  • 11.1.
  • Splitting a Degenerate Conic
  • 11.2.
  • The Necessity of "If" Operations
  • 11.3.
  • Intersecting a Conic and a Line
  • 11.4.
  • 1.6.
  • Intersecting Two Conics
  • 11.5.
  • The Role of Complex Numbers
  • 11.6.
  • One Tangent and Four Points
  • 12.
  • Projective d-space
  • 12.1.
  • Elements at Infinity
  • 12.2.
  • Complex Numbers and Circles
  • Homogeneous Coordinates and Transformations
  • 12.3.
  • Points and Planes in 3-Space
  • 12.4.
  • Lines in 3-Space
  • 12.5.
  • Joins and Meets: A Universal System
  • 12.6.
  • ... And How to Use It
  • 13.
  • 1.7.
  • Diagram Techniques
  • 13.1.
  • From Points, Lines, and Matrices to Tensors
  • 13.2.
  • A Few Fine Points
  • 13.3.
  • Tensor Diagrams
  • 13.4.
  • How Transformations Work
  • 13.5.
  • Finally
  • The δ-tensor
  • 13.6.
  • ε-Tensors
  • 13.7.
  • The ε-δ Rule
  • 13.8.
  • Transforming ε-Tensors
  • 13.9.
  • Invariants of Line and Point Configurations
  • 14.
  • pt. I
  • Working with diagrams
  • 14.1.
  • The Simplest Property: A Trace Condition
  • 14.2.
  • Pascal's Theorem
  • 14.3.
  • Closed ε-Cycles
  • 14.4.
  • Conics, Quadratic Forms, and Tangents
  • 14.5.
  • Machine generated contents note:
  • Projective Geometry
  • Diagrams in RP3
  • 14.6.
  • The ε-δ-rule in Rank 4
  • 14.7.
  • Co- and Contravariant Lines in Rank 4
  • 14.8.
  • Tensors versus Plucker Coordinates
  • 15.
  • Configurations, Theorems, and Bracket Expressions
  • 15.1.
  • 2.
  • Desargues's Theorem
  • 15.2.
  • Binomial Proofs
  • 15.3.
  • Chains and Cycles of Cross-Ratios
  • 15.4.
  • Ceva and Menelaus
  • 15.5.
  • Gluing Ceva and Menelaus Configurations
  • 15.6.
  • Projective Planes
  • Furthermore
  • pt. III
  • Measurements
  • 16.
  • Complex Numbers: A Primer
  • 16.1.
  • Historical Background
  • 16.2.
  • The Fundamental Theorem
  • 16.3.
  • 2.1.
  • Geometry of Complex Numbers
  • 16.4.
  • Euler's Formula
  • 16.5.
  • Complex Conjugation
  • 17.
  • The Complex Projective Line
  • 17.1.
  • CP1
  • 17.2.
  • Drawings and Perspectives
  • Testing Geometric Properties
  • 17.3.
  • Projective Transformations
  • 17.4.
  • Inversions and Mobius Reflections
  • 17.5.
  • Grassmann-Plucker relations
  • 17.6.
  • Intersection Angles
  • 17.7.
  • 2.2.
  • Stereographic Projection
  • 18.
  • Euclidean Geometry
  • 18.1.
  • The points I and J
  • 18.2.
  • Cocircularity
  • 18.3.
  • The Robustness of the Cross-Ratio
  • 18.4.
  • The Axioms
  • Transformations
  • 18.5.
  • Translating Theorems
  • 18.6.
  • More Geometric Properties
  • 18.7.
  • Laguerre's Formula
  • 18.8.
  • Distances
  • 19.
  • 2.3.
  • Euclidean Structures from a Projective Perspective
  • 19.1.
  • Mirror Images
  • 19.2.
  • Angle Bisectors
  • 19.3.
  • Center of a Circle
  • 19.4.
  • Constructing the Foci of a Conic
  • 19.5.
  • The Smallest Projective Plane
  • Constructing a Conic by Foci
  • 19.6.
  • Triangle Theorems
  • 19.7.
  • Hybrid Thinking
  • 20.
  • Cayley-Klein Geometries
  • 20.1.
  • I and J Revisited
  • 20.2.
  • 3.
  • Measurements in Cayley-Klein Geometries
  • 20.3.
  • Nondegenerate Measurements along a Line
  • 20.4.
  • Degenerate Measurements along a Line
  • 20.5.
  • A Planar Cayley-Klein Geometry
  • 20.6.
  • A Census of Cayley-Klein Geometries
  • 20.7.
  • 1.
  • Homogeneous Coordinates
  • Coarser and Finer Classifications
  • 21.
  • Measurements and Transformations
  • 21.1.
  • Measurements vs. Oriented Measurements
  • 21.2.
  • Transformations
  • 21.3.
  • Getting Rid of X and Y
  • 21.4.
  • 3.1.
  • Comparing Measurements
  • 21.5.
  • Reflections and Pole/Polar Pairs
  • 21.6.
  • From Reflections to Rotations
  • 22.
  • Cayley-Klein Geometries at Work
  • 22.1.
  • Orthogonality
  • 22.2.
  • A Spatial Point of View
  • Constructive versus Implicit Representations
  • 22.3.
  • Commonalities and Differences
  • 22.4.
  • Midpoints and Angle Bisectors
  • 22.5.
  • Trigonometry
  • 23.
  • Circles and Cycles
  • 23.1.
  • 3.2.
  • Circles via Distances
  • 23.2.
  • Relation to the Fundamental Conic
  • 23.3.
  • Centers at Infinity
  • 23.4.
  • Organizing Principles
  • 23.5.
  • Cycles in Galilean Geometry
  • 24.
  • The Real Projective Plane with Homogeneous Coordinates
  • Non-Euclidean Geometry: A Historical Interlude
  • 24.1.
  • The Inner Geometry of a Space
  • 24.2.
  • Euclid's Postulates
  • 24.3.
  • Gauss, Bolyai, and Lobachevsky
  • 24.4.
  • Beltrami and Klein
  • 24.5.
  • 3.3.
  • The Beltrami-Klein Model
  • 24.6.
  • Poincare
  • 25.
  • Hyperbolic Geometry
  • 25.1.
  • The Staging Ground
  • 25.2.
  • Hyperbolic Transformations
  • 25.3.
  • Joins and Meets
  • Angles and Boundaries
  • 25.4.
  • The Poincare Disk
  • 25.5.
  • CP1 Transformations and the Poincare Disk
  • 25.6.
  • Angles and Distances in the Poincare Disk
  • 26.
  • Selected Topics in Hyperbolic Geometry
  • 26.1.
  • 3.4.
  • Circles and Cycles in the Poincare Disk
  • 26.2.
  • Area and Angle Defect
  • 26.3.
  • Thales and Pythagoras
  • 26.4.
  • Constructing Regular n-Gons
  • 26.5.
  • Symmetry Groups
  • 27.
  • Parallelism
  • What We Did Not Touch
  • 27.1.
  • Algebraic Projective Geometry
  • 27.2.
  • Projective Geometry and Discrete Mathematics
  • 27.3.
  • Projective Geometry and Quantum Theory
  • 27.4.
  • Dynamic Projective Geometry
  • 3.5.
  • Pappos's Theorem: Nine Proofs and Three Variations
  • Duality
  • 3.6.
  • Projective Transformations
  • 3.7.
  • Finite Projective Planes
  • 4.
  • Lines and Cross-Ratios
  • 4.1.
  • Coordinates on a Line
  • 4.2.
  • 1.1.
  • The Real Projective Line
  • 4.3.
  • Cross-Ratios (a First Encounter)
  • 4.4.
  • Elementary Properties of the Cross-Ratio
  • 5.
  • Calculating with Points on Lines
  • 5.1.
  • Harmonic Points
  • 5.2.
  • Pappos's Theorem and Projective Geometry
  • Projective Scales
  • 5.3.
  • From Geometry to Real Numbers
  • 5.4.
  • The Fundamental Theorem
  • 5.5.
  • A Note on Other Fields
  • 5.6.
  • Von Staudt's Original Constructions
  • 5.7.
  • 1.2.
  • Pappos's Theorem
  • 6.
  • Determinants
  • 6.1.
  • A "Determinantal" Point of View
  • 6.2.
  • A Few Useful Formulas
  • 6.3.
  • Plucker's μ
  • 6.4.
  • Euclidean Versions of Pappos's Theorem
  • Invariant Properties
  • 6.5.
  • Grassmann-Plucker relations
  • 7.
  • More on Bracket Algebra
  • 7.1.
  • From Points to Determinants
  • 7.2.
  • ... and Back
  • 7.3.
  • 1.3.
  • A Glimpse of Invariant Theory
  • 7.4.
  • Projectively Invariant Functions
  • 7.5.
  • The Bracket Algebra
  • pt. II
  • Working and Playing with Geometry
  • 8.
  • Quadrilateral Sets and Liftings
  • 8.1.
Control code
ocn690089198
Dimensions
25 cm
Extent
xxii, 571 p.
Isbn
9783642172854
Other physical details
ill.
System control number
(OCoLC)690089198

Library Locations

    • Manawatū LibraryBorrow it
      Tennent Drive, Palmerston North, Palmerston North, 4472, NZ
      -40.385340 175.617349
Processing Feedback ...