Preface to the Second Edition
Preface to the First Edition
Chapter 1 Projective Planes
1.1 Preliminaries
1.1.1 Transformations
1.1.2 Orthogonal Transformations
1.1.3 Similarity Transformations
1.1.4 Affine Transformations
Exercises 1.1
1.2 Extended Planes
1.2.1 Central Projections
1.2.2 Elements at Infinity
1.2.3 Extended Planes
1.2.4 The Basic Properties of Extended Lines and Extended Planes
Exercises 1.2
1.3 The Homogeneous Coordinates in an Extended Plane
1.3.1 The Equivalent Classes of n-dimensional Real Vectors
1.3.2 Homogeneous Point Coordinates
1.3.3 The Homogeneous Coordinate Equation of a Line
1.3.4 Homogeneous Line Coordinates
1.3.5 Some Basic Conclusions on Homogeneous Coordinates
1.3.6 Homogeneous Cartesian Coordinate System in an Extended Plane
Exercises 1.3
1.4 Projective Planes
1.4.1 Real Projective Planes（Two-dimensional Projective Space）
1.4.2 The Models of a Real Projective Plane
1.4.3 Projective Coordinate Transformations
1.4.4 Real Projective Lines（One-dimensional Real Projective Space）
1.4.5 Real-Complex Projective Planes
1.4.6 Basic Projective Figures and Projective Properties of Figures
Exercises 1.4
1.5 The Plane Duality Principle
1.5.1 The Plane Duality Principle
1.5.2 The Principle of Algebraic Duality
Exercises 1.5
1.6 Desargues Two-Triangle Theorem
1.6.1 Desargues Two-Triangle Theorem
1.6.2 Applications of Desargues Two-rIYiangle Theorem
Exercises 1.6
Chapter 2 Projective Transformations
2.1 The Cross Ratio
2.1.1 The Cross Ratio of Collinear Four Points
2.1.2 The Cross Ratio of Concurrent Four Lines in a Projective Plane
Exercises 2.1
2.2 Harmonic Properties of Complete 4-points and Complete 4-lines
2.2.1 Harmonic Properties of Complete 4-points and Complete 4-lines
2.2.2 The Applications of Harmonic Properties of Complete 4-points and Complete 4-lines
Exercises 2.2
2.3 The Projective Correspondences Between One-dimensional Basic Figures
2.3.1 The Perspectivities
2.3.2 The Projective Correspondences
2.3.3 The Algebraic Definition of Projective Correspondence
Exercises 2.3
2.4 One-dimensional Projective Transformations
2.4.1 One-dimensional Projective Transformations
2.4.2 The Classification of One-dimensional Projective rlyansformations
Exercises 2.4
2.5 The Involutions of One-dimensional Basic Figures
2.5.1 The Definition of Involutions
2.5.2 The Conditions to Determine Involutions
2.5.3 The Invariant Elements of Involutions and Their Properties
2.5.4 Desargues Involution Theorem
Exercises 2.5
2.6 Two-dimensional Projective Transformations
2.6.1 Two-dimensional Projective Correspondences
2.6.2 Two-dimensional Projective Transformations
Exercises 2.6
Chapter 3 Transformation Groups and Geometry
3.1 Projective Affine Planes
3.1.1 Projective Affine Planes
3.1.2 Projective Afline Transformations and Affme Transformations
3.1.3 Projective Similarity Transformations and Similarity Transformations
3.1.4 Projective Orthogonal Transformations and Orthogonal Transformations
Exercises 3.1
3.2 Some Transformation Groups of Plane
3.2.1 Groups and Transformation Groups
3.2.2 Some Transformation Groups of Plane
Exercises 3.2
3.3 Transformation Groups and Geometry
3.3.1 Klein's Thought of Transformation Group
3.3.2 The Comparison of Several Plane Geometries
Exercises 3.3
Chapter 4 Theory of Conics
4.1 Definitions and Basic Properties of Conics
4.1.1 Algebraic Definition of Conics
4.1.2 Projection Definition of Conics
4.1.3 Tangent Lines of a Point Conic and Tangent Points of a Line Conic
4.1.4 The Geometric Unity of Point Conics and Line Conics
4.1.5 Pencil of Point Conics
Exercises 4.1
4.2 Theorems of Pascal and Brianchon
4.2.1 Theorems of Pascal and Brianchon
4.2.2 Special Cases of Pascal's Theorem
4.2.3 Applications of Pascal's Theorem and Brianchon's Theorem
Exercises 4.2
4.3 Polar Transformations
4.3.1 Poles and Polar Lines
4.3.2 Polar Transformations
Exercises 4.3
*4.4 Projective Transformations of a Quadratic Point Range
4.4.1 Projective Correspondences of a Quadratic Range of Points
4.4.2 Projective Transformations of a Quadratic Range of Points
4.4.3 Involutions of a Quadratic Range of Points
Exercises 4.4
4.5 Projective Classifications of Conics
4.5.1 Singular Points of a Point Conic
4.5.2 Projective Classifications of Point Conics
Exercises 4.5
4.6 Affine Theory of Conics
4.6.1 The Relation of a Point Conic and the Infinite Line
4.6.2 The Center of a Point Conic
4.6.3 Diameter and Conjugate Diameter of a Conic
4.6.4 Asymptotes of a Central Point Conic
Exercises 4.6
4.7 Affine Classifications of Conics
4.7.1 Projective Affine Classifications of Non-degenerate Point Conics
4.7.2 Projective Affine Classifications of Degenerate Point Conics
Exercises 4.7
Chapter 5 The Historic Developmental Track of Geometry
5.1 Euclidean Geometry
5.2 Pappus and Projective geometry
5.3 Descartes and Analytic Geometry
5.4 The Fifth Axiom and Non-Eucildean Geometry
5.5 Gauss, Riemaun and Differential Geometry
5.6 Cantor, Poincare and Topology
5.7 Hilbert and His Foundations of Geometry
Bibliographies
Subject Index
