高等几何

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 Tryansformations

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 R，eal 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—Triangle Theorem

Exercises 1.6

Chapter 2 Projective Transformations

2.1 The Cross Ratio

2.1.1 The Cross Ratio of CoUinear 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 Thansformations

2.4.2 The Classification of One—dimensional Projective Transformations

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 Affine Transformations and Affine 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 Application8 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 Coruc

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， Riemann and Differential Geometry

5.6 Cantor， Poincare and Topology

5.7 Hilbert and His Foundations of Geometry

Bibliographies

Subject Index

由Xinghe Zhou和Mingsheng Yang编著的这本《Higher Geometry(SECOND EDITION)》的英文简介如下： This book provides an introduction to plane projective geometry and affine geometry. Its main contents involve projective plane， projective transformations， transformation groups and geometry， the projective and affine theories of conics， the historic development track of geometry and so on. The book is easy both to teach and to learn， which is typical of its well-knit system， refined content and colorful pictures. The book can act as text-book for undergraduates in maths' majors as well as reference book for faculty concerned.《高等几何(第2版)(英文版)》简介：This book provides an introduction to plane projective geometry and affine geometry.Its main contents involve projective plane， projective transformations， transformation groups and geometry， the projective and affine theories of'conics， the historic development trackof geometry and so on.The book is easy both to teach and to learn， which is typical of its weil—knit system， refined content and colorful pictures.The book can act as text—book for undergraduates in maths' majors as well asreference book for faculty concerned.