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Geometry, Second Edition

Book Description

This richly illustrated and clearly written undergraduate textbook captures the excitement and beauty of geometry. The approach is that of Klein in his Erlangen programme: a geometry is a space together with a set of transformations of the space. The authors explore various geometries: affine, projective, inversive, hyperbolic and elliptic. In each case they carefully explain the key results and discuss the relationships between the geometries. New features in this second edition include concise end-of-chapter summaries to aid student revision, a list of further reading and a list of special symbols. The authors have also revised many of the end-of-chapter exercises to make them more challenging and to include some interesting new results. Full solutions to the 200 problems are included in the text, while complete solutions to all of the end-of-chapter exercises are available in a new Instructors' Manual, which can be downloaded from www.cambridge.org/9781107647831.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. 0. Introduction: Geometry and Geometries
  9. 1. Conics
    1. 1.1 Conic Sections and Conics
    2. 1.2 Properties of Conics
    3. 1.3 Recognizing Conics
    4. 1.4 Quadric Surfaces
    5. 1.5 Exercises
    6. Summary of Chapter 1
  10. 2. Affine Geometry
    1. 2.1 Geometry and Transformations
    2. 2.2 Affine Transformations and Parallel Projections
    3. 2.3 Properties of Affine Transformations
    4. 2.4 Using the Fundamental Theorem of Affine Geometry
    5. 2.5 Affine Transformations and Conics
    6. 2.6 Exercises
    7. Summary of Chapter 2
  11. 3. Projective Geometry: Lines
    1. 3.1 Perspective
    2. 3.2 The Projective Plane RP[sup(2)]
    3. 3.3 Projective Transformations
    4. 3.4 Using the Fundamental Theorem of Projective Geometry
    5. 3.5 Cross-Ratio
    6. 3.6 Exercises
    7. Summary of Chapter 3
  12. 4. Projective Geometry: Conics
    1. 4.1 Projective Conics
    2. 4.2 Tangents
    3. 4.3 Theorems
    4. 4.4 Applying Linear Algebra to Projective Conics
    5. 4.5 Duality and Projective Conics
    6. 4.6 Exercises
    7. Summary of Chapter 4
  13. 5. Inversive Geometry
    1. 5.1 Inversion
    2. 5.2 Extending the Plane
    3. 5.3 Inversive Geometry
    4. 5.4 Fundamental Theorem of Inversive Geometry
    5. 5.5 Coaxal Families of Circles
    6. 5.6 Exercises
    7. Summary of Chapter 5
  14. 6. Hyperbolic Geometry: the Poincaré Model
    1. 6.1 Hyperbolic Geometry: the Disc Model
    2. 6.2 Hyperbolic Transformations
    3. 6.3 Distance in Hyperbolic Geometry
    4. 6.4 Geometrical Theorems
    5. 6.5 Area
    6. 6.6 Hyperbolic Geometry: the Half-Plane Model
    7. 6.7 Exercises
    8. Summary of Chapter 6
  15. 7. Elliptic Geometry: the Spherical Model
    1. 7.1 Spherical Space
    2. 7.2 Spherical Transformations
    3. 7.3 Spherical Trigonometry
    4. 7.4 Spherical Geometry and the Extended Complex Plane
    5. 7.5 Planar Maps
    6. 7.6 Exercises
    7. Summary of Chapter 7
  16. 8. The Kleinian View of Geometry
    1. 8.1 Affine Geometry
    2. 8.2 Projective Reflections
    3. 8.3 Hyperbolic Geometry and Projective Geometry
    4. 8.4 Elliptic Geometry: the Spherical Model
    5. 8.5 Euclidean Geometry
    6. Summary of Chapter 8
  17. Special Symbols
  18. Further Reading
  19. Appendix 1: A Primer of Group Theory
  20. Appendix 2: A Primer of Vectors and Vector Spaces
  21. Appendix 3: Solutions to the Problems
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  22. Index