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Geometry

Book Description

This is an undergraduate textbook that reveals the intricacies of geometry. The approach used is that a geometry is a space together with a set of transformations of that space (as argued by Klein in his Erlangen programme). The authors explore various geometries: affine, projective, inversive, non-Euclidean and spherical. In each case the key results are explained carefully, and the relationships between the geometries are discussed. This richly illustrated and clearly written text includes full solutions to over 200 problems, and is suitable both for undergraduate courses on geometry and as a resource for self study.

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
  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
  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
    5. 3.5 Cross-Ratio
    6. 3.6 Exercises
  12. 4. Projective Geometry: Conics
    1. 4.1 Projective Conics
    2. 4.2 Tangents
    3. 4.3 Theorems
    4. 4.4 Duality and Projective Conics
    5. 4.5 Exercises
  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
  14. 6. Non-Euclidean Geometry
    1. 6.1 Non-Euclidean Geometry
    2. 6.2 Non-Euclidean Transformations
    3. 6.3 Distance in Non-Euclidean Geometry
    4. 6.4 Geometrical Theorems
    5. 6.5 Non-Euclidean Tessellations
    6. 6.6 Exercises
  15. 7. Spherical Geometry
    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 Exercises
  16. 8. The Kleinian View of Geometry
    1. 8.1 Affine Geometry
    2. 8.2 Projective Reflections
    3. 8.3 Non-Euclidean Geometry and Projective Geometry
    4. 8.4 Spherical Geometry
    5. 8.5 Euclidean Geometry and Non-Euclidean Geometry
  17. Appendix 1: A Primer of Group Theory
  18. Appendix 2: A Primer of Vectors and Vector Spaces
  19. 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
  20. Index