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Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry

Hans Schwerdtfeger
Publisher: 
Dover Publications
Publication Date: 
1980
Number of Pages: 
200
Format: 
Paperback
Price: 
14.95
ISBN: 
0486638308
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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INTRODUCTION: NOTE ON TERMINOLOGY AND NOTATIONS
  CHAPTER I. ANALYTIC GEOMETRY OF CIRCLES
  § 1. Representation of Circles by Hermitian Matrices
    a. One circle
    b. Two circles
    c. Pencils of circles
      Examples
  § 2. The Inversion
    a. Definition
    b. Simple properties of the inversion
      Examples
  § 3. Stereographic Projection
    a. Definition
    b. Simple properties of the stereographic projection
    c. Stereographic projection and polarity
      Examples
  § 4. Pencils and Bundles of Circles
    a. Pencils of circles
    b. Bundles of circles
      Examples
  § 5. The Cross Ratio
    a. The simple ratio
    b. The double ratio or cross ratio
    c. The cross ratio in circle geometry
      Examples
  CHAPTER II. THE MOEBIUS TRANSFORMATION
  § 6. Definition: Elementary Properties
    a. Definition and notation
    b. The group of all Moebius transformations
    c. Simple types of Moebius transformations
    d. Mapping properties of the Moebius transformations
    e. Transformation of a circle
    f. Involutions
      Examples
  § 7. Real One-dimensional Projectivities
    a. Perpectivities
    b. Projectivities
    c. Line-circle perspectivity
      Examples
  § 8. Similarity and Classification of Moebius Transformations
    a. Introduction of a new variable
    b. Normal forms of Moebius transformations
    c. "Hyperbolic, elliptic, loxodromic transformations"
    d. The subgroup of the real Moebius transformations
    e. The characteristic parallelogram
      Examples
  § 9. Classification of Anti-homographies
    a. Anti-homographies
    b. Anti-involutions
    c. Normal forms of non-involutory anti-homographies
    d. Normal forms of circle matrices and anti-involutions
    e. Moebius transformations and anti-homographies as products of inversions
    f. The groups of a pencil
      Examples
  § 10. Iteration of a Moebius Transformation
    a. General remarks on iteration
    b. Iteration of a Moebius transformation
    c. Periodic sequences of Moebius transformations
    d. Moebius transformations with periodic iteration
    e. Continuous iteration
    f. Continuous iteration of a Moebius transformation
      Examples
  § 11. Geometrical Characterization of the Moebius Transformation
    a. The fundamental theorem
    b. Complex projective transformations
    c. Representation in space
      Examples
  CHAPTER III. TWO-DIMENSIONAL NON-EUCLIDEAN GEOMETRIES
  § 12. Subgroups of Moebius Transformations
    a. The group U of the unit circle
    b. The group R of rotational Moebius transformations
    c. Normal forms of bundles of circles
    d. The bundle groups
    e. Transitivity of the bundle groups
      Examples
  § 13. The Geometry of a Transformation Group
    a. Euclidean geometry
    b. G-geometry
    c. Distance function
    d. G-circles
      Examples
  § 14. Hyperbolic Geometry
    a. Hyperbolic straight lines and distance
    b. The triangle inequality
    c. Hyperbolic circles and cycles
    d. Hyperbolic trigonometry
    e. Applications
      Examples
  § 15. Spherical and Elliptic Geometry
    a. Spherical straight lines and distance
    b. Additivity and triangle inequality
    c. Spherical circles
    d. Elliptic geometry
    e. Spherical trigonometry
      Examples
  APPENDICES
  1. Uniqueness of the cross ratio
  2. A theorem of H. Haruki
  3. Applications of the characteristic parallelogram
  4. Complex Numbers in Geometry by I. M. Yaglom
  BIBLIOGRAPHY
  SUPPLEMENTARY BIBLIOGRAPHY
  INDEX

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