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Geometry of Complex Numbers by Hans Schwerdtfeger

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Geometry of Complex Numbers writthen by Hans Schwerdtfeger . This is an other great free downloadable mathematics book cover the following topics of complex number.

  • Chapter I. Analytic Geometry Of Circles
  • 1. Representation of Circles by Hermitian Matrices
    One circle, Two circles, Pencils of circles, Examples

  • 2. The Inversion
    Definition, Simple properties of the inversion

  • 3. Stereographic Projection
    Simple properties of the stereographic projection, Stereographic projection and polarity

  • 4. Pencils and Bundles of Circles
    Pencils of circles, Bundles of circles

  • 5. The Cross Ratio
    The simple ratio, The double ratio or cross ratio, The cross ratio in circle geometry

  • Chapter II. The Moebius Transformation
  • 6. Definition:
    Elementary Properties, Definition and notation, The group of all Moebius transformations, Simple types of Moebius transformations, Mapping properties of the Moebius transformations, Transformation of a circle, Involutions

  • 7. Real One-dimensional Projectivities
    Perpectivities, Projectivities, Line-circle perspectivity

  • 8. Similarity and Classification of Moebius Transformations
    Introduction of a new variable, Normal forms of Moebius transformations, "Hyperbolic, elliptic, loxodromic transformations", The subgroup of the real Moebius transformations, The characteristic parallelogram

  • 9. Classification of Anti-homographies
    Anti-homographies, Anti-involutions, Normal forms of non-involutory anti-homographies, Normal forms of circle matrices and anti-involutions, Moebius transformations and anti-homographies as products of inversions, The groups of a pencil

  • 10. Iteration of a Moebius Transformation
    General remarks on iteration, Iteration of a Moebius transformation, Periodic sequences of Moebius transformations, Moebius transformations with periodic iteration, Continuous iteration, Continuous iteration of a Moebius transformation

  • 11. Geometrical Characterization of the Moebius Transformation
    The fundamental theorem, Complex projective transformations, Representation in space

  • Chapter III. Two-Dimensional Non-Euclidean Geometries
  • 12. Subgroups of Moebius Transformations
    The group U of the unit circle, The group R of rotational Moebius transformations, Normal forms of bundles of circles, The bundle groups, Transitivity of the bundle groups

  • 13. The Geometry of a Transformation Group
    Euclidean geometry, G-geometry, Distance function, G-circles

  • 14. Hyperbolic Geometry
    Hyperbolic straight lines and distance, The triangle inequality, Hyperbolic circles and cycles, Hyperbolic trigonometry, Applications

  • 15. Spherical and Elliptic Geometry
    Spherical straight lines and distance, Additivity and triangle inequality, Spherical circles, Elliptic geometry, Spherical trigonometry

  • Appendices
    Uniqueness of the cross ratio, A theorem of H. Haruki, Applications of the characteristic parallelogram, Complex Numbers in Geometry by I. M. Yaglom

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    Geometry of Complex Numbers by Hans Schwerdtfeger
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  • English
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