Geometry of Complex Numbers

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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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