Geometry of Complex Numbers
by Schwerdtfeger, HansFormat: Paperback
Pub. Date: 1980-02-01
Publisher(s): Dover Publications
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Summary
Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and two-dimensional non-Euclidean geometries.
Table of Contents
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
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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