Geometry of Complex Numbers

by Hans Schwerdtfeger

Overview

The elements of abstract algebra have almost everywhere found a place in the undergraduate courses of universities, but this has happened to some extent at the expense of courses on geometry. Therefore a book which applies some notions of algebra to geometry, showing in a deliberately restricted domain their interrelation with geometrical ideas, is a useful counterbalance in the present trend to generalization and abstraction.

This book should be of great value to students of mathematics in their second or third year at the university and be used by them concurrently with an introductory course on functions of a complex variable. It should give them a basis for the geometrical aspects of this theory and simultaneously help them to extend their understanding of the connections between some classical branches of geometry. It will also be useful to anyone, from pure mathematician to electrical engineer, who wishes to deepen his knowledge of the complex number system.

Product Details

ISBN-13:	9781487581985
Publisher:	University of Toronto Press, Scholarly Publishing
Publication date:	04/22/2019
Series:	Heritage
Pages:	202
Product dimensions:	6.14(w) x 9.21(h) x 0.46(d)

About the Author

HANS SCHWERDTFEGER studied Mathematics in Germany in the Universities of Leipzig, Göttingen, and Bonn where he obtained his Dr. phil. Degree (Ph. D.) in 1934. He has held the position of lecturer in Mathematics in the University of Adelaide, Australia (1940-47) and Senior Lecturer in the University of Melbourne (1948-1957). He was Visiting Professor of Mathematics in Queen's University, Kingston, Ont. (1954-55). He was appointed an Associate Professor of Mathematics at McGill University, Montreal in 1958 and became a full Professor in 1960.

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