GEOMETRY OF COMPLEX NUMBERS CIRCLE GEOMETRY, MOEBIUS TRANSFORMATION, NON-EUCLIDEAN GEOMETRY

Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and 2-dimensional non-Euclidean geometries.

Since its initial publication in 1962, Professor Schwerdtfeger's illuminating book has been widely praised for generating a deepe understanding of the geometrical theory of analytic functions as well as of the connection s between different branches of geometry. Its focus lies in the intersection of geometry, analysis, and algebra, with the exposition generally taking place on a moderately advanced level. Much emphasis, however, has been given to the careful exposition of details and to the development of an adequatealgebraic technique. In three broad chapters, the author clearly and elegantly approaches his subject. The first chapter, Analytic Geometry of Cyrcles, treats such topics as representation of circles by Hermitian matrices, inversion, stereographic projection, and the cross ratio. The second chapter considers in depth the Moebius transformation: its elementary properties, real one-dimensional projectivities, similarity and classification of various kinds, anti-homographiess, iteration, and geometrical characterizaztion. The final chapter,Two-Dimensional Non-Euclidean Geometries, discusses subgroups of Moebius transformations, the geometry of a transformation group, hyperbolic geometry, and spherical and elliptic geometry. For this Dover edition, Professor Schwerdtfeger has added four new appendices and supplementary bibliography. Advanced undergraduates who possess a working knoeldge of the algebra of complex numbers and of the elements of analytical geometry and linear algebra will gratly profit from reading this book. It will also prove a stimulating and thought-provoking book to mathematics professors and teachers.