Геометрия комплексных чисел: Круги, преобразования Мебиуса и неевклидова геометрия.

Maths textbook
the book

Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non Euclidean Geometry is an undergraduate textbook on geometry, whose topics include circles, the complex plane, inversive geometry, and non Euclidean geometry. It was written by Hans Schwerdtfeger, and originally published in 1962 as Volume 13 of the Mathematical Expositions series of the University of Toronto Press. A corrected edition was published in 1979 in the Dover Books on Advanced Mathematics series of Dover Publications The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.

The book is divided into three chapters, corresponding to the three parts of its subtitle: circle geometry, Möbius transformations, and non Euclidean geometry. Each of these is further divided into sections (which in other books would be called chapters) and sub sections. An underlying theme of the book is the representation of the Euclidean plane as the plane of complex numbers, and the use of complex numbers as coordinates to describe geometric objects and their transformations. The chapter on circles covers the analytic geometry of circles in the complex plane. It describes the representation of circles by Hermitian matrices, the inversion of circles, stereographic projection, pencils of circles (certain one parameter families of circles) and their two parameter analogue, bundles of circles, and the cross ratio of four complex numbers. The chapter on Möbius transformations is the central part of the book, and defines these transformations as the fractional linear transformations of the complex plane (one of several standard ways of defining them). It includes material on the classification of these transformations, on the characteristic parallelograms of these transformations, on the subgroups of the group of transformations, on iterated transformations that either return to the identity (forming a periodic sequence) or produce an infinite sequence of transformations, and a geometric characterization of these transformations as the circle preserving transformations of the complex plane. This chapter also briefly discusses applications of Möbius transformations in understanding the projectivities and perspectivities of projective geometry. In the chapter on non Euclidean geometry, the topics include the Poincaré disk model of the hyperbolic plane, elliptic geometry, spherical geometry, and (in line with Felix Klein's Erlangen program) the transformation groups of these geometries as subgroups of Möbious transformations. This work brings together multiple areas of mathematics, with the intent of broadening the connections between abstract algebra, the theory of complex numbers, the theory of matrices, and geometry. Reviewer Howard Eves writes that, in its selection of material and its formulation of geometry, the book "largely reflects work of C. Caratheodory and E. Cartan".

As background on the geometry covered in this book, reviewer R. P. Burn suggests two other books, Modern Geometry: The Straight Line and Circle by C. V. Durell, and Geometry: A Comprehensive Course by Daniel Pedoe. Other books using complex numbers for analytic geometry include Complex Numbers and Geometry by Liang shin Hahn, or Complex Numbers from A to Z by Titu Andreescu and Dorin Andrica. However, Geometry of Complex Numbers differs from these books in avoiding elementary constructions in Euclidean geometry and instead applying this approach to higher level concepts such as circle inversion and non Euclidean geometry. Another related book, one of a small number that treat the Möbius transformations in as much detail as Geometry of Complex Numbers does, is Visual Complex Analysis by Tristan Needham.