Geometry of Complex Numbers (Dover Books on Mathematics) New edition Edition

by Hans Schwerdtfeger (Author)

6 customer reviews

ISBN-13: 978-0486638300

ISBN-10: 0486638308

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Since its initial publication in 1962, Professor Schwerdtfeger's illuminating book has been widely praised for generating a deeper understanding of the geometrical theory of analytic functions as well as of the connections 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 adequate algebraic technique.
In three broad chapters, the author clearly and elegantly approaches his subject. The first chapter, Analytic Geometry of Circles, 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-homographies, iteration, and geometrical characterization. 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 a supplementary bibliography.
Advanced undergraduates who possess a working knowledge of the algebra of complex numbers and of the elements of analytical geometry and linear algebra will greatly profit from reading this book. It will also prove a stimulating and thought-provoking book to mathematics professors and teachers.

Geometry of Complex Numbers (Dover Books on Mathematics)

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Product Details

Series: Dover Books on Mathematics
Paperback: 224 pages
Publisher: Dover Publications; New edition edition (February 1, 1980)
Language: English
ISBN-10: 0486638308
ISBN-13: 978-0486638300
Product Dimensions: 8.2 x 5.6 x 0.4 inches
Shipping Weight: 8.5 ounces (View shipping rates and policies)
Average Customer Review: 5.0 out of 5 stars See all reviews (6 customer reviews)
Amazon Best Sellers Rank: #1,020,292 in Books (See Top 100 in Books)
- #489 in Books > Textbooks > Science & Mathematics > Mathematics > Geometry
- #974 in Books > Science & Math > Mathematics > Geometry & Topology
- #1476 in Books > Science & Math > Earth Sciences > Geology

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61 of 64 people found the following review helpful

Should be a "must read" for math students

By Carl Mclaren on March 9, 2002

Format: Paperback

This inexpensive book covers material not easily found elsewhere but key in understanding complex functions. The problem with complex functions is they are hard to visualize because the input is a plane and the output is another plane. The book covers Circles, Moebius transforms, and Non-Euclidean Geometry. The level is senior undergraduate, 1st year graduate. The book is easy to understand with good exercises. I really like this book.

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20 of 21 people found the following review helpful

plane geometry and complex numbers

By Gilles Benson on October 6, 2006

Format: Paperback

I discovered this book some twenty years ago while trying to improve my knowledge of plane geometry; I used it especially to work on circle pencils: a part of geometry I had already encountered time and again; setting up circles through two-rowed hermitian matrices and linear transforms {z->(az+b)/(cz+d) }as done in the book is both very pretty and efficient. The appendix (numbered 3) describing the use and applications of the characteristic parallelogram really appealed to me. I was also quite impressed by the way the cross ratio of 4 complex numbers is dealt with in the book; to put icing on the cake, one can find within those 200 pages some knowledge of non euclidian plane geometry ...and dynamical systems associated with linear transforms in the complex plane; very informative and quite refreshing.

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35 of 42 people found the following review helpful

This book contained the stuff I wanted to know

By P. Murray on May 9, 2003

Format: Paperback Verified Purchase

I was interested in projecting a network onto hyperbolic space using the upper half plane projection. This book contained the equations relating to that, particularly the moebius transformation z' = (az+b) / (cz + d), and also stuff on stereographic mapping which I found useful.
I have not taken the trouble to understand much of the more in-depth parts of the book, but it is so clear and step-by-step that even though I am not a math student, I'm fairly confident that I could. The whole thing was fairly mind-opening.
Interestingly, after reading this and developing my own intuitions (eg: that flat translation, rotation and scaling are special cases of parabolic, elliptical and hyperbolic transformations with a fixed point at infinity), a re-reading discovered these conclusions in the book. So you can take the exposition and run with it. What I'd really like is to be able to get the n'th root of a transformation (to animate them). I suspect that that's in there too.
The book does not cover real-world applications (aerodynamics, electrodynamics), but that's cool. It's purely about the math.

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Most Recent Customer Reviews

Five Stars

Excellent book and excellent service!

Published 15 months ago by Thomas R. Schulte

fast shipping

I would have to wait months before I can write a real review about the book because I would have to finish reading it. Read more

Published on October 20, 2013 by hector

a good beginning

Schwerdtfeger's nice little book starts at the beginning with geometry of circles, Moebius transformations (a third of the book), and it covers some selected aspects of complex... Read more

Published on August 5, 2006 by Palle E T Jorgensen

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