Geometry 1st Edition

by David A. Brannan (Author)

13 ratings

ISBN-13: 978-0521597876

ISBN-10: 0521597870

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This textbook demonstrates the excitement and beauty of geometry. The approach is that of Klein in his Erlangen program: a geometry is a space together with a set of transformations of that space. The authors explore various geometries: affine, projective, inversive, non-Euclidean and spherical. In each case they carefully explain key results and discuss the relationship among geometries. This richly illustrated and clearly written text includes full solutions to over 200 problems and is suitable both for undergraduate courses on geometry and as a resource for self study.

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Editorial Reviews

Review

"...an excellent and precisely written textbook that should be studied in depth by all would-be mathematicians." Mathematical Reviews

"There are many things to like about this book. It conveys the beauty and excitement of the subject, avoiding the dryness of many geometry texts. I was very pleased to see the classification of conics developed in Chapters 1 through 4...The effort put in on the problems and answers is very impressive...The language is clear, with little to fear from Anglicisms...presentation is clean and pleasing to the eye...it raised in me a desire to teach elementary geometry again, something I have not felt or done in years." MAA Online

Book Description

This textbook demonstrates the excitement and beauty of geometry. The approach is that of Klein in his Erlangen programme: a geometry is a space together with a set of transformations of that space. The authors explore various geometries: affine, projective, inversive, non-Euclidean and spherical. In each case the key results are explained carefully, and the relationships between the geometries are discussed. This richly illustrated and clearly written text includes full solutions to over 200 problems, and is suitable both for undergraduate courses on geometry and as a resource for self study.

Product details

Publisher : Cambridge University Press; 1st edition (April 1, 1999)
Language: : English
Paperback : 512 pages
ISBN-10 : 0521597870
ISBN-13 : 978-0521597876
Item Weight : 2.01 pounds
Dimensions : 7.4 x 1 x 9.7 inches

Best Sellers Rank: #965,735 in Books (See Top 100 in Books)
- #192 in Topology (Books)
- #392 in Geometry (Books)
- #4,859 in Applied Mathematics (Books)

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13 ratings

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avniu

"Who is who" in geometry

Reviewed in the United States on August 3, 2010

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The book presents clearly several geometries in the spirit of Kleinean unifying view of geometry. It begins with a recapitulation of many important facts on conic sections, since conic sections played an extraordinary role in science over two thousand of years, and on the other hand the authors use conic sections as very representative geometric objects to explain transformations that specify affine, projective, inversive, non-Euclidean and spherical geometry. The key ideas, results and techniques that served as corner stone in the discovery and development of geometries are presented clearly and fluently using algebraic methods combined with rich illustrations, which provide the shortest path to the proofs and relationships between geometries. As an excellent example of this approach is the Joachimstal's notation for plane conics. The book can also serve as `who is who' in geometry. However, the book has no references to the literature, which would have been very useful for further reading.

Avni Pllana

4 people found this helpful

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Murph

interesting book

Reviewed in the United States on August 17, 2011

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a class i took used this book, i sold the book after the class ended, for $70, but i'm looking for an opportunity to buy one back in a few years when the price goes down, so it must be a good book, alright. the graphics stick in your head, and the book develops a sense for geometry, it gives you more of a feel for how the math explains the geometry. prerequisites for understanding it might be just high school math, but then knowing some college math is also helpful of course, it was a while ago, but i remeber multivariable calculus and linear algebra in there

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doodler

Five Stars

Reviewed in the United States on September 29, 2015

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Wow, want a well written book. Provides a clear narrative and motivation behind the ideas and concepts.

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William A. Huber

Good and enjoyable for a wide range of readers

Reviewed in the United States on May 6, 2003

A quarter century ago I noticed that some of the graduate physics students in my university were carrying around copies of Scientific American. Armed with that clue, I dug out every article on the newly discovered fundamental particles. Within the space of a week of fairly easy reading I was able to acquire a good sense of what this subject was all about. These articles explained the basic stuff our professors assumed we must know (but most of us surely didn't).
Brannan, Esplen, and Gray's Geometry accomplish for math what those Scientific American articles did for physics: speaking at a level accessible to anyone with a good high school education, they bring the interested reader up to speed in affine, projective, hyperbolic, inversive, and spherical geometry. They provide the simple explanations, diagrams, and computational details you are assumed to know-but probably don't-when you take advanced courses in topology, differential geometry, algebraic geometry, Lie groups, and more. I wish I had had a book like this when I learned those subjects.
Individual chapters of about 50 pages focus on distinct geometries. Each one is written to be studied in the course of five evenings: a week or two of work apiece. Although they build sequentially, just about any of them can be read after mastering the basic ideas of projective geometry (chapter 3) and inversive geometry (chapter 5). This makes the latter part of the book relatively accessible even to the less-committed reader and an effective handbook for someone looking for just an overview and basic formulas.
The approach is surprisingly sophisticated. The authors do not shy away from introducing and using a little bit of group theory, even at the outset. (Scientific American, even in its heyday, never dared do that.) They present all geometries from a relatively modern point of view, as the study of the invariants of a transitive group of transformations on a set. Many explanations and proofs are based on exploiting properties of these transformations. This brings a welcome current of rigor and elegance to a somewhat static subject long relegated to out of date or sloppy authors (with the exception of a few standouts, such as Lang & Murrow's "Geometry").
One nice aspect is the authors' evident awareness of and appreciation for the history of mathematics. Marginal notes begin at Plato and wind up with Felix Klein's Erlangen program some 2300 years later. Although the text does not necessarily follow the historical development of geometry, its references to that development provide a nice context for the ideas. This is an approach that would improve the exposition of many math texts at all levels.
The authors are British and evidently write for students with slightly different backgrounds than American undergraduates. Obvious prerequisites are a mastery of algebra and a good high school course in Euclidean geometry. Synopses of the limited amounts of group theory and linear algebra needed appear in two brief appendices. However, readers had better be intuitively comfortable with matrix operations, including diagonalization and finding eigenspaces, because matrices and complex numbers are used throughout the book for performing computations and developing proofs. A knowledge of calculus is not needed. Indeed, calculus is not used in the first two-thirds of the book, appearing only briefly to derive a distance formula for hyperbolic geometry (a differential equation for the exponential map is derived and solved). During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses). Definitions of these functions are not routinely provided, but algebraic identities appear in marginal notes where they are needed.
Now for the quibbles. The book has lots of diagrams, but not enough of them. The problems are usually trivial, tending to ask for basic calculations to reinforce points in the text. The text itself does not go very deeply into any one geometry, being generally content with a few illustrative theorems. An opportunity exists here to create a set of gradually more challenging problems that would engage smarter or more sophisticated readers, as well as show the casual reader where the theories are headed.
This book is the work of three authors and it shows, to ill effect, in Chapter 6 ("non-euclidean geometry"). Until then, the text is remarkably clean and free of typographical and notational errors. This chapter contains some glaring errors. For example, a function s(z) is defined at the beginning of a proof on page 296, but the proof confusingly proceeds to refer to "s(0,c)", "s(a,b)", and so on.
The written-by-committee syndrome appears in subtler ways. There are few direct cross-references among the chapters on inversive, hyperbolic, and spherical geometry, despite the ample opportunities presented by the material. Techniques used in one chapter that would apply without change to similar situations in another are abandoned and replaced with entirely different techniques. Within the aberrant Chapter 6, some complex derivations could be replaced by much simpler proofs based on material earlier in the chapter.
The last chapter attempts to unify the preceding ones by exhibiting various geometries as sub-geometries of others. It would have been better to make the connections evident as the material was being developed. It is disappointing, too, that nothing in this book really hints at the truly interesting developments in geometry: differentiable manifolds, Lie groups, Cartan connections, complex variable theory, quaternion actions, and much more. Indeed, any possible hint seems willfully suppressed: the matrix groups in evidence, such as SL(2, R), SU(1,1, C), PSL(3, R), O(3), and so on, are always given unconventional names, for instance. Even where a connection is screaming out, it is not made: the function abstractly named "g" on pages 296-97 is the exponential map of differential geometry, for instance.
Despite these limitations, Brannan et al. is a good and enjoyable book for anyone from high school through first-year graduate level in mathematics.

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Shankar N. Swamy

A Nice Introduction to Geometrys - Precise and Accurate!

Reviewed in the United States on January 2, 2005

This book is at the level of a freshman mathematics course.

Mainly deals with Affine, Projective, Inversive, Spherical and Non-Euclidean geometrys. The beauty of the book is in its accuracy. Someone has done a good job of technical editing! There is always a risk of getting things wrong when attempting to make mathematics accessible at a lower level. The authors seem to have avoided that pitfall with significant success. The subject matter is focused and to the point. At each point, it precisely explains what is intended and moves on without digressions.

I have had significant interests in geometries, and work in a area that uses some elementary projective geometry. At times I get asked some relatively simple questions such as "why do we need 4x4 matrices in Computer Graphics?" Often I just answer such questions to the minimum (" ... it makes applying translations easier ..."). I never proffer a deeper answer because most people I run into either have no background to understand a more technical explanation in terms of the algebra of projective planes or they don't care - they don't need to, for most of their work!. (Many of the computer graphics folks I have met think that the homogeneous coordinates is an ad-hok concept that was invented as a "trick"!)

Occasionally, I do run into some who are interested in knowing the analytical reasoning behind some of the transformations used everyday in computer graphics. This book demonstrated to me how to talk to some of those without having to use very abstract concepts of geometry. I read it first in 1999. I have revisited it since, many times for the nice figures they provide. First time, it took me about three "after work" months to study through the book - not bad at all for a 350+ pages mathematics book!

By looking at the diagrams in the book, I learned how to draw simple diagrams instead of abstract symbols to explain the concepts, theorems and problems. For a book that is as simple, the technical content is remarkably precise and accurate. The book assumes minimal background in mathematics.

Recommended for people interested in computer graphics and want to understand the transformations in there deeper (for whatever reason!), under-graduate students interested in geometry, and for anyone with a casual interest in geometry.

20 people found this helpful

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