Geometry

I have purchased and worked through several geometry books in the past year. This is, by far, the best one I've come across. It is modern, it is fun, and it is enlightening. I love the clear worked out numerical examples. As I write geometric computer programs it gives me a way to check each function as I go along. It has an abundance of wonderful illustrations that help you understand the theorems and concepts. I found this Geometry to be clearly and fully presented for an independent reader, and not just a supplement to some lecture course. There is a minimum amount of mathematical history included, but it is well-chosen and well-written. Plus, I learned how to draw the tessellations of the Poincare disc, ala M.C. Escher. So that alone made it worth the price and the effort. Definitely 5 stars.

It is a beautiful book to support University course "Foundations of geometry" for undergraduate students.

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.

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 neede

This book gives a beautiful overview of geometry of 2 dimensions. All of the book is about many plane geometries I have heard of, but didn't really know. This book changed that. The first chapter treats some basics about conics. The second chapter is on affine geometry. The third and fourth chapters are about projective geometry. In the fifth chapter you will be led through Inversive geometry which functions as a base for the sixth and seventh chapter. The sixth chapter has as itst title Non-Euclidean geometry, but it is in fact the Hyperbolic geometry of Boljay in a formulation of Henry Poincaré. The seventh chapter is about Spherical Geometry. In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself. It is clearly demonstrated that this is less trivial than you would expect.I learned two things from this book. The first is, that you can, in principle, prove every theorem of geometry by just using Euclidean geometry. But if you do this, the amount of work it takes can be very huge indeed. It is a far better strategy to try to determine what geometry is best suited for the problem at hand, and solve it within that geometry. Since the book gives a very clear picture not only of the particular geometries, but also to how the geometries relate to each other, you have, as an extra bonus, insight in the level of abstraction and the scope of your theorem. The second thing I learned is how you can use geometry to make concepts as simple as 'triangle' precise. What I mean is this: a right angle triangle is not the same as an equilateral triangle. But both are the same in the sense that they are both triangles. The question is this: how can two 'things' be the same and at the same time not 'the same'? The book gives an answer to this 'question about the meaning of abstractions'. It gives the following solution. Take a triangle, ANY triangle. Consider the group of all affine transformations A (which consists of an uncountably infinite set of transformations.) If you subject this one triangle Tr to every affine transformation in this group A, you will have created a set consisting of exactly ALL triangles. In other words, the abstract idea of 'triangle' consists of ONE triangle Tr together with the set of ALL affine transformations. You can denote this as the pair (Tr, A). In the same way you can express the abstract idea of ellipse by the pair (El, A), and the abstract idea of parabola by the pair (Par, A). And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations.The book presupposes some group theory and some knowledge of linear algebra. Furthermore you ha