Geometry

Rating:
Summary: Good and enjoyable for a wide range of readers
Review: 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 This brings a welcome current of rigor and elegance to a somewhat static subject long relegated to out of date or sloppy authors (with the exception of a few standouts, such as Lang & Murrow's "Geometry").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.

Rating:
Summary: Good and enjoyable for a wide range of readers
Review: 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.

Rating:
Summary: Best on affine transformations used in computer graphics
Review: I'm trying to understand transformations used in computer graphics, for example world transformation used in Windows GDI API. And I found this book to be the best description on the topic, that is affine transformations

Rating:
Summary: A lovely Introduction to all kinds of Plane Geometries
Review: 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 have to know a little calculus. I have very little knowledge of group theory, and I have just about enough knowledge and skill about linear algebra to know the difference between an orthogonal and unitary matrix, and to know what eigenvectors are. I have studied the first 5 chapters of CALCULUS from Tom M. Apostol, which does not go too deep into linear algebra. This proved to be enough.

I have only one point of critique. Virtually all problems in the book are of the 'plug in type', even those at the end of every chapter (from which, by the way, you cannot find the solutions at the end of the book, while the solutions of those in the text can be found in an appendix). If you have understood the text, you have no difficulties whatsoever to solve them. The problems are not challenging enough to give you a real skill in all of these geometries, although they do become more challenging in later chapters. They are only intended to help you to understand the basic principles of all of these geometries, no more, no less. So if you want to have a tool to help you in obtaining a greater skill in, say, the special theory of relativity by studying hyperbolic geometry, this is not a suitable book. That is why I have given it 4 stars, and not the full 5 stars.

I also have a piece of advise. Although the problems are, from a conceptual point of view, not challenging, a mistake is easily made. Therefore it is best to solve the problems by making use of a mathematical program like Maple or Mathematica. If you then have made a mistake, you can backtrack exactly where you have made it, and let the program take care of all of the tedious calculations. This has also stimulated to try to calculate some outcomes by following a different approach, and then to compare the results.

I have enjoyed studying this book immensely.

Rating:
Summary: A Nice Introduction to Geometrys - Precise and Accurate!
Review: 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.