Geometry: A Comprehensive Course

I purchased this book to provide a second source of information for a college senior/ graduate level geometry class. It was quite helpful at times in giving second explanations of topics covered in class at a price significantly less than the class textbook. The book is comprehensive, and reasonably well written for a math book. One drawback was that it does not discuss Poincare's upper half plane model of hyperbolic geometry, only Poincare's disc model is covered. Inversions are well covered, and one could use this information to map from the disc model to the upper half plane model, so this was not a serious omission. I have no regrets about buying this book.

This text - comprised of detailed accounts of Euclidean, Affine, and Projective geometries, a thorough discussion of the Poincare model of hyperbolic geometry, and a motivating chapter on Algebraic geometry - is remarkable for its clarity and simplicity. Very little is initially expected of the reader - a sound knowledege of linear algebra, complex arithmetic and trigonometry - and the proof style is not too formal. Considering the level of detail of its treatment (it's a little more than twice the length of Coxeter's Geometry Revisited, which shares a similar selection of topics) it is ideal for undergraduates or those looking for a good, inexpensive text for self study. Throughout the text the author emphasizes the use of basic algebraic techniques as an aid to finding clear and simple proofs. In more than one case a result is proved several times, each proof illustrating a different technique. In the first chapter, the utility of the vector approach is highlighted by using vector arithmetic, inner-products and exterior algebra to prove several classic theorems of plane geometry. In later chapters elementary group theory, Moebius transformations and linear algebra are used extensively in the discussions of the mappings of the Euclidean plane, of the mappings of the inversive plane, and of projective geometry respectively. Basically, this is a good, detailed undergraduate introduction to geometry. It's perhaps a little less entertaining than Coxeter's introductory geometry books but it has a much friendlier price-tag.

Be aware that the preface of this book explains that it is based on a course for junior and senior college students and first-year graduate students. This book is much more than the average person will ever need. A simpler approach for people like me who just want to prepare for calculus is Geometry and Trigonometry for Calculus by Peter H. Selby.

A wonderful book, value for money! If you are learning geometry for interest or to prepare for competitions such as IMO, then this book is for you! The theorems and diagrams are complete and straight to the point. The author presents the informations in a succinct manner; thus it is easy for one to follow and comprehend. I guess it may be one of the most complete book on geometry in the market!