Overview

An introduction to geometry in the plane, both Euclidean and hyperbolic, this book is designed for an undergraduate course in geometry. With its patient approach, and plentiful illustrations, it will also be a stimulating read for anyone comfortable with the language of mathematical proof. While the material within is classical, it brings together topics that are not generally found together in books at this level, such as: parametric equations for the pseudosphere and its geodesics; trilinear and barycentric coordinates; Euclidean and hyperbolic tilings; and theorems proved using inversion. The book is divided into four parts, and begins by developing neutral geometry in the spirit of Hilbert, leading to the Saccheri–Legendre Theorem. Subsequent sections explore classical Euclidean geometry, with an emphasis on concurrence results, followed by transformations in the Euclidean plane, and the geometry of the Poincaré disk model.

Full Product Details

Author:   Matthew Harvey (University of Virginia)
Publisher:   Mathematical Association of America
Imprint:   Mathematical Association of America
Dimensions:   Width: 18.50cm , Height: 3.20cm , Length: 26.20cm
Weight:   1.150kg
ISBN:  

9781939512116

ISBN 10:   1939512115
Pages:   558
Publication Date:   30 September 2015
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In stock  
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Table of Contents

Axioms and models; Part I. Neutral Geometry: 1. The axioms of incidence and order; 2. Angles and triangles; 3. Congruence verse I: SAS and ASA; 4. Congruence verse II: AAS; 5. Congruence verse III: SSS; 6. Distance, length and the axioms of continuity; 7. Angle measure; 8. Triangles in neutral geometry; 9. Polygons; 10. Quadrilateral congruence theorems; Part II. Euclidean Geometry: 11. The axiom on parallels; 12. Parallel projection; 13. Similarity; 14. Circles; 15. Circumference; 16. Euclidean constructions; 17. Concurrence I; 18. Concurrence II; 19. Concurrence III; 20. Trilinear coordinates; Part III. Euclidean Transformations: 21. Analytic geometry; 22. Isometries; 23. Reflections; 24. Translations and rotations; 25. Orientation; 26. Glide reflections; 27. Change of coordinates; 28. Dilation; 29. Applications of transformations; 30. Area I; 31. Area II; 32. Barycentric coordinates; 33. Inversion I; 34. Inversion II; 35. Applications of inversion; Part IV. Hyperbolic Geometry: 36. The search for a rectangle; 37. Non-Euclidean parallels; 38. The pseudosphere; 39. Geodesics on the pseudosphere; 40. The upper half-plane; 41. The Poincaré disk; 42. Hyperbolic reflections; 43. Orientation preserving hyperbolic isometries; 44. The six hyperbolic trigonometric functions; 45. Hyperbolic trigonometry; 46. Hyperbolic area; 47. Tiling; Bibliography; Index.

Reviews

...The author succeeds in elevating Euclidean geometry in particular to the level of advanced undergraduate study and in so doing presents it as a counterpart to a first analysis course. At the same time, some readers may prefer to move on to new ideas and results more quickly; to that end, Harvey gives an indication of several paths through the book in his preface. - Choice To my mind, this book stands out from the crowd partly because of the way in which its imaginatively devised illustrations are used to stage the introduction of basic concepts and to guide the reader through various steps in a proof. It's not just a matter of pretty pictures, however, because Matthew Harvey's written commentary is easy going and yet mathematically precise. He has written a truly lovely book, which is now top of my reading list as an introduction to geometry at this level. - Peter Ruane

This expansive, copiously illustrated textbook provides rigorous and detailed treatments of Euclidean and hyperbolic plane geometries. Harvey (Univ of Virginia's College at Wise) begins with a thorough development of neutral geometry (i.e., geometry without the assumption of the parallel postulate or its negation) and then discusses Euclidean and hyperbolic geometry. Fifteen of the book's 47 chapters are devoted to Euclidean transformations and applications and 12 to hyperbolic topics, including the pseudosphere, upper-half plane, Poincare disk models, hyperbolic trigonometry, hyperbolic area, and tilings. Throughout, Harvey maintains a careful yet friendly tone and assumes mostly minimal prerequisites: facility with mathematical proof, some multivariable calculus, linear algebra, differential equations (needed only for the calculation of geodesics), and familiarity with complex numbers. The author succeeds in elevating Euclidean geometry in particular to the level of advanced undergraduate study and in so doing presents it as a counterpart to a first analysis course. At the same time, some readers may prefer to move on to new ideas and results more quickly; to that end, Harvey gives an indication of several paths through the book in his preface. - Choice To my mind, this book stands out from the crowd partly because of the way in which its imaginatively devised illustrations are used to stage the introduction of basic concepts and to guide the reader through various steps in a proof. It's not just a matter of pretty pictures, however, because Matthew Harvey's written commentary is easy going and yet mathematically precise. He has written a truly lovely book, which is now top of my reading list as an introduction to geometry at this level. - Peter Ruane

Author Information

Matthew Harvey is an Associate Professor of Mathematics at the University of Virginia's College at Wise, where he has taught since 2006. He graduated from the University of Virginia in 1995 with a BA in Mathematics, and from Johns Hopkins University in 2002 with a PhD in Mathematics.

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