Geometry Illuminated: An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry (Mathematical Association of America Textbooks)

__Geometry Illuminated__ is an introduction to geometry in the plane, both Euclidean and hyperbolic. It is designed to be used in an undergraduate course on geometry, and as such, its target audience is undergraduate math majors. However, much of it should be readable by anyone who is comfortable with the language of mathematical proof. Throughout, the goal is to develop the material patiently. One of the more appealing aspects of geometry is that it is a very "visual" subject. This book hopes to takes full advantage of that, with an extensive use of illustrations as guides. __Geometry Illuminated__ is divided into four principal parts. Part 1 develops neutral geometry in the style of Hilbert, including a discussion of the construction of measure in that system, ultimately building up to the Saccheri-Legendre Theorem. Part 2 provides a glimpse of classical Euclidean geometry, with an emphasis on concurrence results, such as the nine-point circle. Part 3 studies transformations of the Euclidean plane, beginning with isometries and ending with inversion, with applications and a discussion of area in between. Part 4 is dedicated to the development of the Poincaré disk model, and the study of geometry within that model. While this material is traditional, __Geometry Illuminated__ does bring together topics that are generally not found in a book at this level. Most notably, it explicitly computes parametric equations for the pseudosphere and its geodesics. It focuses less on the nature of axiomatic systems for geometry, but emphasizes rather the logical development of geometry within such a system. It also includes sections dealing with trilinear and barycentric coordinates, theorems that can be proved using inversion, and Euclidean and hyperbolic tilings. 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. -- Provided By Publisher. 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. Matthew Harvey. Includes Bibliographical References And Index. An introduction to Euclidean and hyperbolic geometry in the plane, this book is designed for an undergraduate course in geometry, but will also be a stimulating read for anyone comfortable with the language of mathematical proof. The text is extensively illustrated and brings together topics not typically found together.