Geometry Illuminated
Geometry Illuminated
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.
- Emphasises the logical development of geometries within an axiomatic system
- Brings together a distinctive selection of topics that are not often found together in books at this level
- Extensively illustrated to enhance the reader's journey
Product details
No date availableHardback
9781939512116
558 pages
262 × 185 × 32 mm
1.15kg
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.
