Geometry Illuminated
An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry

An extensively illustrated introduction to Euclidean and hyperbolic geometry in the plane, designed for undergraduate courses in geometry.

Matthew Harvey (Author)

9781939512116, The Mathematical Association of America

Hardback, published 30 September 2015

558 pages
26.2 x 18.5 x 3.2 cm, 1.15 kg

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.

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.

Subject Areas: Euclidean geometry [PBMH], Geometry [PBM], Mathematics [PB]