Spherical Geometry and Its Applications (Textbooks in Mathematics)

Review of three-dimensional geometry -- The sphere in place -- Axiomatic spherical geometry -- Trigonometry -- Astronomy -- Polyhedra -- Spherical mappings -- Quaternions;Spherical Geometry and Its Applications introduces spherical geometry and its practical applications in a mathematically rigorous form. The text can serve as a course in spherical geometry for mathematics majors. Readers from various academic backgrounds can comprehend various approaches to the subject. The book introduces an axiomatic system for spherical geometry and uses it to prove the main theorems of the subject. It also provides an alternate approach using quaternions. The author illustrates how a traditional axiomatic system for plane geometry can be modified to produce a different geometric world - but a geometric world that is no less real than the geometric world of the plane. Features: A well-rounded introduction to spherical geometry Provides several proofs of some theorems to appeal to larger audiences Presents principal applications: the study of the surface of the earth, the study of stars and planets in the sky, the study of three- and four-dimensional polyhedra, mappings of the sphere, and crystallography Many problems are based on propositions from the ancient text Sphaerica of Menelaus Contents......Page 6 Preface......Page 8 1 Geometry in a plane......Page 14 2 Geometry in space......Page 21 3 Plane trigonometry......Page 25 4 Coordinates and vectors......Page 32 5 Great circles......Page 38 6 Distance and angles......Page 48 7 Area......Page 50 8 Spherical coordinates......Page 54 9 Basic axioms......Page 58 10 Angles......Page 72 11 Triangles......Page 85 12 Congruence......Page 97 13 Inequalities......Page 105 14 Area......Page 119 15 Spherical Pythagorean theorem and law of sines......Page 124 16 Spherical law of cosines and analogue formula......Page 131 17 Right triangles......Page 140 18 The four-parts and half-angle formulas......Page 149 19 Dualization......Page 153 20 Solution of triangles......Page 162 21 The celestial sphere......Page 170 22 Changing coordinates......Page 180 23 Rise and set of objects in the sky......Page 188 24 The measurement of time......Page 200 25 Rise and set times in standard time......Page 209 26 Regular solids......Page 212 27 Crystals......Page 225 28 Rotations and re ections......Page 234 29 Spherical projections......Page 245 30 Review of complex numbers......Page 256 31 Quaternions: De nitions and basic properties......Page 260 32 Application to the sphere......Page 268 33 Triangles......Page 280 34 Rotations and re ections......Page 285 Selected solutions to exercises......Page 300 Bibliography......Page 330 Index......Page 336 The author introduces spherical geometry and it practical applications in a mathematically rigorous form. Readers will see how the axiom system for plane geometry can be modified in certain ways to produce a completely different geometric world.