Spherical Geometry and Its Applications

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

Review of three-dimensional geometry Geometry in a plane Geometry in space Plane trigonometry Coordinates and vectors The sphere in space Great circles Distance and angles Area Spherical coordinates Axiomatic spherical geometry Basic axioms Angles Triangles Congruence Inequalities Area Trigonometry Spherical Pythagorean theorem and law of sines Spherical law of cosines and analogue formula Right triangles The four-parts and half angle formulas Dualization Solution of triangles Astronomy The celestial sphere Changing coordinates Rise and set of objects in the sky The measurement of time Rise and set times in standard time Polyhedra Regular solids Crystals Spherical mappings Rotations and reflections Spherical projections Quaternions Review of complex numbers Quaternions: Definitions and basic properties Application to the sphere Triangles Rotations and Reflections Selected solutions to exercises

Marshall A. Whittlesey is an Associate Professor of Mathematics at California State University San Marcos. He received a BS (1992) from Trinity College in Connecticut, and a PhD from Brown University (1997) under the direction of John Wermer. He was a Visiting Assistant Professor at Texas A&M University was SE Warchawski Assistant Professor at University of California San Diego (1999-2001). He has a series of research publications in functions of several complex variables.