Spherical Geometry and Its Applications  book cover
1st Edition

Spherical Geometry and Its Applications

ISBN 9780367196905
Published August 15, 2019 by Chapman and Hall/CRC
347 Pages 123 B/W Illustrations

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Book Description

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.


  • 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

Table of Contents

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


Spherical coordinates

Axiomatic spherical geometry

Basic axioms







Spherical Pythagorean theorem and law of sines

Spherical law of cosines and analogue formula

Right triangles

The four-parts and half angle formulas


Solution of triangles


The celestial sphere

Changing coordinates

Rise and set of objects in the sky

The measurement of time

Rise and set times in standard time


Regular solids


Spherical mappings

Rotations and reflections

Spherical projections


Review of complex numbers

Quaternions: Definitions and basic properties

Application to the sphere


Rotations and Reflections

Selected solutions to exercises

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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.