Spherical Geometry and Its Applications: 1st Edition (Hardback) book cover

Spherical Geometry and Its Applications

1st Edition

By Marshall A. Whittlesey

Chapman and Hall/CRC

316 pages | 123 B/W Illus.

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Hardback: 9780367196905
pub: 2019-08-26
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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. At the same time, the author wrote this book so that 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 and its practical applications in a mathematically rigorous form
  • The book uses the axiomatic method, but also shows how the subject can be approached with quaternions
  • Provides several proofs of some theorems which are appealing 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
  • Poses many problems based on propositions from the ancient text Sphaerica of Menelaus
  • Brings all these ideas under one title in a way that has not been done before

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

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

About the Author

Marshall A. Whittlesey is an associate professor of mathematics at California State University San Marcos, where he has worked since 2001. He received a BS (1992) in mathematics from Trinity College in Connecticut, and a PhD in mathematics from Brown University (1997) under the direction of John Wermer. He was a visiting assistant professor at Texas A&M University (1997-99) and was SE Warchawski Assistant Professor at University of California San Diego (1999-2001) before coming to Cal State San Marcos. He has a series of research publications in functions of several complex variables.

About the Series

Textbooks in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT012000
MATHEMATICS / Geometry / General