Modern Geometry with Applications

By: George Jennings

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Paperback | 1 July 1997

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204 Pages

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23.39 x 15.6 x 1.09

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This is an introduction to the theory and applications of modern geometry. It differs from other books in its field in its emphasis on applications and its discussion of Special Relativity as a major example of a non-Euclidean geometry. Besides Special Relativity, it covers two other important areas of non-Euclidean geometry: spherical geometry (used in navigation and astronomy) and projective geometry (used in art). In addition, it reviews many useful topics from Euclidean geometry, emphasizing transformations, and includes a chapter on conics and planetary orbits. Applications are stressed throughout the book. Every topic is motivated by an application and many additional applications are given in the exercises. The book would be an excellent introduction to higher geometry for those students, especially prospective mathematics and teachers, who need to know how geometry is used in addition to its formal theory.

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Foreword
Euclidean Geometry	p. 1
Euclidean Space	p. 1
Isometries and Congruence	p. 2
Reflections in the Plane	p. 3
Reflections in Space	p. 5
Translations	p. 7
Rotations	p. 9
Applications and Examples	p. 11
Some Key Results of High School Geometry: The Parallel Postulate, Angles of a Triangle, Similar Triangles, and the Pythagorean Theorem	p. 17
SSS, ASA, and SAS	p. 27
The General Isometry	p. 36
Appendix: The Planimeter	p. 39
Spherical Geometry	p. 43
Geodesics	p. 43
Geodesics on Spheres	p. 46
The Six Angles of a Spherical Triangle	p. 48
The Law of Cosines for Sides	p. 54
The Dual Spherical Triangle	p. 55
The Law of Cosines for Angles	p. 59
The Law of Sines for Spherical Triangles	p. 62
Navigation Problems	p. 63
Mapmaking	p. 66
Applications of Stereographic Projection	p. 77
Conics	p. 83
Conic Sections	p. 83
Foci of Ellipses and Hyperbolas	p. 86
Eccentricity and Directrix; the Focus of a Parabola	p. 90
Tangent Lines	p. 93
Focusing Properties of Conics	p. 96
Review Exercises: Standard Equations for Smooth Conics	p. 98
LORAN Navigation	p. 99
Kepler's Laws of Planetary Motion	p. 101
Appendix: Reduction of a Quadratic Equation to Standard Form	p. 110
Projective Geometry	p. 115
Perspective Drawing	p. 115
Projective Space	p. 120
Desargues' Theorem	p. 122
Cross Ratios	p. 126
Projections in Coordinates	p. 133
Homogeneous Coordinates and Duality	p. 136
Homogeneous Polynomials, Algebraic Curves	p. 140
Tangents	p. 143
Dual Curves	p. 145
Pascal's and Brianchon's Theorems	p. 148
Special Relativity	p. 153
Spacetime	p. 153
Galilean Transformations	p. 155
The Failure of the Galilean Transformations	p. 158
Lorentz Transformations	p. 159
Relativistic Addition of Velocities	p. 165
Lorentz-FitzGerald Contractions	p. 167
Minkowski Geometry	p. 169
The Slowest Path is a Line	p. 174
Hyperbolic Angles and the Velocity Addition Formula	p. 176
Appendix: Circular and Hyperbolic Functions	p. 178
References	p. 183
Index	p. 185
Table of Contents provided by Blackwell. All Rights Reserved.

Modern Geometry with Applications

At a Glance

Paperback

Shipping

How to return your order

Defective items

You Can Find This Book In

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Volume I: Symmetric Bimonoidal Categories and Monoidal Bicategories

Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory

Volume II: Braided Bimonoidal Categories with Applications

Basic Modern Theory of Linear Complex Analytic $q$-Difference Equations

Mathematical Surveys and Monographs

Introduction to Lie Algebras

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Sum Stories Equations and Their Origins

Equations and Their Origins

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