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Modern Geometry with Applications : Universitext - George A. Jennings

Modern Geometry with Applications


Paperback Published: 12th June 1997
ISBN: 9780387942223
Number Of Pages: 204

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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.

Euclidean Geometryp. 1
Euclidean Spacep. 1
Isometries and Congruencep. 2
Reflections in the Planep. 3
Reflections in Spacep. 5
Translationsp. 7
Rotationsp. 9
Applications and Examplesp. 11
Some Key Results of High School Geometry: The Parallel Postulate, Angles of a Triangle, Similar Triangles, and the Pythagorean Theoremp. 17
SSS, ASA, and SASp. 27
The General Isometryp. 36
Appendix: The Planimeterp. 39
Spherical Geometryp. 43
Geodesicsp. 43
Geodesics on Spheresp. 46
The Six Angles of a Spherical Trianglep. 48
The Law of Cosines for Sidesp. 54
The Dual Spherical Trianglep. 55
The Law of Cosines for Anglesp. 59
The Law of Sines for Spherical Trianglesp. 62
Navigation Problemsp. 63
Mapmakingp. 66
Applications of Stereographic Projectionp. 77
Conicsp. 83
Conic Sectionsp. 83
Foci of Ellipses and Hyperbolasp. 86
Eccentricity and Directrix; the Focus of a Parabolap. 90
Tangent Linesp. 93
Focusing Properties of Conicsp. 96
Review Exercises: Standard Equations for Smooth Conicsp. 98
LORAN Navigationp. 99
Kepler's Laws of Planetary Motionp. 101
Appendix: Reduction of a Quadratic Equation to Standard Formp. 110
Projective Geometryp. 115
Perspective Drawingp. 115
Projective Spacep. 120
Desargues' Theoremp. 122
Cross Ratiosp. 126
Projections in Coordinatesp. 133
Homogeneous Coordinates and Dualityp. 136
Homogeneous Polynomials, Algebraic Curvesp. 140
Tangentsp. 143
Dual Curvesp. 145
Pascal's and Brianchon's Theoremsp. 148
Special Relativityp. 153
Spacetimep. 153
Galilean Transformationsp. 155
The Failure of the Galilean Transformationsp. 158
Lorentz Transformationsp. 159
Relativistic Addition of Velocitiesp. 165
Lorentz-FitzGerald Contractionsp. 167
Minkowski Geometryp. 169
The Slowest Path is a Linep. 174
Hyperbolic Angles and the Velocity Addition Formulap. 176
Appendix: Circular and Hyperbolic Functionsp. 178
Referencesp. 183
Indexp. 185
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387942223
ISBN-10: 038794222X
Series: Universitext
Audience: General
Format: Paperback
Language: English
Number Of Pages: 204
Published: 12th June 1997
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.55 x 15.72  x 1.19
Weight (kg): 0.3

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