Modern Geometry with Applications

Portada
Springer Science & Business Media, 12 de juny 1997 - 204 pàgines
This book is an introduction to the theory and applications of "modern geometry" ~ roughly speaking, geometry that was developed after Euclid. It covers three major areas of non-Euclidean geometry and their applica tions: spherical geometry (used in navigation and astronomy), projective geometry (used in art), and spacetime geometry (used in the Special The ory of Relativity). In addition it treats some of the more useful topics from Euclidean geometry, focusing on the use of Euclidean motions, and includes a chapter on conics and the orbits of planets. My aim in writing this book was to balance theory with applications. It seems to me that students of geometry, especially prospective mathe matics teachers, need to be aware of how geometry is used as well as how it is derived. Every topic in the book is motivated by an application and many additional applications are given in the exercises. This emphasis on applications is responsible for a somewhat nontraditional choice of top ics: I left out hyperbolic geometry, a traditional topic with practically no applications that are intelligible to undergraduates, and replaced it with the spacetime geometry of Special Relativity, a thoroughly non-Euclidean geometry with striking implications for our own physical universe. The book contains enough material for a one semester course in geometry at the sophomore-to-senior level, as well as many exercises, mostly of a non routine nature (the instructor may want to supplement them with routine exercises of his/her own).
 

Què opinen els usuaris - Escriviu una ressenya

No hem trobat cap ressenya als llocs habituals.

Continguts

Euclidean Geometry
1
12 Isometries and Congruence
2
13 Reflections in the Plane
3
14 Reflections in Space
5
15 Translations
7
16 Rotations
9
17 Applications and Examples
11
The Parallel Postulate Angles of a Triangle Similar Triangles and the Pythagorean Theorem
17
Standard Equations for Smooth Conies
98
38 Keplers Laws of Planetary Motion
101
Reduction of a Quadratic Equation to Standard Form
110
Projective Geometry
115
42 Projective Space
120
43 Desargues Theorem
122
44 Cross Ratios
126
45 Projections in Coordinates
133

19 SSS ASA and SAS
27
110 The General Isometry
36
The Planimeter
39
Spherical Geometry
43
22 Geodesies on Spheres
46
23 The Six Angles of a Spherical Triangle
48
24 The Law of Cosines for Sides
54
25 The Dual Spherical Triangle
55
26 The Law of Cosines for Angles
59
27 The Law of Sines for Spherical Triangles
62
28 Navigation Problems
63
29 Mapmaking
66
210 Applications of Stereographic Projection
77
Conies
83
32 Foci of Ellipses and Hyperbolas
86
33 Eccentricity and Directrix the Focus of a Parabola
90
34 Tangent Lines
93
35 Focusing Properties of Conies
96
46 Homogeneous Coordinates and Duality
136
47 Homogeneous Polynomials Algebraic Curves
140
48 Tangents
143
49 Dual Curves
144
410 Pascals and Brianchons Theorems
147
Special Relativity
153
52 Galilean Transformations
155
53 The Failure of the Galilean Transformations
158
54 Lorentz Transformations
159
55 Relativistic Addition of Velocities
165
56 LorentzFitzGerald Contractions
167
57 Minkowski Geometry
169
58 The Slowest Path is a Line
174
59 Hyperbolic Angles and the Velocity Addition Formula
176
Circular and Hyperbolic Functions
178
References
183
Index
185
Copyright

Altres edicions - Mostra-ho tot

Frases i termes més freqüents

Informació bibliogràfica