# Modern Geometry with Applications

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).

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### 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

### Referències a aquest llibre

 Operations Research Calculations HandbookDennis BlumenfeldPrevisualització limitada - 2001
 Geometría para la informática gráfica y CADJoan Trias PairóPrevisualització no disponible - 2003
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