# Geometry

Springer Science & Business Media, Sep 19, 2002 - Mathematics - 361 pages
Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michčle Audin, professor at the University of Strasbourg, has written a book allowing them to remedy this situation and, starting from linear algebra, extend their knowledge of affine, Euclidean and projective geometry, conic sections and quadrics, curves and surfaces.
It includes many nice theorems like the nine-point circle, Feuerbach's theorem, and so on. Everything is presented clearly and rigourously. Each property is proved, examples and exercises illustrate the course content perfectly. Precise hints for most of the exercises are provided at the end of the book. This very comprehensive text is addressed to students at upper undergraduate and Master's level to discover geometry and deepen their knowledge and understanding.

### Contents

 Introduction 1 2 How to use this book 2 3 About the English edition 3 Affine Geometry 7 2 Affine mappings 14 three theorems in plane geometry We are now in an affine plane 23 a few words on barycenters 26 the notion of convexity 28
 Conies and Quadrics 183 1 Affine quadrics and conics generalities 184 2 Classification and properties of affine conics 189 3 Projective quadrics and conics 200 4 The crossratio of four points on a conic and Pascals theorem 208 5 Affine quadrics via projective geometry 210 6 Euclidean conics via projective geometry 215 7 Circles inversions pencils of circles 219

 Cartesian coordinates in affine geometry 30 Exercises and problems 32 Euclidean Geometry Generalities 43 2 The structure of isometries 46 3 The group of linear isometries 52 Exercises and problems 58 Euclidean Geometry in the Plane 65 2 Isometries and rigid motions in the plane 76 3 Plane similarities 79 4 Inversions and pencils of circles 83 Exercises and problems 98 Euclidean Geometry in Space 113 2 The vector product with area computations 116 3 Spheres spherical triangles 120 4 Polyhedra Euler formula 122 5 Regular polyhedra 126 Exercises and problems 130 Projective Geometry 143 2 Projective subspaces 145 3 Affine vs projective 147 4 Projective duality 153 5 Projective transformations 155 6 The crossratio 161 7 The complex projective line and the circular group 164 Exercises and problems 170
 a summary of quadratic forms 225 Exercises and problems 233 Curves Envelopes Evolutes 247 1 The envelope of a family of lines in the plane 248 2 The curvature of a plane curve 254 3 Evolutes 256 a few words on parametrized curves 258 Exercises and problems 261 Surfaces in 3dimensional Space 269 2 Differential geometry of surfaces in space 271 3 Metric properties of surfaces in the Euclidean space 284 a few formulas 294 Exercises and problems 296 A few Hints and Solutions to Exercises 301 Chapter II 304 Chapter III 306 Chapter IV 314 Chapter V 321 Chapter VI 326 Chapter VII 332 Chapter VIII 336 Bibliography 343 Index 347 Copyright

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### References to this book

 Geometrie: Ein Lehrbuch für Mathematik- und PhysikstudierendeHorst KnörrerNo preview available - 2006
 Geometrie: Ein Lehrbuch für Mathematik- und PhysikstudierendeHorst KnörrerLimited preview - 2006