## GeometryGeometry, 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 |

343 | |

347 | |

### Common terms and phrases

affine mapping affine plane affine space affine transformation angle associated Assume axis basis belongs bisector called Chapter circle collinear complex composition conic consider consists construct contains convex coordinates curvature curve Deduce defined definition denoted describes determine dilatation dimension direct distinct equality equation equivalent Euclidean examples Exercise exists fact Figure fixed point formula four frame geometry given gives hence hyperplane infinity intersection intersection point inversion isometry isomorphism linear linear mapping look matrix measure namely Notice oriented orthogonal orthonormal parallel parametrization pencil plane pole positive preserve projective projective line Proof properties Proposition Prove quadratic form quadric reader reflections regular relation Remark respect result rotation satisfies sides similarity simply sphere subset surface symmetry theorem transformation translation triangle unique unit vector space write

### Popular passages

### References to this book

Geometrie: Ein Lehrbuch für Mathematik- und Physikstudierende Horst Knörrer No preview available - 2006 |