## 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. |

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### 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 Euclidean plane affine frame affine mapping affine plane affine quadric affine space affine subspace affine transformation algebra Assume barycenter bijective Cartesian Chapter circumcircle collinear composition computation consider contains coordinates Corollary cross-ratio curvature curve Deduce defined definition denoted differential ellipse envelope equality equation equivalent Euclidean affine Euclidean vector space Exercise VIII exists Figure fixed point form q formula four points geometric angle geometry hence homography hyperbola hyperplane intersection point Let ABC Let F line at infinity linear mapping matrix midpoint nondegenerate nonzero notation oriented angles orthogonal orthonormal basis parabola parallel parametrization pencil of circles perpendicular bisector polar polyhedron polynomial preserve projective line projective space projective transformation Proof properties Proposition quadratic form quadric radical axis reader real number reflections regular Remark rigid motions rotation satisfies scalar space of dimension subset surface symmetry tangent plane theorem translation triangle ABC unique unit vectors vertex zero