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### Summary

### Summary

Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michle 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.

### Reviews 1

### Choice Review

Relying merely on the title Geometry, one might imagine many rather different books sandwiched between the covers! One thinks first of axiomatic Euclid, so ancient theorems about triangles and circles or perhaps modern gems such as one finds, say, in Ross Honsberger's Episodes in Nineteenth Century and Twentieth Century Geometry (CH, Nov'95). But one might also get (perhaps too dryly) a comparative text about geometries, one that contrasts the Euclidean, elliptic, hyperbolic, and projective and adds a touch of group theory to bring out the role of symmetry. A differential geometry of curves and surfaces will have a completely different flavor still, even more so a tome on the generalities of Riemannian geometry. Audin (Universite Louis Pasteur), writing for undergraduates who have some linear algebra, selects and unifies a range of topics that actually traverses a good deal of this terrain. Though she chooses foundations based on linear algebra rather than axiomatics, she does develop a suite of classic Euclidean theorems, if mostly as exercises. But her development also proceeds up to Gauss's celebrated "Theorema egregium." Withal, a solid introduction to many faces of modern geometry.The mere fame of a classic book does not automatically guarantee the easy availability of a truly useful and adequate modern edition in translation. Now in this republication of Thomas L. Heath's 1908 Cambridge University Press translation, the editors, with classroom use in mind, have indeed succeeded in their aim of providing a "one-volume, clean, student-friendly, Euclid-centered edition of the complete text of all thirteen books of the Elements.'' As "clean'' here means the omission of Heath's commentaries, the Dover edition will remain a necessity for scholars. But this edition does serve a purpose if, by dint of its gracious typography and layout, it makes a direct confrontation with Euclid not too intimidating for those modern undergraduates who might otherwise never even bother to take a look. ^BSumming Up: Both books--Recommended. General readers; lower-division undergraduates through professionals. D. V. Feldman University of New Hampshire

### Table of Contents

Introduction | p. 1 |

1 This is a book | p. 1 |

2 How to use this book | p. 2 |

3 About the English edition | p. 3 |

4 Acknowledgements | p. 3 |

I Affine Geometry | p. 7 |

1 Affine spaces | p. 7 |

2 Affine mappings | p. 14 |

3 Using affine mappings: three theorems in plane geometry | p. 23 |

4 Appendix: a few words on barycenters | p. 26 |

5 Appendix: the notion of convexity | p. 28 |

6 Appendix: Cartesian coordinates in affine geometry | p. 30 |

Exercises and problems | p. 32 |

II Euclidean Geometry, Generalities | p. 43 |

1 Euclidean vector spaces, Euclidean affine spaces | p. 43 |

2 The structure of isometries | p. 46 |

3 The group of linear isometries | p. 52 |

Exercises and problems | p. 58 |

III Euclidean Geometry in the Plane | p. 65 |

1 Angles | p. 65 |

2 Isometries and rigid motions in the plane | p. 76 |

3 Plane similarities | p. 79 |

4 Inversions and pencils of circles | p. 83 |

Exercises and problems | p. 98 |

IV Euclidean Geometry in Space | p. 113 |

1 Isometries and rigid motions in space | p. 113 |

2 The vector product, with area computations | p. 116 |

3 Spheres, spherical triangles | p. 120 |

4 Polyhedra, Euler formula | p. 122 |

5 Regular polyhedra | p. 126 |

Exercises and problems | p. 130 |

V Projective Geometry | p. 143 |

1 Projective spaces | p. 143 |

2 Projective subspaces | p. 145 |

3 Affine vs projective | p. 147 |

4 Projective duality | p. 153 |

5 Projective transformations | p. 155 |

6 The cross-ratio | p. 161 |

7 The complex projective line and the circular group | p. 164 |

Exercises and problems | p. 170 |

VI Conics and Quadrics | p. 183 |

1 Affine quadrics and conics, generalities | p. 184 |

2 Classification and properties of affine conics | p. 189 |

3 Projective quadrics and conics | p. 200 |

4 The cross-ratio of four points on a conic and PascalÆs theorem | p. 208 |

5 Affine quadrics, via projective geometry | p. 210 |

6 Euclidean conics, via projective geometry | p. 215 |

7 Circles, inversions, pencils of circles | p. 219 |

8 Appendix: a summary of quadratic forms | p. 225 |

Exercises and problems | p. 233 |

VII Curves, Envelopes, Evolutes | p. 247 |

1 The envelope of a family of lines in the plane | p. 248 |

2 The curvature of a plane curve | p. 254 |

3 Evolutes | p. 256 |

4 Appendix: a few words on parametrized curves | p. 258 |

Exercises and problems | p. 261 |

VIII Surfaces in 3-dimensional Space | p. 269 |

1 Examples of surfaces in 3-dimensional space | p. 269 |

2 Differential geometry of surfaces in space | p. 271 |

3 Metric properties of surfaces in the Euclidean space | p. 284 |

4 Appendix: a few formulas | p. 294 |

Exercises and problems | p. 296 |

A few Hints and Solutions to Exercises | p. 301 |

Chapter I | p. 301 |

Chapter II | p. 304 |

Chapter III | p. 306 |

Chapter IV | p. 314 |

Chapter V | p. 321 |

Chapter VI | p. 326 |

Chapter VII | p. 332 |

Chapter VIII | p. 336 |

Bibliography | p. 343 |

Index | p. 347 |