Cover image for Geometry
Audin, Michèle.
Personal Author:
Uniform Title:
Géométrie. English
Publication Information:
Berlin ; New York : Springer, [2003]

Physical Description:
vi, 357 pages : illustrations ; 24 cm.
Subject Term:
Format :


Call Number
Material Type
Home Location
Central Library QA445 .A8413 2003 Adult Non-Fiction Non-Fiction Area

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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

Introductionp. 1
1 This is a bookp. 1
2 How to use this bookp. 2
3 About the English editionp. 3
4 Acknowledgementsp. 3
I Affine Geometryp. 7
1 Affine spacesp. 7
2 Affine mappingsp. 14
3 Using affine mappings: three theorems in plane geometryp. 23
4 Appendix: a few words on barycentersp. 26
5 Appendix: the notion of convexityp. 28
6 Appendix: Cartesian coordinates in affine geometryp. 30
Exercises and problemsp. 32
II Euclidean Geometry, Generalitiesp. 43
1 Euclidean vector spaces, Euclidean affine spacesp. 43
2 The structure of isometriesp. 46
3 The group of linear isometriesp. 52
Exercises and problemsp. 58
III Euclidean Geometry in the Planep. 65
1 Anglesp. 65
2 Isometries and rigid motions in the planep. 76
3 Plane similaritiesp. 79
4 Inversions and pencils of circlesp. 83
Exercises and problemsp. 98
IV Euclidean Geometry in Spacep. 113
1 Isometries and rigid motions in spacep. 113
2 The vector product, with area computationsp. 116
3 Spheres, spherical trianglesp. 120
4 Polyhedra, Euler formulap. 122
5 Regular polyhedrap. 126
Exercises and problemsp. 130
V Projective Geometryp. 143
1 Projective spacesp. 143
2 Projective subspacesp. 145
3 Affine vs projectivep. 147
4 Projective dualityp. 153
5 Projective transformationsp. 155
6 The cross-ratiop. 161
7 The complex projective line and the circular groupp. 164
Exercises and problemsp. 170
VI Conics and Quadricsp. 183
1 Affine quadrics and conics, generalitiesp. 184
2 Classification and properties of affine conicsp. 189
3 Projective quadrics and conicsp. 200
4 The cross-ratio of four points on a conic and PascalÆs theoremp. 208
5 Affine quadrics, via projective geometryp. 210
6 Euclidean conics, via projective geometryp. 215
7 Circles, inversions, pencils of circlesp. 219
8 Appendix: a summary of quadratic formsp. 225
Exercises and problemsp. 233
VII Curves, Envelopes, Evolutesp. 247
1 The envelope of a family of lines in the planep. 248
2 The curvature of a plane curvep. 254
3 Evolutesp. 256
4 Appendix: a few words on parametrized curvesp. 258
Exercises and problemsp. 261
VIII Surfaces in 3-dimensional Spacep. 269
1 Examples of surfaces in 3-dimensional spacep. 269
2 Differential geometry of surfaces in spacep. 271
3 Metric properties of surfaces in the Euclidean spacep. 284
4 Appendix: a few formulasp. 294
Exercises and problemsp. 296
A few Hints and Solutions to Exercisesp. 301
Chapter Ip. 301
Chapter IIp. 304
Chapter IIIp. 306
Chapter IVp. 314
Chapter Vp. 321
Chapter VIp. 326
Chapter VIIp. 332
Chapter VIIIp. 336
Bibliographyp. 343
Indexp. 347

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