Geometry

Front Cover
Springer Science & Business Media, 2003 - Mathematics - 357 pages
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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.
  

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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
Bibliography
343
Index
347
Copyright

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About the author (2003)

Michele Audin is professor at Strasbourg (Universite Louis Pasteur).

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