Audin Geometry

2003
ISBN: 978-3-642-56127-6
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 361 Seiten, eBook

Reihe: Universitext

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

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Audin, Michele

Inhaltsverzeichnis

1. This is a book.- 2. How to use this book.- 3. About the English edition.- 4. Acknowledgements.- I. Affine Geometry.- 1. Affine spaces.- 2. Affine mappings.- 3. Using affine mappings: three theorems in plane geometry.- 4. Appendix: a few words on barycenters.- 5. Appendix: the notion of convexity.- 6. Appendix: Cartesian coordinates in affine geometry.- Exercises and problems.- II. Euclidean Geometry, Generalities.- 1. Euclidean vector spaces, Euclidean affine spaces.- 2. The structure of isometries.- 3. The group of linear isometries.- Exercises and problems.- III. Euclidean Geometry in the Plane.- 1. Angles.- 2. Isometries and rigid motions in the plane.- 3. Plane similarities.- 4. Inversions and pencils of circles.- Exercises and problems.- IV. Euclidean Geometry in Space.- 1. Isometries and rigid motions in space.- 2. The vector product, with area computations.- 3. Spheres, spherical triangles.- 4. Polyhedra, Euler formula.- 5. Regular polyhedra.- Exercises and problems.- V. Projective Geometry.- 1. Projective spaces.- 2. Projective subspaces.- 3. Affine vs projective.- 4. Projective duality.- 5. Projective transformations.- 6. The cross-ratio.- 7. The complex proje ctive line and the circular group.- Exercises and problems.- VI. Conics and Quadrics.- 1. Affine quadrics and conics, generalities.- 2. Classification and properties of affine conics.- 3. Projective quadrics and conics.- 4. The cross-ratio of four points on a conic and Pascal’s theorem.- 5. Affine quadrics, via projective geometry.- 6. Euclidean conics, via projective geometry.- 7. Circles, inversions, pencils of circles.- 8. Appendix: a summary of quadratic forms.- Exercises and problems.- VII. Curves, Envelopes, Evolutes.- 1. The envelope of a family of lines in the plane.- 2. The curvature of aplane curve.- 3. Evolutes.- 4. Appendix: a few words on parametrized curves.- Exercises and problems.- VIII. Surfaces in 3-dimensional Space.- 1. Examples of surfaces in 3-dimensional space.- 2. Differential geometry of surfaces in space.- 3. Metric properties of surfaces in the Euclidean space.- 4. Appendix: a few formulas.- Exercises and problems.- VI.- VII.- VIII.- A few Hints and Solutions to Exercises.- I.- II.- III.- IV.- V.