Geometry: (Universitext)

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

Table of Contents:
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.

Review :
From the reviews: "With this book on geometry - an English translation of the French original published in 1998 - the author has without a doubt created a work which will crucially influence many generations of geometers to come. Audin knows how to combine clarity with modern methods of geometric thought, and one cannot help but notice the love for geometry on each and every page. ... As might be expected the author cites many French textbooks in the well thought our bibliography, yet does not shy from including som "humorous" works as well. Congratulations!" H. Sachs in "Mathematical Reviews", 2003 "The book is to be welcomed. There are plenty of exercises and fort pages devoted to hints and solutions, and it is indeed advisable that the student works carefully through them in order to cement understanding. … Final year undergraduates or postgraduates will find this a valuable summary of geometry from an algebraic perspective." (Gerry Leversha, The Mathematical Gazette, March, 2005) "This book, which has been published originally in French in 1998, gives a sound introduction into affine, Euclidean and projective geometry. … The book can be recommended unreservedly for upper undergraduates." (G. Kowol, Monatshefte für Mathematik, Vol. 143 (4), 2004) "Everything is presented clearly and rigorously. 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." (Serguey M. Pokas, Zentralblatt MATH, Vol. 1043 (18), 2004) "This textbook deals with quite a host of classical geometric topics. … As a benefit each chapter ends with numerous interesting exercises and examples to be solved by the reader. Moreover at the end of the book hints forthe solution of these exercises are offered. …The book can be recommended to students at an upper undergraduate level having a good base of linear algebra … ." (A. Gfrerrer, IMN, Vol. 57 (193), 2003) "Audin … writing for undergraduates who have some linear algebra, selects and unifies a range of topics that actually traverses a good deal of this terrain. … a solid introduction to many faces of modern geometry. … Summing Up: … Recommended. General readers; lower-division undergraduates through professionals." (D. V. Feldman, CHOICE, July, 2003) "With this book on geometry … the author has without a doubt created a work which will crucially influence many generations of geometers to come. Audin knows how to combine clarity with modern methods of geometric thought, and one cannot help but notice the love for geometry on each and every page. … Congratulations!" (Hans Sachs, Mathematical Reviews, 2003 h) "The book is addressed to students at upper undergraduate and Master’s level to discover geometry and deepen their knowledge and understanding, starting from linear algebra … . Each property is proved, examples and exercises illustrate the course content. Precise hints for most of the exercises are provided at the end of the book." (Zentralblatt für Didaktik der Mathematik, June, 2002)