Geometry (Paperback, 2003)

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 a plane 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.