| Introduction 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 afflne 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 afllne spaces 2. The structure of isometries 3. The group of linear isometrics Exercises and problems Ⅲ. Euclidean Geometry in the Plane 1. Angles 2. Isometrics and rigid motions in the plane 3. Plane similarities 4. Inversions and pencils of circles Exercises and problems Ⅳ. Euclidean Geometry in Space 1. Isometrics 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 projective line and the circular group Exercises and problems Ⅵ. Conics and Quadrics 1. Affine quadrics and conics, generalities 2. Classification and properties of aifine 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 A few Hints and Solutions to Exercises Chapter Ⅰ Chapter Ⅱ Chapter Ⅲ Chapter Ⅳ Chapter Ⅴ Chapter Ⅵ Chapter Ⅶ Chapter Ⅷ Bibliography Index |
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