1 The Birth of Analytic Geometry
1.1 Fermat's Analytic Geometry
1.2 Descartes' Analytic Geometry
1.3 More on Cartesian Systems of Coordinates
1.4 Non—Cartesian Systems of Coordinates
1.5 Computing Distances and Angles
1.6 Planes and Lines in Solid Geometry
1.7 The Cross Product
1.8 Forgetting the Origin
1.9 The Tangent to a Curve
1.10 The Conics
1.11 The Ellipse
1.12 The Hyperbola
1.13 The Parabola
1.14 The Quadrics
1.15 The Ruled Quadrics
1.16 Problems
1.17 Exercises
2 Affine Geometry
2.1 Affine Spaces over a Field
2.2 Examples of Affine Spaces
2.3 Affine Subspaces
2.4 Parallel Subspaces
2.5 Generated Subspaces
2.6 Supplementary Subspaces
2.7 Lines and Planes
2.8 Barycenters
2.9 Barycentric Coordinates
2.10 Triangles
2.11 Parallelograms
2.12 Affine Transformations
2.13 Affine Isomorphisms
2.14 Translations
2.15 Projections
2.16 Symmetries
2.17 Homotheties and Affinities
2.18 The Intercept Thales Theorem
2.19 Affine Coordinates
2.20 Change of Coordinates
2.21 The Equations of a Subspace
2.22 The Matrix of an Affine Transformation
2.23 The Quadrics
2.24 The Reduced Equation of a Quadric
2.25 The Symmetries of a Quadric
2.26 The Equation of a Non—degenerate Quadric
2.27 Problems
2.28 Exercises
3 More on Real Affine Spaces
3.1 About Left，Right and Between
3.2 Orientation of a Real Affine Space
3.3 Direct and Inverse Affine Isomorphisms
3.4 Parallelepipeds and Half Spaces
3.5 Pasch's Theorem
3.6 Affine Classification of Real Quadrics
3.7 Problems
3.8 Exercises
4 Euclidean Geometry
4.1 Metric Geometry
4.2 Defining Lengths and Angles
4.3 Metric Properties of Euclidean Spaces
4.4 Rectangles， Diamonds and Squares
4.5 Examples of Euclidean Spaces
4.6 Orthonormal Bases
4.7 Polar Coordinates
4.8 Orthogonal Projections
4.9 Some Approximation Problems
4.10 Isometries
4.11 Classification of Isometries
4.12 Rotations
4.13 Similarities
4.14 Euclidean Quadrics
4.15 Problems
4.16 Exercises
5 Hermitian Spaces
5.1 Hermitian Products
5.2 Orthonormal Bases
5.3 The Metric Structure of Hermitian Spaces
5.4 Complex Quadrics
5.5 Problems
5.6 Exercises
6 Projective Geometry
6.1 Projective Spaces over a Field
6.2 Projective Subspaces
6.3 The Duality Principle
6.4 Homogeneous Coordinates
6.5 Projective Basis
6.6 The Anharmonic Ratio
6.7 Projective Transformations
6.8 Desargues' Theorem
6.9 Pappus' Theorem
6.10 Fano's Theorem
6.11 Harmonic Quadruples
6.12 The Axioms of Projective Geometry
6.13 Projective Quadrics
6.14 Duality with Respect to a Quadric
6.15 Poles and Polar Hyperplanes
6.16 Tangent Space to a Quadric
6.17 Projective Conics
6.18 The Anharmonic Ratio Along a Conic
6.19 The Pascal and Brianchon Theorems
6.20 Affine Versus Projective
6.21 Real Quadrics
6.22 The Topology of Projective Real Spaces
6.23 Problems
6.24 Exercises
7 Algebraic Curves
7.1 Looking for the Right Context
7.2 The Equation of an Algebraic Curve
7.3 The Degree of a Curve
7.4 Tangents and Multiple Points
7.5 Examples of Singularities
7.6 Inflexion Points
7.7 The Bezout Theorem
7.8 Curves Through Points
7.9 The Number of Multiplicities
7.10 Conics
7.11 Cubics and the Cramer Paradox
7.12 Inflexion Points of a Cubic
7.13 The Group of a Cubic
7.14 Rational Curves
7.15 A Criterion of Rationality
7.16 Problems
7.17 Exercises
Appendix A Polynomials over a Field
A.1 Polynomials Versus Polynomial Functions
A.2 Euclidean Division
A.3 The Bezout Theorem
A.4 Irreducible Polynomials
A.5 The Greatest Common Divisor
A.6 Roots of a Polynomial
A.7 Adding Roots to a Polynomial
A.8 The Derivative of a Polynomial
Appendix B Polynomialsin Several Variables
B.1 Roots
B.2 Polynomial Domains
B.3 Quotient Field
B.4 Irreducible Polynomials
B.5 Partial Derivatives
Appendix C Homogeneous Polynomials
C.1 Basic Properties
C.2 Homogeneous Versus Non—homogeneous
Appendix D Resultants
D.1 The Resultant of two Polynomials
D.2 Roots Versus Divisibility
D.3 The Resultant of Homogeneous Polynomials
Appendix E Symmetric Polynomials
E.1 Elementary Symmetric Polynomials
E.2 The Structural Theorem
Appendix F Complex Numbers
F.1 The Field of Complex Numbers
F.2 Modulus， Argument and Exponential
F.3 The Fundamental Theorem of Algebra
F.4 More on Complex and Real Polynomials
Appendix G Quadratic Forms
G.1 Quadratic Forms over a Field
G.2 Conjugation and Isotropy
G.3 Real Quadratic Forms
G.4 Quadratic Forms on Euclidean Spaces
G.5 On Complex Quadratic Forms
Appendix H Dual Spaces
H.1 The Dual of a Vector Space
H.2 Mixed Orthogonality
References and Further Reading
Index
