几何三部曲(第2卷几何的代数方法)(英文版) 世界图书出版公司 下载 pdf 百度网盘 epub 免费 2025 电子书 mobi 在线

复投影平面中代数曲线的研究是几何应用如密码技术研究的重要内容，也是线性几何研究向代数几何研究的自然过渡。本书论述了几何空间中的各种不同代数方法，给出了解析几何、仿射几何、欧几里得几何和投影几何研究的具体内容，并详尽地描述了各类几何空间和代数曲线的性质。

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

Francis Borceux是一位比利时数学家。

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书籍介绍

复投影平面中代数曲线的研究是几何应用如密码技术研究的重要内容，也是线性几何研究向代数几何研究的自然过渡。本书论述的就是几何空间中的各种不同代数方法，给出了解析几何、仿射几何、欧几里得几何和投影几何研究的具体内容，并详尽地描述了各类几何空间和代数曲线的性质。本书适合从事数学史、几何学、代数及其相关领域研究生和科研人员阅读和参考。