General Background I first became involved in the teaching of geometry about twenty years ago，when my department introduced an optional second year course on the geometry of plane curves，partly to redress the imbalance in the teaching of the subject。It Was mildly revolutionary，since it went back to an earlier sct of precepts where the differential and algebraic geometry of cuwes were pursued simultaneously,to their mutua！advantage.

List of Illustrations

List of Tables

Preface

1 Real Algebraic Curves

1.1 Parametrized and Implicit Curves

1.2 Introductory Examples

1.3 Curves in Planar Kinematics

2 General Ground Fields

2.1 Two Motivating Examples

2.2 Groups, Rings and Fields

2.3 General Affine Planes and Curves

2.4 Zero Sets of Algebraic Curves

3 Polynomial Algebra

3.1 Factorization in Domains

3.2 Polynomials in One Variable

3.3 Polynomials in Several Variables

3.4 Homogeneous Polynomials

3.5 Formal Differentiation

4 Atfine Equivalence

4.1 Affine Maps

4.2 Affline Equivalent Curves

4.3 Degree as an Affine Invariant

4.4 Centres as Affine Invariants

5 Affline Conics

5.1 Affline Classification

5.2 The Delta Invariants

5.3 Uniqueness of Equations

6 Singularities of Afline Curves

6.1 Intersection Numbers

6.2 Multiplicity of a Point on a Curve

6.3 Singular Points

7 Tangents to Afline Curves

7.1 Generalities about Tangents

7.2 Tangents at Simple Points

7.3 Tangents at Double Points

7.4 Tangents at Points of Higher Multiplicity

8 Rational Afline Curves

8.1 Rational Curves

8.2 Diophantine Equations

8.3 Conics and Integrals

9 Projective Algebraic Curves

9.1 The Projective Plane

9.2 Projective Lines

9.3 Atfine Planes in the Projective Plane

9.4 Projective Curves

9.5 Affine Views of Projective Curves

10 Singularities of Projective Curves

10.1 Intersection Numbers

10.2 Multiplicity of a Point on a Curve

10.3 Singular Points

10.4 Delta Invariants viewed Projectively

11 Projective Equivalence

11.1 Projective Maps

11.2 Projective Equivalence

11.3 Projective Conics

11.4 Afline and Projective Equivalence

12 Projective Tangents

12.1 Tangents to Projective Curves

12.2 Tangents at Simple Points

12.3 Centres viewed Projectively

12.4 Foci viewed Projectively

12.5 Tangents at Singular Points

12.6 Asymptotes

13 Flexes

13.1 Hessian Curves

13.2 Configurations of Flexes

14 Intersections of Proiective Curves

14．1 The Geometric Idea

14．2 Resultants in One Variable

14．3 Resultants in Severa!Variables

14．4 B6zout’S Theorem

14．5 Thc Multiplicity Inequality

14．6 Invariance of the Intersection Number

15 Proiective Cubics

15．1 Geometric Types 0f Cubics

15．2 Cubics of General Type

15．3 Singular Irreducible Cubics

15．4 Reducible Cubics

16 Linear Systems

16．1 Projective Spaces of Curves

16．2 Pcncils of CuiNes

16．3 Solving Quartic Equations

16．4 Subspaces or Projective Spaces

16．5 Linear Systems of Culwes

16．6 Dual CulNes

17 The Group Structure on a Cubic

17．1 The Nine Associated Points

17．2 The Star Operation

17．3 Cubics as Groups

17．4 Group Computations

17．5 Determination of the Groups

18 Rational Projective Curves

18．1 Thc Projective Concept

18．2 Quartics with Three Double Points

18．3 Thc Deficiency of a CHIve

18．4 Some Rational Curves

18．5 Some Non-Rational Curves

Index