1 The Genesis of Differential Methods
1.1 The Static Approach to Curves
1.2 The Dynamic Approach to Curves
1.3 Cartesian Versus Parametric
1.4 Singularities and Multiplicities
1.5 Chasing the Tangents
1.6 Tangent:The Differential Approach
1.7 Rectification of a Curve
1.8 Length Versus Curve Integral
1.9 Clocks,Cycloids and Envelopes
1.10 Radius of Curvature and Evolute
1.11 Curvature and Normality
1.12 Curve Squaring
1.13 Skew Curves
1.14 Problems
1.15 Exercises
2 Plane Curves
2.1 Parametric Representations
2.2 Regular Representations
2.3 The Cartesian Equation of a Curve
2.4 Tangents
2.5 Asymptotes
2.6 Envelopes
2.7 The Length of an Arc of a Curve
2.8 Normal Representation
2.9 Curvature
2.10 Osculating Circle
2.11 Evolutes and Involutes
2.12 Intrinsic Equation of a Plane Curve
2.13 Closed Curves
2.14 Piecewise Regular Curves
2.15 Simple Closed Curves
2.16 Convex Curves
2.17 Vertices of a Plane Curve
2.18 Problems
2.19 Exercises
3 A Museum of Curves
3.1 Some Terminology
3.2 The Circle
3.3 The Ellipse
3.4 The Hyperbola
3.5 The Parabola
3.6 The Cycloid
3.7 The Cardioid
3.8 The Nephroid
3.9 The Astroid
3.10 The Deltoid
3.11 The Limacon of Pascal
3.12 The Lemniscate of Bernoulli
3.13 The Conchoid of Nicomedes
