Plane curves arise naturally in numerous areas of the physical sciences (such as particle physics, engineering robotics and geometric optics) and within areas of pure mathematics itself (such as number theory, complex analysis and differential equations). In this introductory chapter, we will motivate some of the basic ideas and set up the underlying language of affine algebraic curves. That will also give us the opportunity to preview some of the material you will meet in the later chapters.

Parametrized and Implicit Curves

At root there are two ways in which a curve in the real plane ℝ2 may be described. The distinction is quite fundamental.

• A curve may be defined parametrically, in the form x = x(t), y = y(t). The parametrization gives this image a dynamic structure: indeed at any parameter value t we have a tangent vector (x′(t), y′(t)) whose length is the speed of the curve at the parameter t. An example is the line parametrized by x = t, y = t, with constant speed, another parametrization such as x = 2t, y = 2t yields the same image, but at twice the speed.

• A curve may be defined implicitly, as the set of points (x, y) in the plane satisfying an equation f(x, y) = 0, where f(x, y) is some reasonable function of x, y. For instance the line parametrized by x = t, y = t arises from the function f(x, y) = y-x. Such a curve has no associated dynamic structure – it is simply a set of points in the plane.