Plane parametrized curves arise naturally throughout the physical sciences and mathematics. In this introductory chapter we will set up the underlying definitions, and introduce the reader to a small zoo of curves to provide a useful basis for illustration in later chapters. The first step in our geometric development is to associate ‘tangent’ vectors to each parameter on a curve. In particular, that allows us to distinguish ‘irregular’ parameters for which the associated ‘tangent’ vector is zero. Such parameters may correspond to points where the curve is visibly different from other points.

The General Concept

For the purposes of this book a parametrized curve (or just curve if there is no ambiguity) is a smooth mapping z : I → ℝ2, with I ⊆ ℝ an open interval. Thus I is a set of real numbers t (the parameters) which satisfy an inequality of the form a < t < b, where we allow either or both of the possibilities a = -∞, b = ∞.

The meaning of the term ‘smooth’ in the above is as follows. For each parameter t we have a point z(t) = (x(t),y(t)) in the plane: the resulting functions x, y : I → ℝ of a single real variable t are the components of z, and ‘smoothness’ means that at every parameter t both components x, y possess derivatives of all orders.