2 - Curves, and functions on them

Summary

‘I think, Watson, that you have put on seven and a half pounds since I last saw you.’

‘Seven,’ I answered.

‘Indeed, I should have thought a little more. Just a trifle more, I fancy, Watson.’

(A Scandal in Bohemia)

Plane curves arise naturally in all sorts of situations and in many guises. Solutions of Newton's laws of motion give the orbits of the planets as ellipses with the Sun at a focus. A spot of paint on a train wheel describes a cycloid as the wheel rolls. These are examples of curves parametrized by time: for each time t a definite point on the curve is determined. If a solid object (such as Dr Watson) is viewed from a distance its outline, also called its apparent contour or profile, is essentially a plane curve (or a curve on the retina), but this time it is not given dynamically as a moving point (fig. 2.1). It is more reminiscent of curves given by equations f(x, y) = 0; these latter curves are one of the subjects of chapter 4. A curve may be traced by a linkage of bars and gearwheels; the position of the pencil drawing the curve perhaps depends on the angle of some controlling bar, and so is parametrized by this angle. (Alas! We have no space for this beautiful subject.) When the Sun's rays are reflected from the rounded inner surface of a teacup they produce on the surface of the tea a bright ‘caustic’ curve.