Suppose that a bicycle wheel of radius a rolls along a flat surface without slipping. If...

Suppose that a bicycle wheel of radius a rolls along a flat surface without slipping. If a reflector is attached to a spoke of the wheel at a distance b from the center, the resulting curve traced by the reflector is called a curtate cycloid. One such cycloid appears in Figure 1.32, where a = 3 and b = 2.

Using vector methods or otherwise, find a set of parametric equations for the curtate cycloid. Figure 1.33 should help. (Take a low point of the cycloid to lie

on the y-axis.) There is no theoretical reason that the cycloid just described cannot have a < b, although in such case the bicycle-wheel–reflector application is no longer relevant. (When a < b, the parametrized curve that results is called a prolate cycloid.) Your parametric equations should be such that the constants a and b can be chosen independently of one another. An example of a prolate cycloid, with a = 2 and b = 4, is shown in Figure 1.34. Try to think of a physical situation in which such a curve would arise.