As a child, you may have played with a popular toy called a Spirograph ® . With...

As a child, you may have played with a popular toy called a Spirograph ^® . With it one could draw some appealing geometric figures. The Spirograph consists of a small toothed disk with several holes in it and a larger ring with teeth on both inside and outside as shown in Figure 1.120. You can draw pictures by meshing the small disk with either the inside or outside circles of the ring and then poking a pen through one of the holes of the disk while turning the disk. (The large ring is held fixed.)

An idealized version of the Spirograph can be obtained by taking a large circle (of radius a ) and letting a small circle (of radius b ) roll either inside or outside it without slipping. A “Spirograph” pattern is produced by tracking a particular point lying anywhere on (or inside) the small circle. Exercises 34–37 concern this set-up.

Consider the original Spirograph set-up again. If we now mark a point P at a distance c from the center of the smaller circle, then the curve traced by P is called a hypotrochoid (if the smaller circle rolls on the inside of the larger circle) or an epitrochoid (if the smaller circle rolls on the outside). Note that we must have b < a, but we can have c either larger or smaller than b. (If c < b, we get a “true” Spirograph pattern in the sense that the point P will be on the inside of the smaller circle. The situation when c > b is like having P mounted on the end of an elongated spoke on the smaller circle.) Give a set of parametric equations for the curves that result in this way. (See Figure 1.124.)