An idealized version of the Spirograph can be obtained by taking a large circle (of

Chapter 1, Problem 37

(choose chapter or problem)

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 3437 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.)