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

Chapter 1, Problem 36

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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.(a) A cusp (or corner) occurs on either the hypocycloid or epicycloid every time the point P on the small circle touches the large circle. Equivalently, y 246 2 6 4 2 4 4 2 x y 2 1 1 2 3 x 1 2 1 Figure 1.122 Hypocycloids with a = 3, b = 2 and a = 6, b = 5. y 2 4 2 4 2 4 4 2 x Figure 1.123 An epicycloid with a = 4, b = 1. this happens whenever the smaller circle rolls through 2. Assuming that a/b is rational, how many cusps does a hypocycloid or epicycloid have? (Your answer should involve a and b in some way.) (b) Describe in words and pictures what happens when a/b is not rational.