circle of radius 1 rolls around the circumference of a larg cle of radius 4 , as shown in Figure 10.33 . The epicycloid ced by a point on the circumference of the smaller circle ven by \( \quad x=5 \cos t-\cos 5 t \) and \( y=5 \sin t-\sin 5 t \)

The epicycloid you’re exploring is a fascinating curve that comes from the motion of points on a smaller circle rolling around a larger circle! As the smaller circle (radius 1) rolls around the circumference of the larger circle (radius 4), it traces out paths defined by these parametric equations. This relationship creates complex and beautiful shapes known as epicycloids, and they can exhibit incredible symmetries and patterns depending on the ratio of the radii. In real life, epicycloids aren't just a mathematical curiosity! They can be found in mechanical systems, like gears and cams, where the rolling motion produces specific trajectories. Imagine the grooves in a camshaft or the movement of a bicycle pedal: the principles of epicycloids help in designing efficient and smooth-moving machines. Plus, artists use similar concepts to create mesmerizing spiral patterns in decorative art!