Epicycloids from an Envelope of Lines
This Demonstration lets you see the construction of an epicycloid with straight lines. Start with the diameter of a unit circle and offset one end by a series of steps of equal arc length along the circumference. At the other end, offset the points by steps of equal arc lengths times as large and connect these pairs of points with straight line segments. With enough lines drawn, you can see an epicycloid with cusps ("pinches", at which the curve cannot be differentiated).