Rotations and Trajectories

Abstract

In this chapter we present some remarkable curves that are naturally generated as trajectories of points on a circle rolling along a straight line or along another circle. The most interesting properties of these curves are connected with tangents. We will start by investigating cycloids, which are the paths traced by a single point on a circle as the circle rotates along another curve. The reader may recall that, at the end of the Introduction, we revisited Problem 0.1 and encountered a curve real¬ized as the envelope of a family of lines. This envelope was a curve with four cusps, called an astroid. We will examine this fact in greater detail here, and we will also see why a spot of light in a cup formed by reflected rays has a characteristic singularity, a cusp. The devotee of classical geometry will find out about the connections between the nine-point circle of a triangle, its Wallace-Simson lines and their envelope, the Steiner deltoid, which is a cycloid with three cusps.