Rolling Along

Today I worked on exercises in my calculus book concerning parametric curves. I also talked with my father a bit about cycloids, the curves traced by a point on the perimeter of a circle as the circle rolls along a line. (These are simplest to define parametrically.)

As part of our discussion, we imagined a related class of curves that turn out to be called called cyclogons, traced by a point at the vertex of a rolling regular polygon. Just as a circle is the limit of a sequence of polygons as the number of sides approaches infinity, so a cycloid is the limit of a sequence of cyclogons as the number of sides of the polygon approaches infinity. Visualizing this can make it more intuitive that the point that traces a cycloid never moves backward as the circle rolls. Check out this demonstration and notice how the point always traces part of the top half of a circle, passing through each horizontal position only once.

(I’d also like to draw readers’ attention to the series of videos on rolling curves that I shared a while ago: The Wonderful World of Weird Wheels by Morphocular)