Epicycles

There are definitions for the words "deferent" and "epicycle". For the purpose of this article, a deferent is a fixed circle around which another circle known as an "epicycle" can roll. Usually, but not by definition, the epicycle is smaller than the deferent.

In the animation we see an epicycle that has "half" the radius of the deferent being rolled. The epicycle contains a point on the circle and its trace is shown as a dotted curve as it rotates. The shape that is traced has a name, nephroid, although most of the possible traces are not named. The small circle actually makes three complete revolutions from start to finish. The two vectors, $\mathbf{u}$ and $\mathbf{v}$ point to the other two places where the epicycle has just completed a full revolution. You can try it by toggling the animation to a stop right when the epicycle is tangent to one of the vectors. If you do that, you will see that point $B$ is back on the east side of the epicycle, indicating a completed revolution.

In the next animation, we have an epicycle that has 1/3 the radius of the deferent. It will roll around the deferent with 3 revolutions, but will make a 4th revolution by the time it returns. This can be see clearly by toggling the motion when the epicycle (small circle) is tangent to any of the 3 vectors and confirming that the epicycle has made a complete revolution.

Besides rolling the circle around the outside of the fixed circle, we could also roll it around the inside. Doing that will produce a curve called an astroid (not asteroid). Although, if the number of cusps is reduced to two, the curve produced is commonly called a line segment. Even then, it isn't boring! If the point at $B$ were extended out of the plane, say toward the viewer, and the two circles were provided with gear teeth, then we would have changed circular motion into reciprocal linear motion.