The Hypocycloid
by Emily Kennedy

A cycloid is the locus of a point on a circle as the circle "rolls" along a flat surface.

The red curve in this animation is a cycloid:

What if, instead of a flat surface, the circle rolled around the inside of a larger circle?

This is called a hypocycloid.  You may recognize it as the curve traced by a Spirograph.

The red curve in this animation is a hypocycloid:

Let's try to determine parametric equations for the hypocycloid.

Recall that parametric equations (in the plane) are two functions

x() and y()

Such that x() describes the curve's x-coordinate and y() describes the curve's y-coordinate, and is some parameter.

The extent of the curve will depend on the range of .

Let R be the radius of the larger circle,
and let r be the radius of the smaller circle.

For simplicity, assume the center of the larger circle is located at .

Let be the coordinates of the smaller circle's center.

This is an enlargment of the diagram above, showing the smaller circle.

From this diagram, we have

Now let's determine the values of and in terms of .

Note that the center of the smaller circle lies on a circle of radius R-r.

From the first diagram, we have

Now we need to find the value of in terms of .

In the diagram, the dark blue and dark green arcs must be the same length, because the smaller circle is "rolling" along the larger one.

This gives us

So we have

and

Thus, we have found parametric equations for the hypocycloid:

For a GSP file demonstrating the hypocycloid,