Suppose you have a stationary circle, and inside that circle is a second smaller circle, touching the larger circle from within. Pick an arbitrary point on the inner circle, call it $K$. Imagine pushing the inner circle against the outer circle, so that the inner circle revolves around its own center, as it moves inside the larger circle. What path does $K$ describe?

If you saw my earlier blog post on cats on ladders, you would recognize it to be an astroid!

Try setting $\text{Radius}=0.33$ in the slider to see what happens when the radius of the inner circle is a third of the larger circle as well, although the rounding error becomes more visible. (It’s a deltoid)

Now this is the interesting part: what happens if the diameter of the inner circle is exactly the radius of the larger circle? Make a guess!

Here is a more detailed version, emphasizing that the angles $\alpha = \beta$ at all times.

Did you guess that it would be a straight line? The fact that the path described is a straight line is sometimes called Copernicus’s theorem. However, mechanical devices exploiting this fact have been made long before Copernicus: see Tusi couple.

In general, shapes traced by small circles rolling inside of larger ones are called hypocycloids. For a nice exposition of this, as well as some related phenomena (some quite a bit deeper), see this post on John Baez’s blog. For more problems like this, see the book Lines and Curves.

I used the library JSXGraph. For my source code, see here.