Here are two wheels of equal radius, one rolling round the other.

What do you think the path of just one of those pink dots looks like?

Here it is again, but with just one dot showing.

Now can you imagine the path more easily?

And here is the path traced out:

Here’s the curve on its own. It’s called a cardioid.

Why?

If the pink cross starts on top of the blue cross, and the pink disc rolls around the blue till it gets to the point in this diagram, what do you know about the highlighted blue and pink arcs?

What does this tell you about the two line segments?

Draw a circle with the two line segments as radiuses.

What happens to the yellow circle as its centre moves round the blue circumference?

It looks completely obvious that the outline created by the yellow circles is the same as the locus of the pink cross, namely the cardioid. Actually, though, from the point of view of geometrical proof, the argument is actually quite subtle.

If the two pink line segments are reflections of each other on the inside of the circle, what are the shaded angles?

What is the ratio of the lengths of the green and red segments?

They are the same!

What does this tell you about the reflected pink line segment and the yellow circle from earlier?

We already know that the yellow circle traces out the cardioid as its centre moves around the blue circle.

What does this mean for the reflection of the light ray from the left-most point on the outer circle as the point of incidence on the inside of the circle moves around the circle?