Paint can cardioids

As I looked into a nearly empty paint can, I noticed a familiar heart-shaped graph:


I'm used to seeing the cardioid-shaped region of the Mandelbrot set (what my students have sometimes called "the butt crack"), but also noticed it in my Spirograph program, where a family of graphs can be generated that seem similar to the paint can cardioids (or caustics):



You can get the same cardioid graph by drawing a cycloid around a circle of the same diameter (an epicycloid):