Tusi couple

Obviously the center of a circle rolling along a straight line travels a straight line. But I had not imagined that a point on the perimeter of a rolling circle could travel a straight line. It can and does if a circle rolls within another circle twice as big. This was noted by a 13th century Persian astronomer and mathematician. Tusi couple.

The same attribute with eight points on the perimeter of a rolling circle traveling straight lines is shown here, here (YouTube), and here (YouTube). The last gives some mathematical analysis. Guessing, the "illusion" is that the moving circle is imaginary in the two YouTube videos, since the moving dots are not connected to one another. Part of the first YouTube video shows the dots being added one at a time. The illusion is a rolling square while there are only four dots.