This Demonstration draws a roulette of a generator point on an ellipse that rolls without slipping around a circle.

Varying the ellipse semimajor axis or eccentricity will change the circumference ratio between the central circle and the ellipse. A closed curve can be obtained after complete revolutions around the circle. By then the ellipse will have made revolutions around its axis.

Changing the pole offset will further create a variety of curves. The bookmarks and snapshots give some examples.