In the case of coins/circles of equal radius, the moving coin rotates once with respect to the static coin/circle but twice with respect to the observer.

The important point is that were it different, a driven planet gear of radius ‘r’, would rotate more than once when powered by a drive-cog of radius ‘r’. This would involve a creation of energy.

In the case of a coin/circle of radius ‘r’ moving along a line of length 2πr, the moving coin rotates once with respect to the static coin/circle and also once with respect to the observer.

In the case of coins/circles of equal radius, the moving coin rotates once with respect to the static coin/circle but twice with respect to the observer.

The important point is that were it different, a driven planet gear of radius ‘r’, would rotate more than once when powered by a drive-cog of radius ‘r’. This would involve a creation of energy.

In the case of a coin/circle of radius ‘r’ moving along a line of length 2πr, the moving coin rotates once with respect to the static coin/circle and also once with respect to the observer.

In the animation at http://www.geogebratube.org/student/m107691 the black arrow is “with respect to the observer” and the red arrow is “with respect to the static circle.”

The mathematical formula Xπr / Yπr = X/Y remains true in all cases: anything else is an illusion.