Set the ratio to 1:1, set revolve on and move the green point C to 06:00 - clock-position.[br][br][list][*]Then, watch the rolling outer circle: It is at the same horizontal position as starting at 12:00. Hence, it must have been turned around, it revolved completely around C. [/*][*]Then, envision the distance passed by (or rolled over) as a straight line: It's [/*][/list][math]ds=\frac{1}{2}\left(2r\pi\right)=r\pi=10\pi[/math][br]How come?[br]

[b]We compare the circular motion with the circular rotation:[/b][br]The outer circle is moving from 12:00 to 06:00, a fixed distance.[list=1][*][revolve Off] Circular motion: The segment LR stays orthogonally to line AC (g).[/*][*][revolve On] Circular rotation: The segment LR rotates around C.[/*][/list]If we just watch 1, the outer circle is [b]forced to rotate[/b] around C. [br]This fact is hidden to our mind, as we assume that the [b]rollover drives all[/b] the rotation.[br]Watching 1 we can see that the orientation of the outer circle against A does not change. [br][color=#0000ff]But the outer circle is rotating with respect to its orientation at the starting position.[/color][br]

[list][*][color=#ff0000][size=150]Rotation is [u]not[/u] a relative motion.[/size][/color][/*][*][color=#ff0000][size=150]If you walk on a not straight line, you will turn yourself![/size][/color][/*][/list]