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Description
Rope fed uniformly from the height exhibits a perfectly circular coiling on a static plane. Introducing the rotation to the plane breaks the rotational symmetry of circle which gives rise to a variety of the more ordered patterns. For sufficiently fast feeding velocity $v$ the coiling shape laying on a rotating plane is a hypotrochoid, namely a closed curve with exterior loops, when plane rotates slowly with low frequency $f_{0}$. As $f_{0}$ increases, the number of exterior loops decreases and the shape eventually turns to an epitrochoid, namely a closed curve with interior loops. The hypotrochoid-to-epitrochoid transition associates with a change in the sign of angular momentum. As $f_{0}$ increases further, the number of interior loops gradually decreases to zero and the shape thus turns to a circle. Interestingly the circle which is the shape for static plane is restored at fast plane frequency $f_{0}$, rather than at slow plane frequency $f_{0}$ as our intuition would suggest. To elucidate the underlying principles, all the experimentally observed shapes, i.e. hypotrochoid, epitrochoid, and circle, are unified by a geometrical description. The key parameter which controls the shape is the ratio of the circumference velocity of plane relative to the feeding velocity, namely $2\pi f_{0}R/v$ where $R$ denotes the radius of the shape. The feeding velocity-plane frequency phase diagram is presented.
