We present the results of the combined experimental and theoretical investigation of rope coiling arising from the buckling instability. The shape of the rope is perfectly circular in the coiling region and is straight in the region below the feeding point. In between these two distant regions, the rope assumes a catenary-like shape in the limit of slow feeding velocity and a helix-like shape in the limit of fast feeding velocity. When there is an increase in the feeding velocity, the transverse displacement of deformation persists over the long distance far beyond the coiling region. The catenary is associated with the purely imaginary wave number and the helix is associated with the real wave number. The catenary-to-helix shape transition is particularly evident when the rope is fed from a large height.

Coiling of rope, fed from a height onto a rotating plane, progresses through a sequence of shapes, a hypotrochoid to an epitrochoid to a circle as the frequency of the plane increases. Feeding velocity controls the rate of length deposition on a plane and frequency of the plane controls the rate of length transfer from a contact point, where rope first touches a plane. Secondary loops of a hypotrochoid or an epitrochoid are formed when the deposition rate is faster than the transfer rate. When these two rates are comparable, secondary loops disappear and the shape returns to a circle like in rope coiling on a static plane. In a reference frame co-rotating with rope, the Coriolis and centrifugal forces act only at the contact point, not extending to the portion of rope far above a rotating plane. For a small deflection of rope, tension is inferred from the equations of motion by using the radius and frequency of a primary loop measured in experiments. Tension changes continuously at both the hypotrochoid– epitrochoid transition and the epitrochoid–circle transition, reminiscent of the features of a second-order phase transition.