Solutions to the relativistic string equation are found which correspond to rigid body rotation about the z-axis with azimuthal velocity greater than the velocity of light. If the solutions lie entirely in the x-y plane they are rotating epicycloids, complimentary to the hypocycloid solutions found previously. The use of a general solution to the string equation in terms of two arbitrary world-lines with null tangents provides an alternative derivation of the rigidly rotating solutions.