vec r_rotation_1= r*(cos(ωt),sin(ωt)) rotation of the center of the smaller circle
                                        relative to the center of the larger circle 
                                        which is the general origin.
                                        ω is the angular velocity.

  vec r_rotation_2 = r*(cos(-ωt),sin(-ωt)) = (cos(ωt),-sin(ωt))
                                        rotation of the center of the smaller circle
                                        on its own axis.
                                        ω is the angular velocity.

  vec r = vec r_rotation_1 + vec r_rotation_2 = 2r*(cos(ωt),0) net motion of point 
                                        on the surface of the smaller circle. 
                                        For the sinusoidal oscillation,
                                        ω is the angular frequency
                                        and 2r is the amplitude.  

In fact, the Tusi couple sinusoidal oscillation can be seen as just the x component for any x-axis of a rotation: e.g., the x component of rotation vector

            vec r_rotation = 2r*(cos(ωt),sin(ωt))  .  

File: Copernicus file: tusi_couple_1bb.html