Hypotrochoid from Collinear Orbiters
This Demonstration shows the trajectory of a red point on the segment that connects two black points and that travel around two circles. The location of is fixed in terms of the ratio . The phase or offset does not change the shape of the green orbit. The angular velocity of determines the number of lobes of the green curve it lies on.
The green figure is neither a rose curve nor a cycloid. Based on its algebraic form, it can be generated by a Spirograph with two solid gears. It is a hypotrochoid with 90 degree phase offset. Also, the inner product of the and coordinate of a point on the green curve vanishes if the angular velocity is an integer, that is, the integral of their product over one period is zero.