Rolling a Sphere around a Circle without Slipping

Consider a sphere of unit radius rolling (without spinning or slipping) around a circle of radius in the plane. The sphere's orientation can be described by a rotation about the circle center and a rotation about an axis from the sphere center to the circle center. For what angles along the circle is ? For what values of will the sphere periodically return to its initial configuration (position and orientation)?

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The circular locus of contact points on the sphere (shown with a blue dashed line) has radius . The angle for , where is an integer.
The sphere's configuration will be periodic if and only if
,
where is a rational number greater than 1. Also, is the ratio of to the radius . Thus, if , the sphere resumes its initial configuration every times around the circle. As an example, if , a unit sphere rolling around a plane circle of radius returns to its initial configuration every three times around the circle. The smallest circle with a period of one (i.e., for which the sphere's configuration repeats with each trip around the circle) has the radius , whereas the smallest circle with a period of two has the radius .. In general, the smallest circle with period has radius .
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