On cage-plane motion

In a recent paper F. Freudenstein and E. Soylemez investigate the motion of the centers of spheres in rolling contact with two planes having coplanar motions. It appears that the said motion is a rigid one. The authors call it the cage-plane motion. In their analysis of the motion they use the theory of instantaneous invariants to obtain a number of interesting results [1].¹ In this paper, the cage-plane motion is treated by means of vector methods. It turns out that this is an easy way to obtain most of the results of Freudenstein and Soylemez and to extend these results in various respects.