Roll a Sphere without Changing Orientation to a New Location in Two Straight Rolls

A sphere can be moved to any location with no net change in orientation by rolling without slipping in two straight-line rolls. Rolling a sphere in a straight line a distance of , where is any integer, returns the sphere to its initial orientation. By combining two such rolls, the sphere can be moved to any location.

J. M. Hammersley proved that three straight-line rolls in the horizontal plane are sufficient to move a sphere to any desired position and orientation, but that three rolls can lead to arbitrarily long paths. In contrast, the total path length needed to move the sphere from a start position to a desired position at a distance away without changing orientation is , where is the ceiling function. The procedure is to first roll the sphere to a desired orientation with two straight-line rolls, and then move the sphere to the desired position with two straight-line moves.

By applying the Demonstration Re-Orient a Sphere with Two Straight Rolls, followed by this Demonstration, a ball can be rolled to any orientation and position in four rolls.

Reference

[1] J. M. Hammersley, "7. Oxford Commemoration Ball," in Probability, Statistics and Analysis: London Mathematical Society Lecture Note Series, No. 79 (J. F. C. Kingman and G. E. H. Reuter, eds.), Cambridge, UK: Cambridge University Press, 1983.