Re-Orient a Sphere with Two Straight Rolls
It is well-known that a sphere can be rolled in the horizontal plane to any orientation by three sequentially orthogonal rolls. This Demonstration shows that only two straight rolls are necessary. The desired orientation, shown in red, is specified by moving the north pole to a desired latitude and longitude , then twisting about this pole by . Rolling from the initial orientation, shown in green, along the blue and red lines brings the sphere to the desired orientation.
The composite rotation of about the axis, about the axis, and about the axis can be reproduced by two straight rolls in the horizontal plane. For the unit sphere, a roll of at an angle with the axis, followed by a roll of length at an angle with the axis produces the same rotation.
Any desired rotation can be specified by three sequentially orthogonal rotations:
A rotation on the plane of length , making angle with the axis produces the rotation
The two straight rolls reproduce the desired rotation:
 J. M. Hammersley, "7. Oxford Commemoration Ball," in Probability, Statistics and Analysis: London Mathematical Society Lecture Note Series, Number 79, ed. J. F. C. Kingman and G. E. H. Reuter, Cambridge University Press, 1983.