Details

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:

reference

[1] 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.