The minimumnumber of rotations about two axes for constructing an arbitrarily fixed rotation

Abstract

For any pair of three-dimensional real unit vectorsˆ mandˆ nwith |ˆ mT ˆ n|<1 and any rotationU,letNˆ m,ˆ n (U) denote the least value of a positive integerksuch thatUcan be decomposed into a product of krotations about either ˆ morˆ n. This work gives the number Nˆ m,ˆ n (U) as a function ofU. Here, a rotation means an element D of the special orthogonal group SO(3) or an element of the special unitary group SU(2) that corresponds to D. Decompositions of Uattaining the minimum number Nˆ m,ˆ n (U) are also given explicitly.