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

摘要

For any pair of three-dimensional real unit vectors m^ and n^ with |m^Tn^|<1 and any rotation U, let Nm^,n^(U) denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about either m^ or n^. This work gives the number Nm^,n^(U) as a function of U. 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 U attaining the minimum number Nm^,n^(U) are also given explicitly.