Abstract
For any pair of three-dimensional real unit vectors and with and any rotation U, let denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about either or . This work gives the number 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 are also given explicitly.
Highlights
In this work, an issue on optimal constructions of rotations in the Euclidean space R3, under some restriction, is addressed and solved
By a rotation or rotation matrix, we usually mean an element of the special orthogonal group SO(3)
This work has established the least value Nm,n (U) of a positive integer k such that U can be decomposed into the product of k rotations about either mor nfor an arbitrarily fixed element U in SU(2), or in SO(3), where m, n ∈ S2 are arbitrary real unit vectors with |m Tn | < 1
Summary
For any pair of three-dimensional real unit vectors mand nwith |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 mor n. This work gives the number Nm ,n (U) as a function of U. 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 given explicitly
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