The Minimum Number of Rotations About Two Axes for Constructing an Arbitrary Fixed Rotation

Abstract

For any pair of three-dimensional real unit vectors $\hat{m}$ and $\hat{n}$ with $|{\hat{m}}^{\mathrm{T}}\hat{n}|<1$ and any rotation $U$, let $N_{\hat{m},\hat{n}}(U)$ denote the least value of a positive integer $k$ such that $U$ can be decomposed into a product of $k$ rotations about either $\hat{m}$ or $\hat{n}$. This work gives the number $N_{\hat{m},\hat{n}}(U)$ as a function of $U$. Here a rotation means an element $D$ of the special orthogonal group $\mathrm{SO}(3)$ or an element of the special unitary group $\mathrm{SU}(2)$ that corresponds to $D$. Decompositions of $U$ attaining the minimum number $N_{\hat{m},\hat{n}}(U)$ are also given explicitly.

Ver.1. 20 pages, 1 figure. Ver.2. Among the two theorems, Theorem 1 is now ascribed to Lowenthal, and Theorem 2, a stronger result, is emphasized. Accordingly, the title slightly changed; the bibliography was doubled; Proof of Theorem 1 was shortened and moved to an appendix; numbering in sections, corollaries, etc., changed. Some other parts were also shortened. 17 pages

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