构造任意固定旋转的最少两轴旋转数
The Minimum Number of Rotations About Two Axes for Constructing an Arbitrary Fixed Rotation
摘要：
对于具有$|\hat{m}^{\rm T} \hat{n}| < 1$和任何旋转$U$的任何一对三维实单位向量$\hat{m}$和$\hat{n}$，令$N_{\hat{m},\hat{n}}(U)$表示正整数$k$的最小值，使得$U$可以被分解为关于$\hat{m}$或$\hat{n}$的$k$旋转的乘积。这项工作给出了数$N_{\hat{m},\hat{n}}(U)$作为$U$的函数。这里，旋转意味着特殊正交群${\rm SO}(3)$的元素$D$或对应于$D$的特殊酉群${\rm SU}(2)$的元素。给出了$U$的最小分解数$N_{\hat{m},\hat{n}}(U)$.
英文摘要：
For any pair of three-dimensional real unit vectors $\hat{m}$ and $\hat{n}$ with $|\hat{m}^{\rm T} \hat{n}| < 1$ 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 ${\rm SO}(3)$ or an element of the special unitary group ${\rm SU}(2)$ that corresponds to $D$. Decompositions of $U$ attaining the minimum number $N_{\hat{m},\hat{n}}(U)$ are also given explicitly.
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