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arXiv · 1401.0153

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

Mitsuru Hamada
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Abstract

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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Mitsuru Hamada. 2014-06-30. The Minimum Number of Rotations About Two Axes for Constructing an Arbitrary Fixed Rotation. https://doi.org/10.1098/rsos.140145

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