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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Three-dimensional rotation operator ： ウィキペディア英語版 Three-dimensional rotation operator This article derives the main properties of rotations in 3-dimensional space. The three Euler rotations are one way to bring a rigid body to any desired orientation by sequentially making rotations about axis' fixed relative to the object. However, this can also be achieved with one single rotation (Euler's rotation theorem). Using the concepts of linear algebra it is shown how this single rotation can be performed. ==Mathematical formulation== Let :$\backslash hat\; e\_1\backslash \; ,\backslash \; \backslash hat\; e\_2\backslash \; ,\backslash \; \backslash hat\; e\_3$ be a coordinate system fixed in the body that through a change in orientation is brought to the new directions :$\backslash mathbf\backslash hat\; e\_1\backslash \; ,\backslash \; \backslash mathbf\backslash hat\; e\_2\backslash \; ,\backslash \; \backslash mathbf\backslash hat\; e\_3.$ Any vector :$\backslash bar\; x\backslash \; =x\_1\backslash hat\; e\_1+x\_2\backslash hat\; e\_2+x\_3\backslash hat\; e\_3$ rotating with the body is then brought to the new direction :$\backslash mathbf\backslash bar\; x\backslash \; =x\_1\backslash mathbf\backslash hat\; e\_1+x\_2\backslash mathbf\backslash hat\; e\_2+x\_3\backslash mathbf\backslash hat\; e\_3$ i.e. this is a linear operator The matrix of this operator relative to the coordinate system :$\backslash hat\; e\_1\backslash \; ,\backslash \; \backslash hat\; e\_2\backslash \; ,\backslash \; \backslash hat\; e\_3$ is :$$ \begin A_ & A_ & A_ \\ A_ & A_ & A_ \\ A_ & A_ & A_ \end = \begin \langle\hat e_1 | \mathbf\hat e_1 \rangle & \langle\hat e_1 | \mathbf\hat e_2 \rangle & \langle\hat e_1 | \mathbf\hat e_3 \rangle \\ \langle\hat e_2 | \mathbf\hat e_1 \rangle & \langle\hat e_2 | \mathbf\hat e_2 \rangle & \langle\hat e_2 | \mathbf\hat e_3 \rangle \\ \langle\hat e_3 | \mathbf\hat e_1 \rangle & \langle\hat e_3 | \mathbf\hat e_2 \rangle & \langle\hat e_3 | \mathbf\hat e_3 \rangle \end As :$\backslash sum\_^3\; A\_A\_=\; \backslash langle\; \backslash mathbf\backslash hat\; e\_i\; |\; \backslash mathbf\backslash hat\; e\_j\; \backslash rangle$ = \begin 0 & i\neq j, \\ 1 & i = j, \end or equivalently in matrix notation :$$ \begin A_ & A_ & A_ \\ A_ & A_ & A_ \\ A_ & A_ & A_ \end^T \begin A_ & A_ & A_ \\ A_ & A_ & A_ \\ A_ & A_ & A_ \end = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end the matrix is orthogonal and as a "right hand" base vector system is re-orientated into another "right hand" system the determinant of this matrix has the value 1. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Three-dimensional rotation operator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.