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how to transform a cylinder from a coordinate system having orthogonal basis vectors $v_1$, $v_2$ and $v_3$ into another a coordinate system having orthogonal basis vectors $\begin{pmatrix}1\\0\\0\end{pmatrix}$, $\begin{pmatrix}0\\1\\0\end{pmatrix}$ and $\begin{pmatrix}0\\0\\1\end{pmatrix}$.

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migrated from Feb 9 '12 at 22:04

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An example would help. – Dan W Feb 9 '12 at 18:29
So would a better formed question. Are you asking about a rotation of a coordinate system? – ldigas Feb 9 '12 at 19:42
Your question literally asks how to transform a cylinder into a coordinate system. :-\. – Zéychin Feb 9 '12 at 22:29

Make a Rotation Matrix from the vectors $u$, $v$ and $w$ by arranging them in columns such that:


Now any vector $a=\begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix}$ is transformed from the local coordinates to the world coordinates with vectors $\mathcal{i}=\begin{pmatrix}1\\0\\0\end{pmatrix}$, $\mathcal{j}=\begin{pmatrix}0\\1\\0\end{pmatrix}$ and $\mathcal{k}=\begin{pmatrix}0\\0\\1\end{pmatrix}$ by simple matrix multiplication


thats it.

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translation should also be included – INS Feb 18 '13 at 11:10
The OP never mentioned translations. – ja72 Feb 18 '13 at 14:26

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