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- Thread starter owlpride
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tiny-tim

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Hi owlpride!

Those two axes are parallel, so I suppose you could relate them by a translation.

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jasonRF

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It turns out that there is a nice way to do this that isn't too hard. I learned it from working a homework problem in "a brief on tensor analysis" by Simmonds (a really good book). If you draw this out I think it will make sense. Let [tex]\mathbf{\hat{e}}[/tex] denote a unit vector in the direction of your axis of rotation. you can construct the matrix that will take an arbitrary vector [tex]\mathbf{u}[/tex] and rotate it (right handed) an angle [tex]\vartheta[/tex] about the axis to get a new vector [tex]\mathbf{v}[/tex]. To do this you form an orthogonal basis out of [tex]\mathbf{\hat{e}}[/tex] , [tex]\mathbf{\hat{e} \times u}[/tex], and [tex]\mathbf{\hat{e} \times \left( \hat{e} \times u \right)}[/tex]. After normalizing the basis vectors, you just find the projection of [tex]\mathbf{v}[/tex] onto the basis vectors and do a page of algebra to simplify. The Answer you get is

[tex]\mathbf{v}=\cos \vartheta \mathbf{u} + \left(1-\cos \vartheta \right) \mathbf{\hat{e}} \left( \mathbf{\hat{e} \cdot u}\right) + \sin \vartheta \mathbf{\hat{e} \times u} = T \mathbf{u}.[/tex]

Thus,

[tex]T=\cos \vartheta I + \left(1-\cos \vartheta \right) \mathbf{\hat{e}\hat{e}} + \sin \vartheta \mathbf{\hat{e} \times }.[/tex]

The notation above is old-fashioned. [tex]I[/tex] is the identity matrix, and the meaning of [tex]\mathbf{\hat{e}\hat{e}}[/tex] and [tex]\mathbf{\hat{e} \times}[/tex] can be deduced from the equation for [tex]\mathbf{v}[/tex].

The matrix you want is the coordinate rotation matrix, [tex]R[/tex], which is [tex]R=T^{-1}=T^{T}[/tex], since rotation matrices are orthogonal.

This will take care of your first case. The second case is offset from the origin so a matrix won't do the job by itself - you also need an offset.

Jason

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HallsofIvy

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[tex]\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\end{bmatrix}[/tex]

then multiply by the matrix that rotates the z-axis into the given axis of rotation.

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