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We can rotate a point 'circularly' about an arbitrary axis:

the equation is here, but this site doesn't trust me enough yet to post an image.,

But as we walk theta 0 -> 2PI this takes the point around a "unit circle" around the axis you're rotating about

How can we make it so as theta 0 -> 2PI the results are about an ellipse of width a, height b?

I do not want to apply transformation matrices to the points after rotating them about the axis - what I'm looking for is an "elliptical" rotation matrix, if anyone knows of such a thing!

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closed as off topic by Scott Morrison Nov 7 '09 at 17:54

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I'm closing this as insufficiently interesting to mathematicians. Please see the FAQ, and bring any discussion over to tea.mathoverflow.net –  Scott Morrison Nov 7 '09 at 18:01
    
are you sure? This isn't very friendly. –  bobobobo Nov 8 '09 at 19:58

2 Answers 2

Sure, you can conjugate the rotation matrix by a matrix which carries the unit circle to the ellipse in question, e.g., the diagonal 2x2 matrix with entries a and b.

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Sure. In 2 dimensions:

$$\begin{pmatrix}\cos\theta&k\sin\theta\\\\ -k^{-1}\sin\theta&\cos\theta\end{pmatrix}$$

The idea: Scale the $x$ axis by $k$, rotate, then scale back. Now pick $k$ appropriately (left as an exercise).

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Hah! Foiled by Firefox, which crashed while I was adding this answer. So Reid got there first. Oh well. 8-) –  Harald Hanche-Olsen Nov 7 '09 at 16:56

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