Question

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!

Answer 1

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.

Answer 2

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).