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Here you can see how to calculate the rotated ellipsoid's new points: What is the general equation equation for rotated ellipsoid?
My question is for this part:

"Exactly what rotation they represent depends on several things: the sequence in which you apply the rotations..."

I know that rotating in $x,y,z$ order won't give the same result(for a fixed input) as rotating in $x,z,y$ or $y,x,z$ or $y,z,x$ or $z,x,y$ or $z,y,x$ order, but does all(any) of them represent the same set of all possible rotation? I mean, do you get all possible rotation if you rotate in x,y,z order or do you need to combine the 6 possible orders somehow? Is there a rotated position which you can't reach by rotating in $x,y,z$ order, but you can in some other order?

user128576
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  • The axis of an orthonormal frame form a symmetrical configuration. So if all rotations can be achieved with the order $x,y,z$ (which is the case), they can also be achieved with other orders. –  Jul 29 '19 at 18:43

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You can obtain any rotation of the ellipsoid by first rotating along the $x$-axis, then the $y$-axis and then the $z$-axis. You do not need to consider all the different orders in which you can choose the axes; each choice of ordering covers all possible rotations. See also this Wikipedia page for more details on these Tait-Bryan angles and the differing conventions for the ordering of the axes.

Servaes
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  • this is true for higher dimensions too, right? For example, in 4D you would have 6 angles - combination(4,2) - and you are rotating around 2D planes. Rotating in any order is fine, right? – user128576 Jul 29 '19 at 19:14
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    The situation is the same in higher dimensions; any ordering of the axis yields all possible rotations, but performing the same rotations in a different order (generally) results in a different rotation. – Servaes Jul 29 '19 at 19:17
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    However, in for dimensions it is not true that you are rotating around 2D planes in general; there are rotations in four dimensions that leave no subspace invariant. You can indeed choose to rotate only around the six 2D coordinate planes, and it is not a difficult exercise to prove that this produces all rotations. – Servaes Jul 29 '19 at 19:18