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?
Best Answer
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.