# Ellipsoid rotation(3D)

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?

• 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. – Yves Daoust Jul 29 '19 at 18:43

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.