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When speaking about the orientation of a rigid body, Symon (Mechanics, 3rd ed.) writes: It turns out that no simple symmetric set of coordinates can be found to describe the orientation of a body, analogous to the coordinates x, y, z which locate the position of a point in space.

The standard demonstration of the problem is to take some object, like a book, and first rotate 90 degrees around a first axis and then 90 degrees around a second axis, and then to do the same thing but in the reversed order and note that the position of the body is not the same in both instances.

One usually studies rotations about a fixed axis first and then, when it comes time to study rotation around a point, Euler's angles are introduced.

My question here is how does the use of Euler's angles resolve the issue mentioned above? Is it just simply because Euler's angles are based upon two sets of axes - one fixed and one that rotates with the body? Some deeper insight would be appreciated.

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    $\begingroup$ I don't know what Symon's definitions of "symmetric" and "simple" are but you define 'xyz' Euler angles with respect to either intrinsic (always rotate about intermediate rotated coords) or extrinsic (always rotate about lab frame coords) rotations and get 3 numbers that uniquely determine the rotation. Whether such a scheme is simple or symmetric I don't know. $\endgroup$
    – Jagerber48
    Commented May 6, 2022 at 0:25

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Euler angles do not solve this problem 100%. There are cases where two or more sets of angles result in the same final orientation. We call these the degenerate cases, or gimbal lock, or singularities.

The idea behind Euler angles is that any orientation can be described by at most three rotation sequences each about a local axis perpendicular to the previous one. Hence three angles about the known axis are needed.

But because of the many possible combinations of sequences and the tricky math involved to find Euler angles from a rotation matrix (since singularities need to be expected) they have limited appeal.

In theory, you only need one angle about an arbitrary axis. This is the idea behind quaternions. They do a much better job of uniquely defining orientations if they are normalized (their magnitude equals one). And it is pretty straightforward to go back and forth from a rotation matrix.

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