# Euler's Angles and Uniquely Defining the Orientation of a Rigid Body

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.

• 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. May 6, 2022 at 0:25