Why are Euler's angles picked exactly that way?

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SUMMARY

Euler's angles are defined through a specific sequence of rotations: first around the z-axis, then around the x-axis, and finally around the z-axis again. This xyz-x'y'z'-x''y''z'' sequence is preferred due to its ability to represent three-dimensional rotations without ambiguity. Alternative sequences, such as rotating around the x-axis first, would result in different transformation equations, complicating the representation of motion. The discussion highlights the importance of conventions in parametrizing rotations, with Euler angles being the most commonly utilized method.

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illidan4426
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TL;DR
I'm wondering why exactly those angles are picked to describe the orientation of the rotating body.
So the Euler's angles are described like this:
xyz-x'y'z' (first rotation around z axis)
x'y'z'-x''y''z'' (second rotation around x')
x''y''z''-XYZ (third rotation around z'')
So I've been thought it goes like this, now I'm wondering why? Why exactly these angles and why this order? Why can't it go like this for example:
xyz-x'y'z' (rotate around x)
x'y'z'-x''y''z'' (rotate around y')
x''y''z''-XYZ (rotate around z'')
Can the motion be described this way? The equations of transformation of xyz-XYZ would be different for sure.
 
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There are all kinds of conventions around to parametrize the rotations. The Euler angles are just the most often used ones.
 
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