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> The equivalent of polar coordinates in 3d are quaternions

I’d disagree. Quaternions are the 3D equivalent (yes, I’m aware of the subtleties here, but speaking in a rough sense) to using complex numbers to represent rotations.

Euler angles are much closer in spirit to polar coordinates, but unfortunately have gimbal lock.

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A single number to represent a rotation in 2D, vs a single number to represent rotations in 4D for which 2D, and 3D are special cases.

Compared to 3 numbers and a series of operations which happen to rotate a 3D vector.

Regardless I think euler angles should not be taught except as a dont do this, its a bad idea. Mostly because when used in practice, in order to interpret them you must convert them to quaternion to do so, which is very telling. Gimbal lock is a secondary problem to that.

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