## Which is the real Euler’s Rotation theorem?

There are two quite different, but useful, theorems related to 3D rotation that are referred to as “*Euler’s Rotation Theorem*“.

The **first definition** is given as a quotation in the book by Kuipers

*Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis.*

There is a similar, but more pithy definition, from Wolfram MathWorld

*An arbitrary rotation may be described by only three parameters.*

This is related to the three-angle representation of an arbitrary rotation by Euler, roll-pitch-yaw, Tati-Bryan angles etc.

The **second definition** is from Wikipedia

*When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position.*(Also given in latin)

This related to the angle-axis representation of an arbitrary rotation. This version of the Rotation Theorem is found in many other places as well, including here on Math Overflow.

Both definitions are related to rotation, and the second at least has a reference to Euler 1776. The first is related to Euler axes and Euler angles, seems like the sort of thing Euler might have figured out, but Kuipers does not provide a reference.

Euler was notorious for not publishing much. Somebody, post Euler, has declared these theorems as Euler’s Rotation Theorem. Which one is it? Is there a more nuanced way to refer to them?

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