Decomposition of 3D rotation (in analogy to translation)

Question

Any displacement along a line can be written as the sum of two perpendicular displacements, which then form a closed triangle with the total displacement vector.

My question is: Can something similar be done with rotation?

Consider a rotation about a fixed axis. Can this always be decomposed into two rotations about two perpendicular axes, which again form a triangle (or similar) with the total axis of rotation? In other words, if the axis of rotation is inclined such that it has components in $x$ and $y$ with respect to a global coordinate system, will the rotation it permits never involve a rotation about the $z$-axis?

If it is indeed possible to consider rotations in this way, what is the simplest way to decompose the rotation into fractional contributions from these two putative axes?

EDIT (in response to comment):

A "sharpened" version of my "theorem" may read as: Any rotation about a fixed arbitrary axis can be decomposed into a rotation about three orthogonal axes, but not into a rotation about two orthogonal axes."

From the (very helpful) first reply, I surmise this is correct?

Linked. Be mindful you need *three* , not two, Euler angles. — Cosmas Zachos, Mar 15 '23 at 14:37
The wording of your "never involve" proposed theorem is worded ambiguously. Can you sharpen it as a "true or false" such? The axis of rotation of any great circle on the globe can be arrived at by unique longitude and latitude rotations. — Cosmas Zachos, Mar 15 '23 at 14:45
@CosmasZachos do we need three angles for rotation about a fixed axis? — Roger V., Mar 15 '23 at 15:35
@Roger Vadim. Of course not. What's taking you there to ask that? Who would ever ask that? — Cosmas Zachos, Mar 15 '23 at 15:49
No he is not! That's why I asked him to parse out his "theorem"... What could you imagine my point to be? He obviously needs to review the Gibbs composition formula. — Cosmas Zachos, Mar 15 '23 at 16:02
What do you mean by rotations? Becasue for infinitesimal rotations, or for rotational speeds yes you can decompose the vector into two components, but for finite rotations the answer is not exactly. — John Alexiou, Mar 15 '23 at 17:11

Cosmas Zachos · Accepted Answer · 2023-03-15T17:23:20.593

You are probably asking about the Gibbs composition of rotations (1884), an idiosyncratic non-abelian law. If you describe a rotation by a Gibbs vector $$ \mathbf{g} = \hat{\mathbf{e}}\tan\frac{\theta}{2},$$ where $\hat{\mathbf{e}}$ is the unit direction of the rotation and $\theta$ the rotation angle, then the Gibbs vector of two such successive rotations g and f is $$(\mathbf{g},\mathbf{f}) = \frac{\mathbf{g}+\mathbf{f}-\mathbf{f}\times\mathbf{g}}{1-\mathbf{g}\cdot\mathbf{f}} \,. $$

So, if these vectors are in the x- and y- directions, the composed rotation will necessarily have a component in the z- direction as well. Do you see this, regardless of which constituent rotation goes first?

The elementary su(2) composition law is normally taught like this.

score 0 · Answer 2 · answered Mar 15 '23 at 14:20

Yes, a rotation can be decomposed into several successive rotations (3 in usual 3-dimensional space) using, e.g., Euler angles. The catch is that these successive rotations are do not commute with each other, i.e., the order in which they are done matters.

Decomposition of 3D rotation (in analogy to translation)

2 Answers2