Maybe I just suck at googling but I’m having a hard time in general finding information specifically about axis angles
Also I mean in the more general case where the direction vector is arbritrary
I would assume there is (a proof would be nice) because you can convert to rotation matrix then multiply them then convert back to axis angle (since that Euler theorem says so)
The more important question I think is how would you find this axis angle without going to a different rotation representation such as rotation matrix and quaternions?
For a proof, you can show that for any two orientations, there is a path between them involving a rotation about a single axis. Then you can let the first orientation be the the original orientation, the second the composition of the two rotations.
In practice, finding this axis is easy, but finding the new angle is complicated. There is a formula which gives the desired axis-angle rotation, but it’s a mess and involves square roots as well as several trig functions and their inverses. Such is the downside of axis-angle representation.
The reason for this is that the axis-angle representation is a linearization of the space of rotations. Rotations can’t be added together like numbers, but axis-angle vectors can, and if you’re doing things that involve some sort of differential control, then this property is useful. However, because the axis-angle representation is constructed in this way, the geometry is obscured and we lose the nice composition properties that quaternions and rotation matrices have.
What is the result when two axis angles are added together? I thought you could only add the angles of same axis axis angles
Would a way to derive it is to simplify the math when you expand converting the 2 axis angles to matrix form and multiplying them then converting them back?
If the two axis-angle vectors are considered as angular velocities instead of rotations, then it makes sense to add them in the same manner as spatial velocities.
That could be a way, but I think converting them to quaternions and then back is probably mathematically simpler.
Yes, it’s provable, and for a much more general case than just the composition of two 3d rotations. It holds for the multiplication of any number of N-dimensional rotation matrices. Step 1, the proof of Euler’s Rotation theorem is on the wiki page, so I won’t repeat that. The second piece is to show that orthogonal matrices are closed under multiplication:
For orthogonal matrix A, Aᵀ=A⁻1 and therefore AAᵀ = I. This is a property of orthogonal matrices, that the transpose is the inverse. Using this, along with the rule (AB)ᵀ=BᵀAᵀ, you can take any two arbitrary orthogonal matrices A and B, and show:
(AB)(AB)ᵀ = ABBᵀAᵀ
=A(BBᵀ)Aᵀ
=AIAᵀ
=AAᵀ
=I
This proves that the composition AB is also orthogonal. You need an additional proof to show that AB is a special orthogonal matrix (has a determinant of +1 like all rotation matrices do) when both A and B are, in order to show that multiplication is closed for rotations, i.e. that multiplying any two matrices of determinant 1 gives you a matrix with determinant 1. That’s covered by proving more generally that detA * detB = det(AB). That’s long, so I’ll link it here: https://proofwiki.org/wiki/Determinant_of_Matrix_Product
This post is illustrative, but it should be said that Euler’s rotation theorem does not generalize to higher dimensions, at least not in the same form as in 3 dimensions. I feel that this was not made clear.