Talk:3D rotation group

Exponential Map section

The sentence "This follows from the fact that every R ∈ SO(3), since every rotation leaves an axis fixed (Euler's rotation theorem), and is conjugate to a block diagonal matrix of the form..." doesn't make sense, because it doesn't say what property every R has. If they meant that every R is in S0(3), then they need to define R. I not sure of the original intent and thus how to fix it. Maybe: "... every since every rotation R ∈ SO(3) leaves an axis fixed ... Chris2crawford (talk) 02:28, 31 August 2020 (UTC)

Surjective property. R is a generic rotation matrix throughout the article.Cuzkatzimhut (talk) 03:05, 31 August 2020 (UTC)

Missing Rotation Group

I realize that this subject has been studied and documented for years, and TL;DR What OTHER groups are there SU(2) and SO(3) are the only ones with a connection? Otherwise, this is a little what I know...

So there's the beginning of a mention of an axis of rotation.... 3D_rotation_group#Axis_of_rotation

this might be just another 'orthogonal' group, since I'm proposing that using this rotation method, that all axii minus 1 are rotated around the axis of rotation; and that rotation is not itself properly reflected as 2D compositions... although the mechanics of conversion to various scalars remains the same. It does have to know what is orthogonal to the axii. A consequence of this is that a rotation in (W,X,Y,Z) around (W) would affect (X,Y,Z) moving them toward 0. (I read lots of books on hypergeometries growing up)

Rotations in 3D can be done by picking an axis (a line) and rotating the space around that line; that there is a computed tangent and bi-tangent on a perpendicular plane is incidental... and simply the composite of $A+B+C$ as in Lie_product_formula of the rotations around X Y and Z axii. The length of this vector is the angle of rotation, and the scalar to normalize the vector to a direction normal, and have a proper unit vector for the axis of rotation and angle. This can be simply encoded in the $D*(A,B,C)$ or $(AD,BD,CD)$. Every point becomes a valid SU(2) quaternion with the cos/sin of the angle (D) and the direction vector (A/D,B/D,C/D) (sorry, inconsistent); and that in turn can map to a rotation matrix; though there is some computation saved going directly from axis-angle instead to matrix instead of through quaternion.

This is also called Euler Axis ( https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula )

https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Rodrigues_vector

reformulation... https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions#Rodriguez_Rotation_(Rotation_Composition)

D3x0r (talk) 00:54, 4 April 2021 (UTC)

Lead needs to be more accessible

The lead is not very accessible to the general reader and could be improved. — Preceding unsigned comment added by ScientistBuilder (talk • contribs) 15:56, 9 February 2022 (UTC)

Agreed! Are there any practical applications for this concept which could help give context? -- Beland (talk) 23:47, 10 December 2025 (UTC)

please fix the commutation relations --

some uncorrupted souls might get confused 192.12.184.7 (talk) 03:54, 5 March 2023 (UTC)

Please edit this paragraph for clarity.

Please edit the paragraph referenced below for clarity. It appears to have some random missing words, or perhaps it's just too technical for me (with a BS in Socìology) to make sense of it.

Paragraph in question: "By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition."

I don't have enough expertise in this topic to edit the unclear grammar while ensuring that the technical language is as accurate as possible.

Thanks! CDUpchurch (talk) 14:16, 25 July 2025 (UTC)

Is the Lie algebra always 3 x 3 matrices?

Under "Lie Algebra" it says:

The Lie algebra of SO(3) is denoted by ${\mathfrak {so}}(3)$ and consists of all skew-symmetric 3 × 3 matrices.

So I'm no mathematician, but is it even true that all of these matrices have to be 3 × 3? What about higher-dimensional representations of ${\mathfrak {so}}(3)$? ~2026-11692-83 (talk) 12:40, 25 February 2026 (UTC)

You're right, it is a common abuse of language. The 3x3 representation is the standard (i.e. conventional) representation, not the only one. Jähmefyysikko (talk) 14:26, 25 February 2026 (UTC)