Talk:3D rotation group

Rotations and inner product spaces
In "properties" it is said that the standard inner product can be written purely in terms of lengths; this is (the so-called polar identity (?)) valid for any inner product space (in fact, for any symmetric bilinear form, I think).

Btw, I didn't find much on this on any of the pages I searched, just some quite well hidden remarks on quadratic form. If s.o. has an idea, it would be nice to have this added somewhere (eg in bilinear form). &mdash; MFH: Talk 21:32, 21 Jun 2005 (UTC)


 * I don't understand the question. Can you rephrase it? What is it that you want more information on? In a standard real vector space, the "purely in terms of lengths" is just some simple vector space math, trigonometry, nothing deeper than that? linas 22:55, 21 Jun 2005 (UTC)

The length of a vector is given by its norm, which is given by the scalar product (or dot product), which in turn is defined by a symmetric positive definite bilinear form. But reciprocally, a norm defines a quadratic form which has an associated bilinear form (which I knew as its "polar form", but this term does not seem common here).

So one could define the rotation group w.r.t. any other (positive?) symmetric bilinear form. It seems to me if we take the bilinear form of index (3,1) i.e. with matrix diag(1,1,1,-1) we get the Lorentz group SO(3,1) pertaining to special relativity. &mdash; MFH: Talk 17:44, 22 Jun 2005 (UTC)


 * Yes, all of those statements appear to be correct; I detect no question. Traditional, narrow formal usage has the rotation group being SO(n) only, although you will occasionally find the broader usage you refer to, e.g. SO(3,1) might be called the "group of hyperbolic rotations".  I've seen other non-SO(N) things called roatations as well, although the usage is informal and speaker-listener-context-dependent.  linas 00:29, 23 Jun 2005 (UTC)

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

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)

please fix the commutation relations --
some uncorrupted souls might get confused 192.12.184.7 (talk) 03:54, 5 March 2023 (UTC)