Talk:3D rotation group/Archive 1

Archive 1

Request for technical explanation

• The first meaning of "rotation group" should probably just be merged with symmetry group, or at least direct readers there for a fuller explanation. Otherwise, it's a bit confusing.
• It's at least a little sad that this article about geometrical relationships doesn't have any diagrams. It would be a lot more accessible if each distinct concept were illustrated with a diagram. For example, the basic concept of what a rotation group is could be illustrated by showing some rotations which are and some rotations which are not in the rotation group. These pictures could demonstrate which properties of the same are and are not preserved. An example of an improper rotation could also be diagrammed; the "properties", "axis of rotation" ,and "topology" sections are definitely ripe for obvious illustrations.
• It's not clear in the introduction whether "the rotation group" and "SO(3)" are the same thing.
• Even having studied matrices in college well enough to be able to find determinants and eigenvectors and whatnot, I find the "orthogonal matrices" quite dense, and in some places very difficult to understand. I was not familiar with the term standard basis, and trying to read that article did not help. But come to basis, and it turns out that this is essentially the notion of an XYZ coordinate system, which most readers are actually already familiar with. This connection should be explained. This will also help give them an example to understand what it's saying about orthonormality, at a glance. The discussion of matrices would definitely benefit from a concrete example showing a specific matrix representing a specific rotation (preferably with a picture showing which properties the numbers in the matrix correspond to).

-- Beland 16:27, 18 December 2005 (UTC)

Improper rotation

Regarding this from the introduction: "A length-preserving transformation which reverses orientation is called an improper rotation. Every improper rotation of three-dimensional Euclidean space is a reflection in a plane through the origin." If I have a transformation T that reflects in the z-axis (negates z) and translates 2 units along in the x-axis, then according to the first sentence T is an improper rotation. The second sentence tells me that T, being an improper rotation, is a reflection in a plane through the origin. Yet I can think of no such reflection that reproduces the effect of T. Where am I going wrong here? Oioisaveloy (talk) 10:44, 9 August 2009 (UTC)

Topology Section

In the article it is mentioned that SO(3) is homeomorphic to $RP^{3}$ and $S^{1}xS^{2}$.
However, the first homology group of $RP^{3}$ is ${\frac {Z}{2Z}}$, and for the product, it's just $Z$, because S^2 is simply connected.
Can someone clarify that? —Preceding unsigned comment added by 77.127.154.232 (talk) 21:38, 1 November 2009 (UTC)

Next question: why a topological space homeomorphic' to the rotation group, but this manifold is diffeomorphic to the rotation group? why we use in one case the word homeomorphic, but in other - diffeomorphic? Explane, please! Vyrivykh (talk) 14:50, 23 October 2010 (UTC)

Here are some answers:

1. The topology section states that SO(3) is homeomorphic to $RP^{3}$, which is correct. Where is it stated that it is homeomorphic to $S^{1}xS^{2}$?
2. The word homeomorphic makes sense for any pair of topological spaces, while the stronger notion of being diffeomorphic is only defined for differentiable manifolds. Because the first construction given is inherently topological (involving gluing along a boundary), it makes sense to say that the result is "homeomorphic" to $RP^{3}$. This gluing also induces a diffeomorphism between manifolds, but this requires additional proof. Jim (talk) 19:16, 23 October 2010 (UTC)

Thank you very much! Vyrivykh (talk) 20:04, 23 October 2010 (UTC)

Lie algebra section

For A(t), a one-parameter subgroup of SO(3) parametrised by t, differentiating $AA^{T}=I\,$ with respect to t gives

$A'(t)A(t)^{T}+A(t)A'(t)^{T}=0\,$

Matekosor (talk) 05:57, 1 August 2012 (UTC)

Rename ?

Shouldn't this page be renamed something like

• Rotation Group SO(3), or
• [[Rotation Group of R³]], or
• Rotation Group (3D)

because there are lots of rotation groups out there, but this article is about one very specific one. Jheald (talk) 07:55, 13 September 2008 (UTC)

I'm neutral about the proposed change. Other rotation groups are covered in the article orthogonal group. I have added a note to the top of the article to this effect. siℓℓy rabbit (talk) 11:41, 13 September 2008 (UTC)

There are many rotation groups, SO3 is just one of them. It may be considered the most important, as it is the one acting on the R³ in which most Wikipedia users happen to be embedded – but mathematically, it is just one among many. So I believe

1. That "Rotation Group" and "SO3" should have separate articles
2. That while they don't have separate articles, this article, which is entirely about SO3, should be renamed to "SO3", or "Rotation Group SO(3)". Maproom (talk) 11:41, 23 January 2012 (UTC)

This has been done, but with proper capitalization, Rotation group SO(3). YohanN7 (talk) 15:53, 13 December 2014 (UTC)

Economy of notation?

If one had done any checking/intuiting of Engø's formulas for compositions of arbitrary rotations, he would quickly supplant his ponderous notation for a and b with the much simpler, and intuitive, a=c cotφ/2, b=c cot θ/2. This would immediately shorten several of his formulas, notably for d, now proportional to c, and illuminate α,β,γ. I don't see any compulsion to stick to his original notation. But it is a purely aesthetic point... Cuzkatzimhut (talk) 14:08, 3 January 2015 (UTC)

A change to a better notation is of course fine with me. I'm still trying to find a reasonable formulation (that could go into the article) of the equivalence between the Engo BCH formula and the one in Pauli matrix, but progress will be slow the upcoming days. YohanN7 (talk) 15:07, 4 January 2015 (UTC)

Shortened the second hidebox too. The Pauli matrix gymnastics is done in the Pauli article, and only the answer, the composition law is transcribed here. Actually, The overall (scale) coefficient has the correct small angle behavior, but is not computed here, but surely the motivated reader can whip it out... heh! Cuzkatzimhut (talk) 02:59, 5 January 2015 (UTC) OK, the motivated reader turned out to be me, so, indeed, to prove γ = γ' , one needs the valid identity d = sin 2c' . The way Engo has written d, one has to factor out a 2cos c' , to then note that what remains inside the square root is just a sin c' . Classic case of hiding simplicity under a bushel... Cuzkatzimhut (talk) 21:18, 5 January 2015 (UTC)

You changed "a similar" to "the same". This must mean that the Pauli version is universal in the same sense as Engo's, i.e., they do not just coincide in su(2), but also in all its representations? (This is the same question as I had on the other talk page.) By extension then, by crafting BCH formulas using the closed expressions for exp (referenced paper), one would obtain an infinite series of BCH formulae, all the same in any representation, just with trigonometric factors appearing vastly different, but actually being the same. YohanN7 (talk) 11:13, 5 January 2015 (UTC)

Yes, I think that α,β,γ would all turn out to be the same, for spin 3/2, 2, etc..., on su(2); but that solving for c' when a' = 2π, for b' would but alternatively yield the same for 2π—b', so I would expect all these formulas to develop multivaluedness peculiarities after a while... I haven't explored Engo's, and rotations of the sphere which should clarify things, but things go flakey for θ or φ past π...

On a similar tack, it is not that the group composition law is simpler for su(2) versus so(3), since they all share the same α,β,γ up there in the exponent... It is that the method to arrive at this exponent is simpler for the doublet representation... But if you could steer in the subtle waters with few gentle words... Cuzkatzimhut (talk) 11:59, 5 January 2015 (UTC)

Importance

I added a physics project template in order for someone to rate the article. Someone did, but the importance of the topic was set to low. This is something I find ridiculous. The importance lies somewhere between high and top. Little Nothing in physics is more important than symmetries, rotational symmetry being the most important symmetry of them all. (Important = useful + much more (physical truth as far as we know)) I reverted the complete rating because of this misjudgement, hoping that someone else drops by to rate the article. YohanN7 (talk) 11:21, 8 January 2015 (UTC)

I agree, in strong terms, with the above assessment that the comically flawed assessment, now reverted, is completely out of touch. The 3D rotation group is the mainstream bridge between classical rotations, the introduction of virtually all physicists to symmetries and Lie groups and quantum angular momentum, rightly assessed to be of high importance. Again, the bulk of the model building particle physics and nuclear physics community have the origin of their computational habits, Lie group education, and reality check crutch traceable to SO(3) and SU(2) and their relation. I have had a lifetime of E₈ bloviator whippersnappers' inanities deflated by confronting them with basic SU(2) features and Pauli matrices (well, having watched Wigner practice this with impeccable politeness and sparse pithiness, early in life).

I insist that a major gaping defect in physics education is a decent, concrete, introduction to rotations Lie-theoretically and their concrete connections to SU(2). In fact, I have been put off by repetitive reinventions of the wheel, "spinor maps", in blithe ignorance of the basics. Listening to a theoretical physicist talk who has not mastered this stuff cold immediately leads me to dismiss them and downgrade their competence. This is the proverbial "required reading".More so than the classic texts of Biedernharn and Louck, Rose, and Edmonds, this article grabs the bull by the horns and gets the horse to the water, so to speak. The level of responsibility in the treatment would be a B−, but definitely above C, and the importance is incontrovertibly higher than mid, arguably high. I desist from rating this, since I have contributed to it, and I am unaware of the WP pseudo-conflict-of-interest practices, but it seems evident to me, and knowledgeable, well-meaning practitioners in my cohort, that the article deserves higher importance.

On the other hand, the physics rating landscape in WP is so willfully skewed and ineptly configured, that one more misjudgment would not be out of character, and that big a deal, at the end of the day. Cuzkatzimhut (talk) 14:44, 8 January 2015 (UTC)

Angular momentum is actually assessed to be of top importance, not merely high. YohanN7 (talk) 15:11, 8 January 2015 (UTC)

I desist from rating too because of the same reason, and don't object very strongly to a low quality rating (while I might object mildly and silently to myself). The article is far from perfect yet so far. But it is pretty damned important. YohanN7 (talk) 15:11, 8 January 2015 (UTC)

I rerated it as High. Sorry for the error. --Meno25 (talk) 22:56, 8 January 2015 (UTC)

Suggested clarification re topological path shrinkage

Summary: Extend the final sentence of the fourth para under the subheading Topology, with the part following the first comma here below:

"This circle can be shrunk to the north pole without problems, as an "antipodal circle" forms and simultaneously shrinks to the south pole." [revised again, 30-8-08]

Detail: As formerly shown in the main article, I don’t see how a single great circle can shrink, unless the antipodal point of every point on the shrinking circle described is also included. The article doesn’t seem to hint at this. However, after looking at Greg Egan’s applet:

http://gregegan.customer.netspace.net.au/APPLETS/21/21.html

... I am emboldened to suggest [a suitable] extension of the final sentence. Unless someone can explain comprehensibly (to someone at my level) how I’m in error, then after seven days I’ll post my suggested extension. OK? PaulGEllis (talk) 19:46, 4 August 2008 (UTC)

that particular sentence in the article is fine. it's not clear what it is exactly that you have a problem with. the precise meaning of "shrink" here is homotopy, with the starting point of a loop fixed throughout the homotopy. the great circle is shrinkable to a point for the same reason any circle in the plane is shrinkable. Mct mht (talk) 21:44, 4 August 2008 (UTC)

Thank you for your reply. What I have a problem with is why you can shrink the great circle, as if it was on a plane. In my understanding (which I admit is not graduate mathematician level - I'm an interested physical chemist, thinking purely geometrically ): I can see how you get to a great circle including the north and south poles. And for that great circle, every point still has its equivalent antipodal point on the great circle. But when you start to shrink the great circle away from the south pole, then no point on [a smoothly; inserted 30-8-08] shrinking circle has an antipodal point represented. Since every point on the surface of the pi-radius ball is equivalent to its antipodal point, I can't see how you can start to shrink the great circle, unless there is a corresponding circle (representing all the points antipodal to the original path). Moreover, Greg Egan's applet appears to represent a "side view" of the same idea.

Still having had no further reply, it seems none can or will explain in simple terms why my proposed short clarificatory insertion is invalid, so I believe I am free to insert it (some parts of my contribution on this page edited out as redundant 30-8-08, having inserted the suggested clarification) PaulGEllis (talk) 13:30, 30 August 2008 (UTC)

I don't know that it is invalid, but I do find the entire description could use an overhaul. If you have a path going from the north pole to the south pole and back, the *obvious* thing to do turns out to be the right one: just move the bottom point of the loop up through the interior of the ball along the polar axis until it hits the north pole. There is your homotopy. The present description is needlessly complicated, irrespective of your contribution to it (which, I should say, adds to rather than reduces the complexity of the description). siℓℓy rabbit (talk) 21:13, 30 August 2008 (UTC)

Neat! Thankyou.

As a non-mathematician, I agree with your recommendation of an overhaul, but appeal for any such to be helpful to the non-expert. All too often, IMHO, the Wikipedia maths entries (which along with the science entries generally seem to be of the highest standard, as compared with entries on other areas) are written more for the expert than to help the student and interested enquirer whom I would expect to be a larger part of the readership (and potentially able to benefit far more) than fellow experts. PaulGEllis (talk) 06:17, 31 August 2008 (UTC)

Late update (2017), just in case anyone else has a similar difficulty seeing how one can start shrinking the great circle (from 4-pi rotation) *without* maintaining the antipodal points: Imagine it in reverse by starting from the central identity point of the ball and expanding a loop outwards until it touches a boundary point. If it can be expanded to the boundary, it can also be contracted from the boundary.PaulGEllis (talk) 13:54, 19 May 2017 (UTC)

Preservng a dot product

In the article there is a sentence : "Hence, any length-preserving transformation in R3 preserves the dot product" which is not exactly true ! The transformation has to satisfy more constraints ! ( linearity ? ... it has to preserve the length of the sum too!) —Preceding unsigned comment added by 62.121.113.97 (talk) 10:02, 27 December 2008 (UTC)

No. This is true as stated. Linearity follows from the fact that the transformation preserves lengths. (Additivity follows essentially by the parallelogram law, and homogeneity follows because any continuous additive transformation is also homogeneous.) siℓℓy rabbit (talk) 15:54, 27 December 2008 (UTC)

I doubt that linearity follows from conservation of the length. As a counter example, consider the transformation from $u$ to $u'=(\lVert u\rVert ,0,0)$. This map preserves the norm but is not linear. The map in this example is from $\mathbb {R} ^{3}$ into $\mathbb {R} ^{3}$, but one may probably be able to construct even another example of a nonlinear map onto $\mathbb {R} ^{3}$. NorioTakemoto (talk) 19:25, 8 January 2018 (UTC)

Obscure

You know that this page is almost impenetrable for anyone not already familiar with rotation matrices? — Preceding unsigned comment added by 131.181.33.96 (talk) 02:26, 13 June 2018 (UTC)

Correct notation

I found the first notation below (coded as \frac{\langle X,Y\rangle}{||X|| ||Y||}) in this article and changed it to the second (coded as \frac{\langle X,Y\rangle}{\|X\|\|Y\|}):

${\begin{aligned}&{\frac {\langle X,Y\rangle }{||X||||Y||}}\\[8pt]&{\frac {\langle X,Y\rangle }{\|X\|\|Y\|}}\end{aligned}}$

This should make it clear why the latter is standard usage. Michael Hardy (talk) 19:45, 9 May 2019 (UTC)