# Talk:Rotation

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## Pitch and yaw

Anybody but me notice that the pitch and yaw are mixed up in that 3d model?

Seems ok, same as e.g. [1].--Patrick 23:23, 19 September 2005 (UTC)

The colors are important here. X, Y, Z is associated with Roll, Pitch, Yaw and with Red, Green, Blue. However, -X, -Y, -Z is associated with Cyan, Magenta, Yellow. It would be a good hint to put arrow heads on the lines. 86.81.228.168 (talk) 16:40, 2 November 2008 (UTC)

## Intro

Old intro:

The rotation of a body is the movement (or turning) of that body around a line, known as the axis, which runs through the body and is perpendicular to the direction of rotational motion.

## Linear transformations

The article says Mathematicians consider rotations to be norm-preserving linear transformations on a vector space with an inner product. These form a group of so called special orthogonal matrices. Is this correct? Consider the matrix

$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Norm-preserving, right? Orthogonal, yes. But special orthogonal? No, the determinant is -1. Right? Josh Cherry 14:44, 31 July 2005 (UTC)

That is no rotation, but a reflection in the line x=y.--Patrick 23:27, 19 September 2005 (UTC)
Right, it's not a proper rotation (it's an improper rotation). But the article says that any norm-preserving linear transformation is a rotation, and my example preserves norms. So if rotation means "proper rotation", the article is wrong. Josh Cherry 00:14, 20 September 2005 (UTC)
You are right, I fixed that.--Patrick 02:09, 20 September 2005 (UTC)

## Order of sections

I believe it is wrong that in this general purpose rotation article the mathematical rotation concept is such proeminent — being almost at the beginning of the article and with a lot of math terminology. I would belive that the right order in this article would be (1) hinges, (2) astronomy (3) physics (4) math and (5) amusement rides.

Also, one should use "three dimensional space" instead of 3D, that one is math and computer graphics jargon, not suitable for this general article.

The "see also" section needs radical trimming and rearrangement. Oleg Alexandrov (talk) 10:33, 2 November 2005 (UTC)

The list of links is a bit long, but the links are all useful, and nicely in alphabetic order.--Patrick 22:16, 2 November 2005 (UTC)
I believe this article needs a thorough rethink. I find the stuff in here very arbitrary. I don't think this article can get much worse. Please edit with extreme prejudice. --MarSch 17:13, 2 November 2005 (UTC)
I plan to work on it. I might make big cuts in place. To reply to Patrick above, you don't need to put all the links containing the word "rotation" in this article. Some are more relevant, and some less. Also, now that thanks to MarSch we have some section headings (which do improve the artilce), some links can go in small "see also"s in their most appropriate sections. Oleg Alexandrov (talk) 01:41, 3 November 2005 (UTC)

## Old version better?

If you want to read a sweet nice article about rotation, just go to an older version of this article. See here. Oleg Alexandrov (talk) 04:41, 3 November 2005 (UTC)

Right from the start that formulation is poor and has been improved.--Patrick 08:32, 3 November 2005 (UTC)
I don't see anything in there which isn't in the current version.--MarSch 11:41, 3 November 2005 (UTC)

You are right in one thing Patrick, that being that the introduction in that old version is incorrect; for example, the symmetry also satisfies that property. However, please note that your "improved" version states:

Rotation is the change of orientation of an object.

while at orientation you state:

Orientation of a rigid body in the three dimensional space changes by rotation.

That is, you have a circular definion, which is not helpful in explaining what rotation is all about. The rest of the introduction at rotation tries to attempt to explain what rotation about axis is, but is hard to follow.

The current verion of this article suffers from a weakness I saw in other places. It is an huge amalgam of facts, put together in a very tasteless way.

I would suggest to make the introduction easier to understand, split off most of the math technicalities to its own article, and kill half of the "physics" and "precession" sections. That should leave us with a short overview of all things rotation without endless ranting about particular instances. Oleg Alexandrov (talk) 13:20, 3 November 2005 (UTC)

That is the most complete b...now wait a minute...I euh... agree completely ;) --MarSch 17:16, 3 November 2005 (UTC)
What a s*t*y piece of humour. Oleg Alexandrov (talk) 19:02, 3 November 2005 (UTC)
Thanks :D --MarSch 16:03, 4 November 2005 (UTC)

## Propeller

An example of rotation of a planar figure around a point is the movement of the propeller of an aircraft.

It does not seem to make sense to model a propeller by a 2D figure.--Patrick 02:44, 5 November 2005 (UTC)

If I make it a spinning wheel, will that be better? An aircraft (or, if you wish, a helicopter) propeller is essentially flat. Oleg Alexandrov (talk) 04:16, 5 November 2005 (UTC)
If it is flat it does not work. For real objects. even if they are flat, a 3D model is more helpful, with a perpendicular axis; your image will do.--Patrick 09:42, 5 November 2005 (UTC)
Of course a flat propeller will not work! But that's not the point. It is much easier to imagine a 2D rotation than a 3D rotation. I believe it is essential the reader understands 2D rotation first. And yes, a flat propeller is not going to push the airplane forward, but I ask the reader to imagine that there is no airplane or time to start with, just a silly 2D spinning propeller. Oleg Alexandrov (talk) 17:58, 5 November 2005 (UTC)
It is a confusing example.--Patrick 20:58, 5 November 2005 (UTC)

## Orientation or process

Please give me example of rotation w/o a fixed point.

For example a rolling object.--Patrick 09:42, 5 November 2005 (UTC)

Above you seem to imply I am not rigurous enough. Now is my turn to say the same to you. As a mathematician to mathematician, a rotation is an isometry of an Euclidean space. A rolling object is a function
$t\to \gamma(t); \, \gamma: \mathbb R \to M$
where M is the (five dimensional?) manifold $SO(3)\times \mathbb R^3$, an element of which is any rotation $x\to Ax+b.$
And as we all know, a path on a manifold is not the same thing as a point on a manifold. Oleg Alexandrov (talk) 17:58, 5 November 2005 (UTC)
This is cleared up by "The term may either refer to the process, or to the resulting change in orientation relative to the starting or reference orientation.". Since you mentioned rotation about a point I concluded you were talking about the process. Now it turns out (below) that you thought that rotation about a point was a possible change of orientation.--Patrick 20:56, 5 November 2005 (UTC)

A rotation is not a process. A rotation is an isometry of R^3. No more, no less. Yes, you can take about functions which have value rotations, and by extension you can call them rotations too, but mentioning that straight in the introduction obscures the point of the article.

If you disagree that rotation is just an isometry, I will ask for references. Oleg Alexandrov (talk) 22:55, 5 November 2005 (UTC)

The process is even the more common meaning, something turning around an axle, etc.--Patrick 00:12, 6 November 2005 (UTC)

This article is a huge mess, without explaining properly what a normal rotation is (isometry in 3D). Once that is out of the way, we can write a paragraph somewhere explaining that for the process one just has a rotation at each time.

Let me put in a different way. Imagine I try to explain what a matrix is, and you keep on insisting that a matrix is a process, because you can have matrix-valued functions. Oleg Alexandrov (talk) 00:48, 6 November 2005 (UTC)

Most of the article is about rotation as a process. The part about rotation as a direct isometry can perhaps be merged with Euclidean group, for a better separation.--Patrick 07:50, 6 November 2005 (UTC)
Your own formulation "travel in circular trajectories" also describes a process.--Patrick 08:02, 6 November 2005 (UTC)

One obvious problem with this page is: Who is it for, who is its intended audience? Is this page for layman who are satisfied with a layman's definition? Is it for mathematicians?

"Notice that the crux of the matter is the difference between 'to spin' and 'to rotate'. The latter is a coordinate transformation and thus it has no history and no time, whereas to spin an object is a process realized in time and it thus has a history" (Altmann [1] p.23). Is that helpful?

In its most formal sense a rotation (of an actual object or of a mathematical entity) is concerned only with an initial and final state. In its most formal sense a spin (of an actual object or of a mathematical entity) is a process: the initial and final states are important but so is the "path" by which the spin proceeded. One can use matrices or not; one can use a path integral formulation or not; blah blah blah; but if one writes of a spin and yet ignores the path--the process--by which the entity moved from the initial to final state, one is no longer using the most formal definition of spin and is instead implicitly using at best the formal definition of rotation. Similarly, if one writes of a rotation and claims it has a history (that it has a process) one is no longer using its most formal definition. Why bother to have two words to mean the same thing? Why bother to have two pages for the same concept?

Of course all kinds of less formal, even layman, definitions of rotation and spin are fine and of immense utility (in theory and practice). Ad hoc definitions also have their use. Certainly the QM definition is fine. (That word was applied to angular momenta because it seemed a good word to use at the time. But as real and formal (and important) as angular momenta are, the use of the word "spin" in QM is very informal, even metaphorical.)

What is on this page is utterly confusing and thus almost worthless. You see what happens when one moves away from the formal definitions (or is unaware of them to begin with). One ends up with discussions like these and a page with all sorts of problems. The creators of this page don't seem able to settle on any definition. And the definitions it does have are, uh, non optimal.

From paragraph 1 "A rotation is a circular movement of an object around a center (or point) of rotation." I don't know what that means. It defines rotation in terms of a point of rotation. And what is a point of rotation? Obviously it is what a rotation goes around on a circle. And what is a circle? Why, obviously it's a circular movement--"locus" mathematicians call it--around a point. I.e. it's a rotation. Humbug. And what is a center? Etc. Why not just stick with coordinate transformations? Or why not make everything for layman and not worry about "point of rotation"? Precision about fuzzy things is pointless.

From paragraph 1: "If the axis passes through the body's center of mass, the body is said to rotate upon itself, or spin" seems purely colloquial to me. Worse: it claims a spin is a special type of rotation. It ignores the clear and simple distinction between a coordinate transformation and a process. And were I to teach someone a spin is a special type of rotation, how would I then discuss coordinate transformations (and demonstrate how fundamentally different rotation and spin are) without looking foolish? (If anything (and I cringe here, don't quote me on this), a rotation is a type of spin, one in which we ignore the process and only are concerned with the initial and final states. I.e. a rotation is a simplified spin. More humbug. The two are not the same at all.)

Do any of you teach math? It is amazing the questions students will ask. (E.g. "If rotations and spins are the same thing, or just two different words for the same thing, why do I need to learn about both?") They find every hole in a definition, wittingly or unwittingly. Whenever this comes up I need to tell them that there are formal definitions and informal ones. Which is the rotation page supposed to discus? Maybe that is the first thing to figure out.Royfleming (talk) 22:34, 5 November 2014 (UTC)

## 3D rotation about a point

This is somewhat related to the previous section. I want to know what "3D rotation about a point" is supposed to mean. In my understanding of 3D rotations, they are always rotations around an axis, for the folliwng reason. If we define rotation as "rigid motion that keeps one point fixed, call it the origin, and preserves (mathematical) orientation", then a rotation is given by a matrix $R\in SO(3)$. Now every 3x3 matrix has an eigenvector, so R has an eigenvector to its only possible eigenvalue 1, and this eigenvector is the axis of rotation which is also kept fixed by R. (In the normal space to this you have an ordinary 2D rotation).

So "rotation about a point" must be a different concept, or a misunderstanding (possibly on my side) Kusma 18:29, 5 November 2005 (UTC)

Well, rotation around a point which is not around an axis is the one with the matrix
$\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}.$
Yes, this article needs more work. Oleg Alexandrov (talk) 18:33, 5 November 2005 (UTC)
That is not a proper rotation (the determinant is -1), but point reflection at the origin. Kusma 18:41, 5 November 2005 (UTC)
You are right. If that matrix is four dimensional, you would get a weird rotation with eigenvalue -1, but in 3D it does not work. So I guess every rotation must be around an axis. I will think more about it. Oleg Alexandrov (talk) 19:38, 5 November 2005 (UTC)
Actually this is already in the article, see the reference to Euler's rotation theorem. I said it before, this article suffers from wealth of detail but lack of clear point. Oleg Alexandrov (talk) 19:45, 5 November 2005 (UTC)

## Removing one paragraph

There is a lot of discussion on this page recently, see the last three sections above. Now I want to start another one. I would like to cut off the paragraph:

Rotation is the change of orientation of an object. The term may either refer to the process, or to the resulting change in orientation relative to the starting or reference orientation. If it refers to the process, the simplest case is rotation of a rigid body about a fixed axis of rotation: a line such that each point of the body moves in a plane perpendicular to that line, in a circle centered at the intersection of the plane and the line.

Here are the reasons:

• The first sentence is circular. It defines rotation by means of change of orientation, and at that article it defines orientation as result of rotation. It does not give any insight into the article.
It is not a formal definition but an introductory remark. It mentions the relation with a related concept, and a link to the corresponding article. Those should be kept, of course.--Patrick 20:56, 5 November 2005 (UTC)
I think orientation should be fixed so as not to rely on this article. A rotation of an object is a change in the orientation of that object. An orientation is a direction from some fixed point.--MarSch 14:11, 6 November 2005 (UTC)
• The second sentence about process or resulting change is one of those things which instead of illuminating the concept forces the reader to stop and wonder what the hell one is talking about.
See above. Process means this function $t\to \gamma(t); \, \gamma: \mathbb R \to M$. The two meanings should be distinguished.--Patrick 20:56, 5 November 2005 (UTC)
I don't see any difference between the two meanings. What I think this might be about is that if you turn a rock in your hands first around one axis and then around another, you might _as well_ have turned it around one axis to arrive at the same final orientation.--MarSch 14:11, 6 November 2005 (UTC)
• I am not sure about the last sentence, it is too wordy and might need rephrasing. Oleg Alexandrov (talk) 19:51, 5 November 2005 (UTC)

## Request for comment

Rotation is in my view a poorly written article which has seen better days. I am deadlocked with Patrick about it who would rather prefer to see rotation not as an isometry of R^3 but primarily like the process of rotation; think of the time-dependent continuous transformation undergone by a door when you open it.

He also defines rotation as "change of orientation", with orientation (rigid body) explaining what he means by "orientation". In short, I would like the first paragraph in the section "rotation#More details" gone or radically rewritten, as I find it utterly confusing.

Comments and actual edits very welcome. Oleg Alexandrov (talk) 01:00, 6 November 2005 (UTC)

I have a sense of what Patrick may be wanting to talk about, and also of what Oleg wants. I suggest a path forward that may work better for all: separate Rotation (mathematics) from Rotation (physics). The contents of the current Rotation article should go into the first, with a revised disambig notice and a cleanup; and the Rotation around a fixed axis article should become (part of) the second, along with relevant material removed from Rotation.
A mathematician expects this kind of definition:
• A rotation is a proper rigid motion leaving one point fixed.
From there we go on to standard topics in geometry, providing background that will support the uses in physics and astronomy and aeronautics and so on.
On the other hand, the Earth spinning on its axis is a continuous function of time. Mere geometric spinning does not cause precession; that involves angular momentum. Once we have established the mathematician's definition in one article, we can refer to it in the other, while cleanly distinguishing the two. I believe there is enough to say about both topics to warrant two articles, and the separation is natural.
Physics taste and mathematics taste are known to differ. But a clean mathematics article also supports more sophisticated physics, that which needs more sophisticated mathematics. Quantum (field) theory comes to mind.
As it is, the current state of affairs serves nobody well, and verges on a protracted slow-motion edit war. That's not helpful for readers and not productive for editors.
So, is there a consensus to accept a split? I'll start the voting. --KSmrqT 08:24, 7 November 2005 (UTC)
Support mathematics/physics split
Support, for reasons listed above. --KSmrqT 08:24, 7 November 2005 (UTC)
Oppose
Oppose. All articles should be merged. I see no clear distinction at all. Central development will be the most beneficial. The elements of SO(n), rotations, can also be viewed as discrete rotations. Then a continuous rotation is a map R → SO(n), that is, an orientation at each point in time. Both of this is rotation and closely related. Remember that there is also already orthogonal group for a mathematical treatment. --MarSch 10:17, 7 November 2005 (UTC)
Point taken; there's also rotation group. However, neither one has much geometry (in the style of Coxeter); and in affine space the rotations do not form a rotation group (see below)! --KSmrqT 12:18, 7 November 2005 (UTC)
Oppose. per MarSch. If daughter articles are needed they should be linked and summarized here. There is no difference between rotation in physics and rotation in math. Only different points of view and semantic nuances for which we must find a compromise. Vb14:22, 7 November 2005 (UTC)
• Oppose. I do plan to fork off most of the math section into its own article (too much intimidating stuff in there), but there's got to be a (preferably short) article explaining what the word rotation is all about. However, I find MarSch's remark about "all articles must be merged" to be naive and used on principle and not based on merits of concrete case. Oh, I don't think I am in an edit war with Patrick. I belive we are converging somewhere. The hard part is to keep in check Patrick's zeal at adding each and every fact related to rotation without regard of the soup one gets at the end. Oleg Alexandrov (talk) 18:00, 7 November 2005 (UTC)
• Oppose. Agree w/ Oleg, the split should be simple vs. complexity of topic, rather than by subject. linas 14:11, 9 November 2005 (UTC)

Some remarks:

• I started Rotation around a fixed axis (with limited scope, but enough, it seems, for a separate article) because some people do not like long articles with lots of content. Thus we may need more physics articles on rotation.
• I think saying that in mathematics a rotation is about a fixed point is a bit odd. It creates the impression that mathematics cannot handle the Euclidean group, and functions from time to it, and therefore only considers rotations with a fixed point.

--Patrick 09:07, 7 November 2005 (UTC)

In the Euclidean group the definition I gave is still correct. You should reconsider the source of your impression, because it is mistaken. (See, for example, Coxeter's Introduction to Geometry, 2/e, ISBN 0471504580.) However, the composition of two rotations is not guaranteed to be a rotation unless they share a common fixed point. In a vector space, SO(n) happens automatically because there is a unique origin; not so in an affine space. To handle functions of time, we begin with this (geometric) definition, then speak of paths in SO(n), as MarSch noted. --KSmrqT 12:18, 7 November 2005 (UTC)
It is not my impression that in mathematics a rotation is about a fixed point. On the contrary, I said that it is odd to create that impression. Thus we have a path in E+(n ), not in general in SO(n ).--Patrick 15:30, 7 November 2005 (UTC)
Your words reveal a confusion in your thinking. What I said was that a rotation "leaves one point fixed", not "is about a fixed point". We are, once again, using the word rotation in two different senses: mine, the standard mathematics one, and yours, more of a physics one. In the mathematics sense, a rotation is a single static linear transformation, captured by a before and after picture with no continuous transition. In this regard, it is exactly like "reflection": It is impossible to get from the identity to a reflection by a continuous rigid motion. Your sense would be considered a path in rotation space. The confusion is a common one. Again, consult Coxeter, as unimpeachable a geometer as one is likely to find, and easy to read. --KSmrqT 00:34, 8 November 2005 (UTC)
I know the distinction between a single transformation and a continuous transition, I was the first who pointed that out in the article. What I meant was, in mathematics we may either consider rotations in the sense of linear or in the sense of affine transformations (the latter in the sense of not keeping the origin fixed, not in the sense of "angles do not matter").--Patrick 11:31, 8 November 2005 (UTC)
Do rotations on affine space form a groupoid? --MarSch 14:43, 7 November 2005 (UTC)
No. The situation is as follows. Take two sheets of graph paper, aligned one on top of the other. Stab a pin through both sheets. Twist the top sheet. Stop. Compare the two sheets to get a before-and-after view of the effect of the transformation. This is a rotation, a proper rigid motion with one point fixed. Without disturbing the position of the sheets, remove the pin and stab a different point. Twist again. This, too, is a rotation. However, comparing the bottom sheet with the top sheet after both twists have been performed, we find that no point remains fixed. In fact, if the first rotation is 90° clockwise and the second rotation is 90° counterclockwise, the net effect is a pure translation. Repeat the exercise with 3×3 homogeneous matrices:
$R_1 = \begin{bmatrix}0&1&x_1\\-1&0&y_1\\0&0&1\end{bmatrix}$
$R_2 = \begin{bmatrix}0&-1&x_2\\1&0&y_2\\0&0&1\end{bmatrix}$
$R_2 R_1 = \begin{bmatrix}1&0&x_2-y_1\\0&1&x_1+y_2\\0&0&1\end{bmatrix}$
In group theory terms, we are seeing the effects of a semidirect product. A pure rotation around an arbitrary point in this example is an element of SE(2), the special Euclidean group of the Euclidean plane. The first matrix leaves fixed (x1+y1,−x1+y1)/2; the second, (x2y2,x2+y2)/2. Considering the transformation as a pair (g,t), the multiplication differs from that of the direct product in that the g components (here, SO(2) elements) simply multiply, but the t components (the translations, a normal subgroup) do not.
Essentially, we've been spoiled by vector spaces to think we always have a common origin. Nevertheless, what I describe is classic geometry. Those fluent in German might consult Grundzüge der Mathematik, Bd. 2. Geometrie, Vandenhoeck & Ruprecht, Göttingen. (Available in English translation from MIT Press as Fundamentals of Mathematics, Vol. II Geometry, ISBN 026252094x.)
Nor is this subtlety of only mathematics interest. Robotic manipulators grip mechanical parts using the theory of screws and wrenches, as pioneered by Ken Salisbury drawing on classical mathematics described by Ball in A treatise on the theory of screws, ISBN 0521636507. A web search for "screw wrench salisbury roth ball" will turn up many examples. In such an application the existence of different pivot points on the gripped object is unavoidable. --KSmrqT 00:34, 8 November 2005 (UTC)

## inventory of rotation articles

To facilitate reorganizing the rotation articles, it should first be clear what exists.

I'm not saying all of this should be merged into one article. Probably not. Please add any articles you think also belong on this list.--MarSch 14:52, 7 November 2005 (UTC)

Okay, I think that

I do not mind combining rotation and orientation (rigid body), and combining Rotation around a fixed axis and axis of rotation, but I oppose combining all four, and oppose merging in Improper rotation, it is quite a different subject. They can be summarized in rotation, while details are in those articles. It is normal practice that articles are split when they get too long.--Patrick 13:39, 8 November 2005 (UTC)

Agree with MarSch, mostly: combine rotation, orientation (rigid body), Rotation around a fixed axis, axis of rotation into one. However, Improper rotation should indeed be kept separate. Agree with Oleg, the discussion of SO(n) should be moved to another article or shortened or something. In particular, Rotation around a fixed axis should never have been split off. It is also rather bizarre and broken in various ways: for example: According to the right-hand rule, moving away from the observer is associated with clockwise rotation Huh ???? linas 14:19, 9 November 2005 (UTC)
Have you never heard of that rule?--Patrick 00:05, 10 November 2005 (UTC)
• Orbital revolution should be listed as a type of rotation. I also feel that the rotation article should reflect that the term "revolution" is more commonly used for movement of one celestial body around another. 12.106.111.10 19:41, 7 December 2005 (UTC)

I would argue that these details belong in some astronomy-specific articles. And it seems that orbital revolution is already linked from this article. Oleg Alexandrov (talk) 22:39, 7 December 2005 (UTC)

## What is rotation?

So, we have rotation as a process, and rotation as a transformation, a distinction which I think is not so important to make as it is clear from the context which one is meant.

I have another question. What exactly is rotation? I would not classify a ball rolling down a hill as rotation, I would think that this movement is more complicated than that. Or am I mistaken? Oleg Alexandrov (talk) 02:05, 9 November 2005 (UTC)

Not sure if this is a rhetorical question. In physics terms, the ball is rotating (or pivoting) "instantaneously" about the point of contact (Its also rotating "instantaneoulsy" about any other point as well, but the point of contact is useful because that is where the friction is acting, and where the torque is being imparted.) In math terms, "instantaneously" just means "tangent space vector" or "derivative". In real life, almost all rotations are constrained in some way: besides rolling and pivoting, there's precessions and torque. Spinning electrons in a magnetic field feel a force; the cosmos, oddly, seems to not be rotating. So its inappropriate to talk about rotation without talking about these other constraint effects (even though they don't map cleanly onto concepts such as so(n)).linas 14:32, 9 November 2005 (UTC)
Hey Linas, maybe you can write a rotation (physics) article explaining all that, and see how to put a blurb at rotation about this. That would be much appreciated. Oleg Alexandrov (talk) 17:52, 9 November 2005 (UTC)
I will not agree with rotation around a fixed axis being merged into here until we clarify what rotation is all about. I still hope to be convinced that arbitrary rolling or moving around in space is still classified as rotation. Oleg Alexandrov (talk) 01:23, 10 November 2005 (UTC)

## My recent removal

I just did a revert. Let me explain why. The "Mathematics" section has exactly four paragraphs, each designed to highlight some of the very important properties for rotation.

• Paragraph 1 says what a rotation is, and what else exists beyond rotation as rigid body transformation.
• Paragraph 2 says that rotations form a group, and only if one looks at rotation around common center/axis.
• Paragraph 3 deals with the very important property that in 3D all rotations are around axis, a fact not at all obvious.
• Paragraph 4 is again about 3D (the most important case anyway) saying that any rotation is a decomposition of three principal rotations. The picture with the airplane illustrates that very nicely.

Patrick added a paragraph 5 listing all possible rigid motions. That is very reduntant, as paragraph 1 already says that. The only new piece of information in here is that a combination of rotation and tranlation is a rotation around an axis followed by translation along that axis.

As such, we have four paragraphs highlighting importat themes, and the fifth paragraph is having a go at the first paragraph again going into big details (with a list). I find that bad taste. In the best case it should have been merged with paragraph 1, but even there it would be reduntant.

Each of the four paragraphs can be expanded with some stuff, but what will we get then? Instead of clearly stating what rotation is about it would become a wealth of detail hard to follow. Yes, this is one of those cases when putting more stuff does not help the reader, it forces the reader to think locally instead of viewing the nature of rotations from a bigger perspective. It is like driving nicely along a highway looking at the view and then being stuck in a traffic jam.

Please put extra details at rotation (mathematics). And even there, please do not add arbitrary pieces of information. Please keep the article readable. There is the right moment of going into detail and the wrong moment. I don't know when is which, just make sure the reader feels as if he/she is gliding through the article, each detail is tastefully chosen, and helps the reader instead of confusing him. Again, an article is not the same as piece of information 1, followed by piece of information 2, ...., followed by piece of information n. Oleg Alexandrov (talk) 16:58, 10 November 2005 (UTC)

If however one performs rotation around a point followed by rotation around another point, the overall movement may not be a rotation anymore.
This suggests that vagueness is unavoidable, because it is too difficult to explain what the combination is, while in fact it is easy to be specific and clear. Also you suggest that my addition is an arbitrary bit of info, while it is an orderly overview showing the big picture. Another vague part is "a combination of the two". This depends on the direction of translation and is either itself a rotation or a screw operation. If we talk about combinations we should mention that (i.e., a little extra over what I added first).--Patrick 21:32, 10 November 2005 (UTC)
I agree that it could be put higher up and integrated with the rest. Having vagueness first and being specific only later is not necessary here.--Patrick 21:02, 10 November 2005 (UTC)

## Request for explanation

Partrick, can you explain this edit? I don't really understand what you mean in there. Also please note that you moved the sentence "Any rigid body movement is in fact either a rotation, or a translation, or a combination of the two." from the first paragraph into the second paragraph. I don't like that, as that sentence is very coupled to the definition of rotation as a rigid body movement in the first paragraph. Thanks. Oleg Alexandrov (talk) 23:21, 10 November 2005 (UTC)

It is better to have a clear idea of a rotation before talking about combinations. Also, talking about rotation around a point and then saying there is only rotation around a line is odd and confusing.--Patrick 00:02, 11 November 2005 (UTC)
That part can now be revised and talk about rotation around a line.--Patrick 00:05, 11 November 2005 (UTC)
OK, let us deal with what is in the previous section later.
There is nothing confusing. The way my text was, left part, first and second paragraph are about rotation in any dimension. Third and fourth paragraphs are about rotations in 3D only, and that is clearly stated. I belive you interpretted the whole text to be about 3D, and this is why you think it is confusing. Now the text is way too much biased for 3D rotations, starting with the first paragraph.
Also, I would not like to have anything else about rotation about a line. Two paragraphs in this four paragraph section is already about that, and I feel more would be too much. This text is meant to be an elementary introduction to rotations in any dimension. Oleg Alexandrov (talk) 00:58, 11 November 2005 (UTC)
A rigid body is usually meant to be 3D. Therefore, indeed, I had 3D in mind. Also, elementary introduction to rotations in any dimension seems a contradiction, especially if you describe it geometrically instead of as a matrix operation. It requires understanding 3D rotations as a starting point.--Patrick 03:05, 11 November 2005 (UTC)

I would think that talking about rigid motion in 2D is not such a farfetched proposition. You are right that this definition cannot bring us higher than 3D, and I agree that tal about higher dimensions better take place latter. I plan to modify the article a bit however to mention both 2D and 3D, I think you are having a POV with 3D. :) Oleg Alexandrov (talk) 01:30, 12 November 2005 (UTC)

2D is fine. Anything non-trivial in 2D should be considered in 2D first before proceeding to the more complicated 3D case.--Patrick 11:57, 12 November 2005 (UTC)
However, real physical objects are 3D, so a 3D body is very common. A thin 3D object should be clearly distinguished from a 2D object, even though we may have the same mental image for both.--Patrick 12:23, 12 November 2005 (UTC)

## Image caption

In the caption for the first image, do the two mentions of the word "plane" refer to "airplane" or plane (mathematics)?--GregRM 19:25, 17 February 2006 (UTC)

The first, I clarified it.--Patrick 00:52, 18 February 2006 (UTC)

## Split?

Could this article be split? It seems like there's 2 (or maybe even 3) pages jammed in here together. The math section really doesn't seem like it belongs here; and I'm not sure what "Allowed rotations" is trying to convey. The rest seem to be elaborations on physics. Maybe some or all of that could even merge into the Spin (physics) page? (See also Talk:Spin (physics)#Split?) Ewlyahoocom 12:58, 2 April 2006 (UTC)

I merged in axis of rotation.

About splitting. It is good to split a page into multiple pages and make it into a disambig when the meanings are different. Here, we are talking about various meanings of rotation, so it is good that they are discussed in the same text, so that the reader sees the similiarity and difference between how the concept of "rotation" is used in various places. Oleg Alexandrov (talk) 15:36, 18 April 2006 (UTC)

I don't think splitting is necessary, too. Being able to discuss the same topic (rotation) from different perspectives is a strength that can be offered by an encyclopaedia.--Wingchi 17:01, 18 April 2006 (UTC)

Part of the issue, as I see it, is some not insignificant number of people mistakenly show up at Spin (physics) which is 99% about quantum/particle physics, looking for information about rotation and I think the current Rotation page is incomplete/confusing.
I agree 100% that a number of different "examples" can be contained in this page. Maybe some other pages could even be merged, or parts of this page in a "summary style" i.e. sections with "Main article" links, see the above section #inventory of rotation articles for some that may be appropriate. To that list I'd like to add Angular momentum and Billiards#English.
And getting rid of that "Allowed rotations" section was good -- Thanks, Oleg. But I still think the math section is out of place, it could be moved to its own article ala Reflection (mathematics), Translation (geometry), Glide reflection, Scaling (geometry). Ewlyahoocom 18:39, 18 April 2006 (UTC)

I am against splitting, for the same reasons cited by others. LjL

## Composition of rotations

(Copied from my talk page). Oleg Alexandrov (talk) 19:53, 15 May 2006 (UTC)

Hi Oleg,

About compositions of rotations: you wrote

If however one performs rotation around a point (axis) followed by rotation around another point (axis), the overall movement may be a translation rather than a rotation.

I think our editing reverts have been because we agree but have opposite points of view: you want to point out that two rotations may be a translation (or rotation), but I want to point out that two rotations in general do not represent a translation or rotation.

Thus, perhaps

If however one performs rotation around a point (axis) followed by rotation around another point (axis), the overall movement may be a translation rather than a rotation (but in general is neither).

216.232.222.122 15:03, 15 May 2006 (UTC), formerly MrMoto, but this account seems to have been eaten up. :(

Can you prove that the composition of two rotations around different centers in the plane may be something else than a rotation or a translation? Because I can prove that it is either a rotation or a translation. Oleg Alexandrov (talk) 15:08, 15 May 2006 (UTC)
Never mind, I think I am wrong. (As always, best ideas come in the shower :). Oleg Alexandrov (talk) 15:24, 15 May 2006 (UTC)
Heh. ok :) You are correct in the context of planar rotations, but in R3, for example, the axes of rotation should intersect. 216.232.222.122 15:41, 15 May 2006 (UTC)
If I may interject, rotations are direct isometries that leave a point fixed. In 2D we have only two choices for direct isometries (aside from the identity): rotations and translations. The latter leave no point fixed. In 3D we get a new option, screws. These can always be cast in the form of a rotation around an axis parallel to a translation. In 4D a rotation generally does not have an axis, and leaves only a single point fixed. Translations and screws still exist, of course. In a vector space we always rotated around a fixed origin, with no possibility of translation or screws. Most folks forget that this is not the only option, since it's the most common way to describe rotations. With a fixed origin, rotations form a group; without one, they are not even closed under composition.
Perhaps it would help readers to see explicit examples in 2D and 3D. I'd suggest pictures and words and matrices, to make the point memorable.
$\begin{bmatrix} 0 & -1 & x_1 \\ 1 & 0 & y_1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 & x_2 \\ -1 & 0 & y_2 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & x_1-y_2 \\ 0 & 1 & y_1+x_2 \\ 0 & 0 & 1 \end{bmatrix}$
$\begin{bmatrix} 0.8 & 0 & -0.6 & x \\ 0 & 1 & 0 & 0 \\ 0.6 & 0 & 0.8 & z \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0.48 & 0.8 & -0.36 & 0 \\ 0.6 & 0 & 0.8 & 0 \\ 0.64 & -0.6 & -0.48 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & x \\ 0.6 & 0 & 0.8 & 0 \\ 0.8 & 0 & -0.6 & z \\ 0 & 0 & 0 & 1 \end{bmatrix}$
The fixed points for the leftmost 3D rotation have the form (x−3z,y,3x+z), while the composition of the two rotations is a screw unless z = −2x. --KSmrqT 18:23, 15 May 2006 (UTC)

Thank you KSmrq. Would be nice to have this somewhere. Maybe not the full formulas in the general purpose rotation article, but at least the ideas. Oleg Alexandrov (talk) 19:53, 15 May 2006 (UTC)

(Copied from my talk page). Oleg Alexandrov (talk) 19:53, 15 May 2006 (UTC)

Furthermore, why is this being fought out here instead of at that sections "Main article" at Rotation (mathematics). (Although, for whatever reason, that article, which is is actually a redirect to Coordinate rotation, bears very little resemblence to what's included here.) Ewlyahoocom 10:45, 16 May 2006 (UTC)

Rotation (mathematics) bears little resemblance to what is here because I took the math section of this article and replaced the lie group, spinor group, orthogonal group and symmetry group crap with normal English explaining the basic math behind rotations. Oleg Alexandrov (talk) 14:50, 16 May 2006 (UTC)

# Suggest distinguish physical process vs equivalent final orientation

Dear esteemed and respected Rotators, Nice job - thanks. I believe one key statement may be rather misleading. As far as I can see, this statement is at best ambiguous and at worst wrong. (Ah - OK - related issues have been raised before but this simple take may help anyway.The verbiage below is just to give a thorough audit of the viewpoint - if accepted, the point and the proposed change are very simple (see below) Comments welcome. The article currently says:

• It turns out that a rotation in the three-dimensional space keeps fixed not just a single point, but rather an entire line; that is to say, any rotation in the three dimensional space is a rotation around an axis.

There is a profound physical sense in which this is untrue and misleading, Conversly, there is an interesting though highly constrained mathematical sense in which it is true. It is vital to distinguish these two senses. Let's consider the physical sense first, then the more constrained mathematical sense as follows.

## As a continuous physical process

In one clear physical sense, the above statement from the current version is not true, and will mislead some readers. Single axis continuous rotation is only a special case physically. This is easily tested/demonstrated empirically as follows.

Take a pencil and rotate it between your fingers along its long axis, like rolling a log. Now at the same time rotate the pencil vertically like a propeller or clock hands.

From a fixed frame of reference the pencil now has two simultaneous independent axes of rotation. There is only a single fixed point. From moment to moment there is demonstrably no fixed axis, only a fixed point at the centre of the pencil.

One the other hand, a statement earlier in the article is physically bang on.

• a rotation is a rigid body movement which keeps a point fixed; unlike a translation. This definition is applicable both for rotations in a plane (two dimensions) and in space (three dimensions).

### Thought experiment to explore other reference frames

Ok – that was an empirical test – now here is a thought experiment to test matters further, from an arbitray inertial frame of reference – imagine the pencil was hollow and full of liquid. If it was spun on just one axis like a centrifuge, the water would adhere to the cylindrical surface like on a spin dryer. If, on the other hand, the pencil was spun like a propeller, the liquid would go to the two ends of the pencil. If you combine these two rotations by starting with a propeller rotation and then adding a centrifuge action, you will get a tendency to get lumps of water shaped like two rings at the two ends. This physical effect has two axes of symmetry when viewed from any inertial frame of reference , due entirely to the rotation of the pencil. These two axes have only a single point in common. This could not be caused by any rotation around a single axis in three dimensions – even an axis continuously moving wrt to the pencil.

Explain to me then how is it possible that every rotation matrix can be transformed into quaternion or axis/angle representation. I can assure you that matrices and quaternions work perfectly, every 3D video game uses them to make nice realistic rotations. Your thought experiment proves nothing, because it seems that the laws of physics that you imagine work differently that the real ones. The water would not gather into two rings, but would lump at two opposite points at the ends of the cylinder - those two points would lie exactly on the axis of rotation. Anyway, any rotation in any number of dimensions can be described as a rotation of a plane around a fixed point, 'dragging' along the points on planes parrelel to it. That's why in 3D it has to leave one-dimensional axis fixed. 94.101.25.225 (talk) 21:22, 19 April 2009 (UTC)

## Mathematically in terms of equivalent final orientations

OK – now let us consider the sense in which the statement is mathematically provable and important, due to Euler ( I would imagine).

Starting from some static orientation of a rigid body, you can compose any number of arbitrary rotations to reach some static final orientation, and the whole process can be expressed as a single rotation about some craftily chosen axis. I don’t know a proof but that sounds right and I would be surprised if it were false.

But that would just be a statement about endpoints. During the actual sequence of physical rotation there could be a single fixed point but there would be no need for a fixed axis of rotation from moment to moment, or even a continuously rotating axis. In the pencil example, there would not even need to be single a fixed axis of rotation in the limit instantaneously

So in physical terms I believe it is misleading and false to say

• any rotation in the three dimensional space is a rotation around an axis.

Where rotation refers to the physical process of rotation.

But no doubt it is true to say that

• The end position produced by any sequence of rotations of a rigid body in a three dimensional space could in every case be effected by a single rotation in the three d space around some appropriately chosen axis.

You can combine these two truths is a really trivial sense as follows

• Any rotation of a rigid body can expressed as a sequence of rotations about a rigid axis, but the axis of rotation may need to move from moment to moment relative to the body. And the movement of the axis over the rotating body may not even be continuous.

That would be an awfully weak and vacuous claim since it would be vacuously true if we just serially timeshared infinitesimal rotation in three independent axis with discontinuous jumps of axis at each step. Apologies about all the above - I needed to think it through even if no-one else did. Do I have a proposal? OK – how would this be?

## Proposed re-statement

• Mathematically, a rotation is a rigid body movement which keeps a point fixed; unlike a translation. This definition is applicable both for rotations in a plane (two dimensions) and in space (three dimensions). It turns out that the final orientation produced by any sequence of rotations in three dimensions around a fixed point can always be achieved equivalently by a single rotation around some appropriately chosen fixed axis (euler's theorem). However, when considering continuous physical rotation, as opposed to initial and final orientations alone, there need be no fixed axis.

This is not a stand off between mathematics and physics - mathematics can no doubt deal with both of these cases. Its just that the restricted case is the one that historically mathematicians have focused on. But innocent visitors cant be expected to know that. A slight restatement would help to avoid non-experts being misled while remaining true to the mathematics.

I'm not editing this in yet (or maybe ever) since I'm rusty wikipedia-wise & I'd like to see what others think first, Also there is evidently a whole bunch of page history which I havent read yet. But as currently stated the sentence could be misleading to innocent visitors. Cheers all & thanks again. -- Reflection 21:12, 2 June 2006 (UTC)

I find the wording
It turns out that the final orientation produced by any sequence of rotations in three dimensions around a fixed point can always be achieved equivalently by a single rotation around some appropriately chosen fixed axis (Euler's Theorem).
to be rather clumsy and combines too many things into one (composition of rotations being rotation and that each rotation around a point in 3D is routation around an exis are different things). Oleg Alexandrov (talk) 18:49, 3 June 2006 (UTC)

Much appreciated, Oleg - I'll give it some more thought :-) Reflection