Talk:Rotation (mathematics)

Name/redirect

It is wrong to name the article "coordinate rotation", since a mathematical rotation — of all things — can be defined and used quite well without coordinates, thank you very much. A rotation is a geometric idea, a direct isometry with a fixed point. Coordinates are handy in applications, but there is no excuse for letting them dominate the name of the article.

In fact, there is a beautiful way to approach geometry using involutions, which is to say, reflections. Discussions can be found in Coxeter's Introduction to Geometry, 2/e (ISBN 0471504580) and in Behnke et al.'s Fundamentals of Mathematics, Volume II:Geometry (ISBN 026252094X). Rotations in 2D or 3D are merely two intersecting reflections, which clearly produces a direct isometry with a fixed point at the intersection.

Therefore I propose to move the article where it belongs, to "Rotation (mathematics)", and redirect "Coordinate rotation" (and links here) to there. --KSmrq^T 12:55, 16 May 2006 (UTC)

I also believe that "coordinate rotation" redirecting to "rotation" is misleading. Point in case: the matrix algebra and subsequent equations for "x'" and "y'" describe the rotation of a point in a fixed coordinate system. The algebra is different for a coordinate rotation and the resulting x' and y'. Jdewald (talk) 17:59, 23 March 2010 (UTC)

Looking at the history of "Rotation (mathematics)", first it was split out from the Rotation page (which I fully support). Later, I suggested a merge with Coordinate rotation (see that version), thinking co-ordinate rotation could be a section on a larger page about mathematical rotations, but didn't really get merged, just redirected. I thought "coordinate rotation" was just an alternate way to describe a rotation, e.g. in computer graphics ray-tracing where sometimes it's easier to move the co-ordinates around than to move the camera or the items. Ewlyahoocom 14:35, 16 May 2006 (UTC)

I would support moving this to rotation (mathematics). You'd be most welcome if you would like to merge this and the text at rotation (mathematics) before I did the redirect. Oleg Alexandrov (talk) 15:04, 16 May 2006 (UTC)

I'm probably not qualified to do the actual merge cause I'm not a mathmagician. And I prefer your version which (to borrow your phrase) replaced the crap with normal English explaining the basic math behind rotations. Ewlyahoocom 22:49, 18 May 2006 (UTC)

Well, I moved the coordinate rotation article to rotation (mathematics). Any improvements to this article are welcome. Oleg Alexandrov (talk) 02:20, 19 May 2006 (UTC)

point, axis, hyper-plane?

As I see it, only in 2D a rotation can be said to be a transformation around a point.

In 3D shouldn't it be around an axis?

In 4D shouldn't it be around an hyper-plane????

Can anyone put some light on this? Mitus08 (discussion) —Preceding comment was added at 13:59, 13 January 2008 (UTC)

The 3D rotation is also around a point. It so turns out though, that any 3D rotation is not only around a point, but around an entire axis. In 4D one can again find rotations which leave just one point fixed, no more. But of course, you can always find 4D rotations which leave an entire axis fixed, or even an entire plane or hyperplane. Oleg Alexandrov (talk) 16:20, 13 January 2008 (UTC)

When it comes to higher dimensions, it might be easier to understand rotations to happen not 'around' points, axes, planes, whatever, but 'in' a plane. Every rotation has a plane in which it happens. If the said plane is defined by two axes of the coordinate system we use (or, alternatively, we define a coordinate system that makes the plane be in the 'right' orientation), the rotation modifies two coordinates (the ones which define the plane) of every point of the rotating object. So, in 2D we have only one possible rotation: in the XoY plane. In 3D the plane on which the rotation happens 'takes' two of the three coordinates of every point, which creates an illusion of a rotation around an axis. In 4D every plane has another plane completely orthogonal to it, so any 4D object can perform two independent rotations at once. This pattern continues, and generally in a (2*n)-dimensional space an object can perform n independent rotations which may leave only one fixed point, and in a (2*n + 1)-dimensional space it can also perform n independent rotations, but always leaving at least a whole axis (and thus one coordinate of every point) fixed. I'm neither a mathematician nor an native English speaker, but I hope you get what I mean. 94.101.25.225 (talk) 21:35, 18 April 2009 (UTC)

New section on 4D

I've added a section on fourth dimensional rotations. It could probably do with some expansion and maybe a few more links, but it covers the main points I hope. I think the last section most needs tidying up now, to better focus on e.g. rotations in five and higher dimensions. JohnBlackburne (talk) 18:34, 7 November 2009 (UTC)

I've started on this, moving the relativity sub-section as it's an example/application of rotations in 4D, at the same time rewriting so it more closely relates to the rest of the article, has more links and a good reference. This shortens and so simplifies what needs to be done in the last section, and I think I know what needs doing so may tackle it soon if no-one else does JohnBlackburne (talk) 00:13, 11 November 2009 (UTC)

2D section rewrite

Just overhauled the 2D section. Not massively, more lots of little tweaks so it reads better and has more information and connections while taking no more space. I think it makes sense to fix the rest of the article before doing the last section as it sort of follows from what goes before it.

We could do with some references for this section. I had a look but most of my books are on more advanced topics - this is all maths I learned at high school. At least the added wikilinks make it easier for readers to dig deeper if they want. JohnBlackburne (talk) 18:53, 19 November 2009 (UTC)

3D section rewrite

A bit longer as it's bigger and there was less there to work with. I've been working on this on and off at User:JohnBlackburne/3D so there's a short history there but nothing very interesting, and if I'd not been so distracted it could have been done in one go anyway. It's fairly light on maths, but it's got lots of links especially to main articles for more detail.

The section that most needs some attention now is generalisations, though the 4D section could be expanded a little. --JohnBlackburne (talk) 18:02, 27 December 2009 (UTC)

Confusing

In section it is said "The degrees of freedom of a rotation matrix is always less than the dimension. A rotation matrix in dimension 2 has only one degree of freedom. Given an angle of rotation the whole matrix is defined. A rotation matrix in dimension 3 has three degrees of freedom." which contradicts itself, since 3<3 is false. ¿Anybody cleaning up? Kjetil Halvorsen 15:40, 23 November 2009 (UTC) —Preceding unsigned comment added by Kjetil1001 (talk • contribs)

Done, or at least I've fixed that particular paragraph. JohnBlackburne (talk) 17:11, 23 November 2009 (UTC)

The degrees of freedom in n dimensions is the number of distinct pairs of coordinate axes, since every rotation can be derived as a product of principal rotations, each of which takes place in the plane defined by a pair of coordinate axes. The degrees of freedom is therefore $n \choose 2$ = ${\frac {n(n-1)}{2}}$.—Tetracube (talk) 17:50, 23 November 2009 (UTC)

Vector space vs affine space rotations

Look at article's very beginning:

“ In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. ”

Does anybody see here the word "vector"? I see only a hint to vector spaces through "linear algebra", and the word "fixed" is also ambiguous: either for each rotation exists a point… or there exists a point such that for each rotation…? The rest of article effectively discusses rotations in vector spaces (the latter choice), not affine (the former). Of course, an [S]O group is a factor of affine [orientation preserving] transformations by a normal subgroup of translations, but… the article says nothing about the difference.

Which statements are wrong for the affine case? For example, that in 2 dimensions any two rotation commute. Generally, such commutator is a translation, and is not identity unless rotations have the same centre. Incnis Mrsi (talk) 20:02, 25 February 2012 (UTC)

Enumerative disease

It’s rather typical phenomenon in Wikipedia. What shall we say about SO(2)? And SO(3)? Not to forget SO(4)! But all these groups (over reals) have distinct articles. Meanwhile, this article (about the concept) mentions matrix multiplication only briefly, and does not say a single word about the essential thing it describes. A poor stuff. Incnis Mrsi (talk) 14:10, 22 January 2014 (UTC)

I don't know what you're proposing. This article is in summary style so has only brief mentions of topics that have their own articles. The problem with saying much more is that the article is already long enough, though changes could be made without adding too much content I'm sure.--JohnBlackburne^words_deeds 16:27, 22 January 2014 (UTC)

The article elucidates several important cases, but it says next to nothing about the mathematical framework where the word “rotation” makes some sense in general. Namely:

1. Rotation is a general concept of geometry and is defined in various ways according to the structure of our space.
2. Rotations are maps.
3. Rotations of vector (affine) spaces are linear maps, but not all linear map (GL is not a “rotation group”).
4. Rotations of a vector space shall conserve some things, necessary to distinguish rotations from other linear maps.
5. In an affine space, rotations are not translations.
6. Without translations (in the case of vector, and others), rotations form a group.
7. For real vector spaces appropriate rotation groups are Lie groups, usually connected Lie groups as we consider identity components of non-connected groups.
8. Lie algebra of infinitesimal rotations.
9. For metric spaces rotations form a subgroup of isometries.
10. Metric spaces are the most important, but not the only case.
11. Rotations can define some classes of symmetry (I mean not the rotational symmetry only, but also the circular symmetry).
12. Rotation invariance in a broader context.

Also some specific facts shall be listed:

• The rotation group of a Euclidean space is SO(n).
  • It is compact.
  • Improper rotations are usually excluded from rotations.
  • Rotations of n-sphere are the same as rotations of Euclidean (n+1)-space.
  • Rotations and orthonormal bases.
• Spin groups.
• What is a “rotation” in a hyperbolic space? Search sources for authoritative definitions.
• Rotations in pseudo-Euclidean spaces. Isomorphism between SO⁺(1,3) and the Möbius group.
  • They are not compact, but include Euclidean rotations (trivial for 1+1 dimensions) as compact subgroups.
• Cyclic subgroups in various rotation groups. Non-trivial in SO(n) for n ≥ 4 (hint: isoclinic rotation).
• Generalizations: complex (U and SU) and infinite-dimensional (unitary operators).

Do you see how many interesting things one can express without a silly enumeration of various n = cases? Incnis Mrsi (talk) 16:31, 23 January 2014 (UTC)

A direct isometry of E³

One can realize that, in the general position, it is a composition of a rotation with a translation along its axis, i.e. in appropriate Cartesian coordinates:

(x, y, z) ↦ (x cosθ − y sinθ, y cosθ + x sinθ, z + c).

Would it be a rotation for c ≠ 0? It hasn’t any fixed point. Incnis Mrsi (talk) 14:08, 7 February 2014 (UTC)

I agree with you. Take it out and see who complains!--guyvan52 (talk) 17:24, 7 February 2014 (UTC)

Lolwut? The article currently avoids the question of general-position E³ isometries. I asked in a hope someone would suggest what the article should say about it. Incnis Mrsi (talk) 18:36, 7 February 2014 (UTC)

P.S. I just learned from the Euclidean group article that this thing is called a “screw operation”. IMHO the fact that for n > 2 isometries in general position cease to be rotations is mention-worthy. The article was heavily biased towards rotations of vectors and didn’t present significant facts about rotations in the context of Euclidean group. Incnis Mrsi (talk) 18:57, 7 February 2014 (UTC)

Actually, not for n > 2, but just for odd n, including 1. Interesting… I didn’t know it earlier. This is an example of a non-trivial fact that an encyclopedic article should communicate. Incnis Mrsi (talk) 19:13, 7 February 2014 (UTC)

By the way, there is a discussion about the “rotation group” term at talk: Rotation group. If nobody will object, I’ll redirect the title here. Incnis Mrsi (talk) 18:36, 7 February 2014 (UTC)

Hyperbolic pararotation

Example Coxeter group fundamental domains

This paper [1] talks about pararotations in hyperbolic space, like translation in Euclidean space, except the center of rotation is at a specific infinity point (seen on the ideal sphere of a Poincare disk projection). I've not seen this term elsewhere and have no other names for it. Tom Ruen (talk) 03:11, 29 March 2014 (UTC)

"The product of two reflections is a rotation, a pararotation, or a translation according as the mirrors intersect, are parallel, or have a common perpendicular."

One verbal source say this is called a limit rotation, which currently has one red link at SL2(R)#Parabolic_elements. Tom Ruen (talk) 22:49, 29 March 2014 (UTC)

Confirmed here [2], Tom Ruen (talk) 23:09, 29 March 2014 (UTC)

p.198 "A limiting case of rotation is where the two lines of reflection do not meet in the half plane, but have a common end P on the boundary R∪{∞} at infinity. Here P is a fixed point, each non-Euclidean line ending at P is moved to another line ending at P, and each curve perpendicular to all these lines is mapped onto itself. This kind of isometry is called a limit rotation, and each curve mapped onto itself is called a limit circle or horocycle." — Preceding unsigned comment added by Tomruen (talk • contribs) 23:12, 29 March 2014 (UTC)

Where are you headed with this? This article at present barely mentions rotations in hyperbolic geometries. —Quondum 23:55, 29 March 2014 (UTC)

I'm looking for what are the most common terms, and adding as a new section here, if that's best. Tom Ruen (talk) 00:09, 30 March 2014 (UTC)

I think there is a lot of writing to be done to satisfactorily introduce and define rotations in noneuclidean geometries before worrying about detailed terms. For example, I don't expect that point-fixing motions will even form a group in general noneuclidean geometries. Trying to stretch the term "rotation" to these contexts in a notable fashion may be tricky. My inclination would be to limit it to generalizations using the respective orthogonal groups, and focus on details of rotations in articles more specifically aimed at the respective geometries. —Quondum 02:14, 30 March 2014 (UTC)

It seems to me this little section could benefit by expansion, but for now I'm just noting this missing definition here while I look at sources. Rotation_(mathematics)#In_non-Euclidean_geometries Tom Ruen (talk) 03:18, 30 March 2014 (UTC)

Where to find alternate mathematical forms to specify a rotation?

I came here looking for an explanation to why mathematicians prefer to specify rotations RxR^-1 rather than Tx. R and T are matrices, x is a vector. I looked around at some related pages: Linear Transformation and rotation matrix in particular. I vaguely remember the problem having to do with generalizing to arbitrary dimension, and involving the non-commutativity of matrix multiplication. Dajonesy (talk) 23:05, 23 December 2014 (UTC)

Duplication

Please see Talk:2D computer graphics#Duplication. fgnievinski (talk) 17:56, 16 May 2022 (UTC)