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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.
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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,
1492:
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
441:
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,
299:
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
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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
565:
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
456:
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
514:
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.
479:
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.
249:
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.
1493:"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. —
327:), 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
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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
1353:
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.
518:
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.
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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.
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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.
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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.
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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.
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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
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I came here looking for an explanation to why mathematicians prefer to specify rotations RxR rather than Tx. R and T are matrices, x is a vector. I looked around at some related pages:
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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:
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741:(the former). Of course, an O group is a factor of affine transformations by a normal subgroup of translations, but… the article says nothing about the difference.
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in particular. I vaguely remember the problem having to do with generalizing to arbitrary dimension, and involving the non-commutativity of matrix multiplication.
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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:
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Therefore I propose to move the article where it belongs, to "Rotation (mathematics)", and redirect "Coordinate rotation" (and links here) to there. --
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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.
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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
287:). Rotations in 2D or 3D are merely two intersecting reflections, which clearly produces a direct isometry with a fixed point at the intersection.
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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.
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The section that most needs some attention now is generalisations, though the 4D section could be expanded a little. --
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didn’t know it earlier. This is an example of a non-trivial fact that an encyclopedic article should communicate.
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Rotations of a vector space shall conserve some things, necessary to distinguish rotations from other linear maps.
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contradicts itself, since 3<3 is false. ¿Anybody cleaning up? Kjetil
Halvorsen 15:40, 23 November 2009 (UTC)
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Where are you headed with this? This article at present barely mentions rotations in hyperbolic geometries. —
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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
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I'm looking for what are the most common terms, and adding as a new section here, if that's best.
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where sometimes it's easier to move the co-ordinates around than to move the camera or the items.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Do you see how many interesting things one can express without a silly enumeration of various
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It is wrong to name the article "coordinate rotation", since a mathematical rotation — of
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As I see it, only in 2D a rotation can be said to be a transformation around a point.
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according as the mirrors intersect, are parallel, or have a common perpendicular."
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and I think I know what needs doing so may tackle it soon if no-one else does
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asked in a hope someone would suggest what the article should say about it.
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Without translations (in the case of vector, and others), rotations form a
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these lines is mapped onto itself. This kind of isometry is called a
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By the way, there is a discussion about the “rotation group” term at
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It’s rather typical phenomenon in
Knowledge. What shall we say about
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Where to find alternate mathematical forms to specify a rotation?
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In fact, there is a beautiful way to approach geometry using
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Metric spaces are the most important, but not the only case.
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Done, or at least I've fixed that particular paragraph.
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I agree with you. Take it out and see who complains!--
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For real vector spaces appropriate rotation groups are
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I don't know what you're proposing. This article is in
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only briefly, and does not say a single word about the
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1508:Rotation_(mathematics)#In_non-Euclidean_geometries
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829:and is defined in various ways according to the
416:In 4D shouldn't it be around an hyper-plane????
384:. Any improvements to this article are welcome.
1604:Knowledge level-5 vital articles in Mathematics
995:? Search sources for authoritative definitions.
278:Fundamentals of Mathematics, Volume II:Geometry
1024:in various rotation groups. Non-trivial in SO(
224:This page has archives. Sections older than
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934:Also some specific facts shall be listed:
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261:. Discussions can be found in Coxeter's
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1384:One verbal source say this is called a
442:or even an entire plane or hyperplane.
234:when more than 10 sections are present.
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1388:, which currently has one red link at
861:In an affine space, rotations are not
707:Vector space vs affine space rotations
413:In 3D shouldn't it be around an axis?
1609:C-Class vital articles in Mathematics
1558:Talk:2D computer graphics#Duplication
1369:"The product of two reflections is a
911:Rotations can define some classes of
18:Talk:Rotation operator (vector space)
7:
1228:article that this thing is called a
959:are usually excluded from rotations.
117:This article is within the scope of
711:Look at article's very beginning:
684:{\displaystyle {\frac {n(n-1)}{2}}}
419:Can anyone put some light on this?
317:history of "Rotation (mathematics)"
60:It is of interest to the following
1619:High-priority mathematics articles
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1342:Coxeter group fundamental domains
1194:the question of general-position
825:Rotation is a general concept of
731:for each rotation exists a point…
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137:Knowledge:WikiProject Mathematics
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1296:’ll redirect the title here.
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901:rotations form a subgroup of
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556:18:02, 27 December 2009 (UTC)
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263:Introduction to Geometry, 2/e
131:and see a list of open tasks.
1614:C-Class mathematics articles
1306:18:36, 7 February 2014 (UTC)
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1267:, including 1. Interesting…
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1047:) and infinite-dimensional (
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782:essential thing it describes
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452:16:20, 13 January 2008 (UTC)
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855:is not a “rotation group”).
640:{\displaystyle n \choose 2}
352:before I did the redirect.
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1039:Generalizations: complex (
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605:The degrees of freedom in
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1390:SL2(R)#Parabolic_elements
1292:. If nobody will object,
1232:. IMHO the fact that for
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938:The rotation group of a
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163:project's priority scale
1311:Hyperbolic pararotation
1000:pseudo-Euclidean spaces
892:infinitesimal rotations
120:WikiProject Mathematics
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1002:. Isomorphism between
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350:rotation (mathematics)
346:rotation (mathematics)
231:Lowercase sigmabot III
1531:Linear Transformation
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1290:talk: Rotation group
1014:They are not compact
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917:rotational symmetry
885:identity components
760:Enumerative disease
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321:Coordinate rotation
257:, which is to say,
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