# Talk:Rotation formalisms in three dimensions

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Field: Geometry

## Fleshing it out

I am now (slowly) adding the meat to the different headings, but this article is actually revolving (excuse the pun) about rotation representations, not kinematics as such. There is currently no proper comparative page on rotation representations (only on selected single representations), so I suggest that the title might be moved to something similar? Fmalan 20:56, 12 November 2006 (UTC)

Hi, Fmalan, I really appreciate your work here! I was overly ambitious, thinking, "oh, I'll just dash off a note or two about kinematics while I shepherd a few articles through FA" — silly! :( I do intend to add stuff about the rotational velocity and acceleration, though, so perhaps we should fork the article or create a redirect from Representation of orientation (mathematics) or some similar title? Good work! Willow 10:52, 13 November 2006 (UTC)
Sure thing! Thank you for the positive feedback, Willow! This is the first article I am writing (almost) from the ground up. Only a few (very) minor edits to other articles before this, so I am still feeling my way, trying to learn the syntax and such. I am all for a fork. I'm not very knowlegeable about angular rates and accelerations, so I'll stick with the rotation representation (name?) for now. I also have a nifty list of conversion formulas between the representations in this article lined up. Maybe under something like conversion between rotation representations?

It is true that the term 'representation's is annoyingly overloaded in math and physics. But I just took a look through a few classical books (Appel, Goldstein, Lanczos) on rigid-body mechanics, and noticed to my chagrin that they did not use the term 'representation' for the concept you describe: they somehow managed to avoid using any word for the concept at all! Instead, they just talk about different choices for sets of equations to describe rigid body motion, e.g. Euler's equations, Euler's parameters, Euler angles, Rodrigues angles...

But there is one word that could be used for the same fundamental concept in each of these, it mystifies me that no one uses it: coordinatization. For most of these examples implicitly use generalized coordinates in a configuration space, the whole character of the representation is determined by that choice of coordinates.

Likewise, each one of these is a coordinatization in the Lie algebra sense too, a coordinatization of SO(3,R). Yet the reader does not have to know anything about Lie algebras to appreciate the term. So my suggestion is to use 'coordinatization' instead of 'representation'.

64.105.137.254 (talk) 22:34, 15 July 2013 (UTC)

## Done!

OK - I've entered everything that I have direct knowledge on... Please feel free to improve from here onwards. Fmalan 18:58, 21 November 2006 (UTC)

Hi Fmalan,
I just logged on and found your "Done!" message — congratulations! :D
The article looks very good! :) I'll probably have to wait a few days to really look it over, though, since I'm trying to get Laplace-Runge-Lenz vector ready for FAC and the objection to Cyclol is still hanging in the air...One thing that strikes me, though, is the plural "representations" — could we change it to "representation"? I'm still a relative newbie, but my impression is that Wikipedia article titles tend to be singular. If you decide that you want to, you can change the title using the "move" button at the top. Many kudos and talk to you soon, Willow 19:34, 21 November 2006 (UTC)

Thanks again for the compliment, Willow. I've got no problem for changes to better adhere to Wikipedia conventions. I created a few redirects to my page. Should I go and manually edit those if I move the page's title (to prevent double redirecting), or will these be automatically changed by Wikipedia's rules? FMalan 08:56, 23 November 2006 (UTC)

You'll have to change them yourself, but only for efficiency. Wikipedia leaves the old page as a REDIRECT to the new page, so clicking on the old link takes you to the right place, cf. Hamilton-Jacobi equations and Hamilton-Jacobi equation. Hoping this helps, :) Willow 12:44, 24 November 2006 (UTC)

## DCM: degrees of freedom vs. rank

Great article. I found the wording of this part a bit confusing: "As Euler's rotation theorem dictates, the DCM has only three degrees of freedom (rank of three), and is a real orthonormal matrix." It seems to imply that the rank is equivalent to or determined by the degrees of freedom, which I don't believe it is. (A 3-DOF 3x3 matrix could also have rank 1 or 2.) The bits about rank and orthonormality, and about linear independence in the next sentence, should probably be moved into the list of DCM properties just below (orthonormality and linear independence are already included in that list). I could be wrong about this though, so I wanted to get some feedback. Hansee 23:56, 28 February 2007 (UTC)

Thanks, Hansee! I agree with your statements. My sentence is rather confusing. A matrix with three degrees of freedom can also be rank 1 or 2, but in this case we are dealing with a rank 3 matrix. It is therefore an "and" rather than an "implies". Feel free to edit, if I don't get around to doing it soon. It is only when one tries to explain a concept to other people that one really understands it well, in my experience. FMalan 09:36, 1 March 2007 (UTC)

## Euler angles - possible confusion

My section on Euler angles have since been edited by a contributor (with paragraphs and matching illustrations removed). I am a bit unhappy about this, since I feel that it illuminated a point of possible confusion. The editor seems to have updated some equations, but not all.

The definition of what I meant with "x-convention" is now no longer there, and care must be taken to ensure that the equations involving the Euler angles are correct. With "x-convention" I referred to non-rotating (intertially fixed) axes of rotation. See the example at Uni Stuttgart.

Wolfram Mathworld seems to decribe the same definition of the x-convention, but then illustrates the example by using co-moving axes. This I find a bit confusing. Please take care! FMalan 09:54, 1 March 2007 (UTC)

## Rodrigues Combined Rotations

When you use the Rodrigues parameterisation (ie. vector with same direction as axis and length of tan of half the angle of rotation), combined rotations have a nice formula:

    w1 + w2 - w1xw2
w = ---------------
1 - w1.w2


Surely, the Rodrigues parameterisation's main use is for this formula, and so if it deserves a section at all then this should be in it. I'm only just learning 3D rotations anyway, but thought this was an important point to make. Hope to see it in a Wikipedia page.... —The preceding unsigned comment was added by 80.43.120.247 (talk) 00:59, 7 May 2007 (UTC).

## Other conversion article

I don't have time to do it now, but there should probably be some merging/linking between this article and Conversion between quaternions and Euler angles. —BryanD 17:06, 1 October 2007 (UTC)

## Connections to the rest of the math body, and expert advice

I would seriously advocate splitting up, merging and linking this article to the rest of the math knowledge on Wikipedia. Representation as a term is here badly overloaded with group representations of SO(3), the ordinary rotation group in 3D Euclidean space. Most of the "representations" here are instead "parametrizations" of ordinary rotations, or "immersions" of the rotation group. Thank fully "atlas" is linked here, which remedies some of the confusion, but IMO, this article still needs serious attention. Decoy 01:35, 10 November 2007 (UTC)

I was having much the same problem with the word "representation", so I've renamed the article to Rotation formalisms in three dimensions. I did think about Rotation parametrizations in three dimensions; but this article is so much about the methematical machinery, not just the variables, that I thought "formalism" caught that better. But I'm sure it could be improved upon still, if anyone has any good ideas. Jheald (talk) 18:37, 5 February 2012 (UTC)

## UVW axes as rows or columns of the DCM

In an engineering contex, as very well put in the beginning of this artice, the DCM defines an object- or camera-frame, or else the orientation of a rigid body in space, as an orthogonal right-handed triad of unit vectors (u, v and w), in terms of the refernce coordinate frame xyz. Of course in a more abstract mathematical context, the DCM is simply a rotation matrix. Indeed very intuitive definition, which could be topped up by some explanation about the ordering of the DCM elements.

There are two possible ways to layout the DCM. One has the UVW axes as columns:

( ux vx wx )
( uy vy wy )
( uz vz wz )


(this is probably the convention used by most textbooks) and the other has the UVW axes as rows:

( ux uy uz )
( vx vy vz )
( wx wy wz )


To transform vectors from the UVW frame to the XYZ frame we need the first:

       ( ux vx wx )          ( ux vx wx )   ( pu )
Pxyz = ( uy vy wy ) . Puvw = ( uy vy wy ) . ( pv )
( uz vz wz )          ( uz vz wz )   ( pw )


To transform vectors from the XYZ frame to the UVW frame we need the second:

       ( ux uy uz )          ( ux uy uz )   ( px )
Puvw = ( vx vy vz ) . Pxyz = ( vx vy vz ) . ( py )
( wx wy wz )          ( wx wy wz )   ( pz )


Obviously, one DCM is the transpose of the other since:

Pxyz = DCM . Puvw <=> Puvw = inverse(DCM) . Pxyz = transpose(DCM) . Pxyz


--xerm (talk) 12:56, 12 March 2008 (UTC)

## Euler axis angle direction

--88.195.119.122 (talk) —Preceding comment was added at 08:32, 4 April 2008 (UTC)

## Eigenvalue of the rotation matrix

Hi, You said "The angle θ which appears in the eigenvalue expression corresponds to the angle of the Euler axis and angle representation." Isn't it more appropriate to say that θ corresponds to the angle of rotation about the Euler axis, rather than the angle that the Euler axis makes (with respect to some other axis). 139.121.201.196 (talk) 19:45, 6 June 2008 (UTC)Mike Carroll

## DCM → Euler angles

The mentioned arctan function is usally referred to as atan2 (in computer science), maybe link to the article atan2? Also, maybe it should be mentioned that in the ZYZ there is a border case when theta = 0 since it will reduce to a single rotation around Z, and the atan2 result is undefined thus this case must be handled explicitly.

145.18.20.20 (talk) 17:37, 3 February 2009 (UTC)

## conversion of axis-angle to DCM, use atan2?

On page 584 of multiview geometry in computer vision by Hartley and Zisserman, says that the atan2 function should be used rather than the acos/asin because it is more accurate and the acos/asin fails when the angle eq PI. —Preceding unsigned comment added by 68.116.196.249 (talk) 01:44, 25 February 2009 (UTC)

### atan2 is used throughout

Note that ${\displaystyle \arctan }$ used in the article takes two parameters. By this we indicate that ${\displaystyle \arctan 2}$ is indeed used, as it should be. FMalan (talk) 10:44, 12 July 2009 (UTC)

## Geometric Algebra and Spinors

I believe Geometric algebra and spinors also deserve attention in this article. As far as I heard (and I'm still studying to understand) rotations have a very clean (coordinate free) representation in any dimensions using spinors and Geometric algebra. I will come back and add them myself if I ever feel confident enough in the subject, but the article begs for someone to do it. Enisbayramoglu (talk) 08:11, 9 September 2009 (UTC)

## Errors in section Rotation matrix ↔ Euler angles

Following some detailed work in the article Euler angles I corrected signs in the individual matrices and the order in the combined A matrix. Look for details in the discussion of that article.Chessfan (talk) 19:49, 20 May 2010 (UTC)

## Geometric algebra

I inserted a section introducing the geometric algebra tool, in relation with quaternions and Euler rotations. Chessfan (talk) 15:42, 23 May 2010 (UTC)

This section is word salad. And the notation is awful. What's with the ellipses? Someone please fix or delete. —Preceding unsigned comment added by 76.94.117.44 (talk) 01:23, 29 May 2010 (UTC)

Seconded. Completely unencyclopaedic writing, very badly formatted, but mostly I can't imagine anyone reading through what seems like two pages of dense trigonometry working and learning anything. Rotors are just quaternions in 3D and there's already a section on quaternions which covers them well enough, and further information on how they relate to the other representations. It is completely out of place in this article even if all the writing and formatting were fixed, so I've removed it.--JohnBlackburnewordsdeeds 09:02, 29 May 2010 (UTC)

Thats what I expected ! Poor Hestenes ! Good-bye my friends .... Chessfan (talk) 13:49, 29 May 2010 (UTC)

One last word : readers who would like to see what motivated that, partially anonymous, censorship will find my text on User talk:Chessfan/work1. Chessfan (talk) 22:39, 29 May 2010 (UTC) See also http://www.geometricalgebra.net/quaternions.html . Chessfan (talk) 22:17, 30 May 2010 (UTC)

## Rotations as Projective Space

I'd like to enter into the wiki another representation of the rotations -- as a 3-dimensional real projective space (this is a well-known fact, by identifying antipodal points of the quaternion unit sphere). This allows one to use the affine patches of projective space as coordinates for the rotations, as I do in my paper [arXiv:1005.4661v1] (http://arxiv.org/abs/1005.4661v1). Therein lies the rub, that wiki, appropriately, is not a forum for original research. However, I think the research aspect of my paper is not the emphasis, but more that this representation of the rotations has been overlooked by the Computer Science community. In the paper, I give an example that uses this projective representation of the rotations to advantage. So what do you think: Enter this into wiki, now, or wait until it may become more familiar in the Computer Science community? Normvcr (talk) 03:30, 30 June 2010 (UTC)

Sorry, I am not at all an expert in projective geometry and its applications, and I dont feel entitled to express a valid opinion on your question. But in a more general point of view I know that the geometric algebra opens enormous possibilities in projective and conformal geometry, which are already studied by a lot of people. It is the only domain where I am sure that GA will be developped and successfully used in the next decennies. So my advice would be perhaps to pursue your research, but also to carefully inform yourself on that new domain. See the work of Leo Dorst (University of Amsterdam). Chessfan (talk) 17:24, 20 July 2010 (UTC)

## Is your rotation matrix the right one ?

A few months ago I took a look at the article "Euler angles" . I quickly discovered that there were within it a lot of errors or at least misleading ideas. I then wrote a tentative new article in User:Chessfan/work1, with commentaries in User talk:Chessfan/work1. I am not sure that my ideas were accepted, and as there is still a lot of editing to do, which I cannot undertake, I suppose that my article will never be substituted to the existing one, but I dont care ...

When looking for other rotation articles, I noticed - the same causes producing the same effects - that the Wiki situation in that domain seems very confusing. I think that is a consequence of several factors, difficult to correct :

-- the encyclopedical orientation of wikipedia, which seems to exclude textbook-like articles, where the reader could verify the theory and the hypothesis in accordance with the relevant practical examples he studies ;

-- the multiplicity of editors in a same article, and their variability in time ;

-- the non-existing of referees, which is often badly compensated by self-proclaimed "watchers" (I try to be polite), situation which either quickly discourages newcomers or generates edit wars.

Well, let us try to give a helping hand for the rotation articles, beginning with the present one. Take a look at the section "Rotation matrix". There we seemingly define such a matrix. But nobody tells us for what it will be used : shall we describe an "active" rotation, that is moving an object defined by the fact that its final position in the new basis is the same than its initial position in the old basis, or shall we describe a "passive" rotation, that is defining the coordinates in the new basis of an unmoved object as functions of its coordinates in the old basis ?

The Euler axis, "left multiplied by the matrix" seems to indicate an active rotation.And the analytical defintion given for the A matrix is coherent with that. But some lines later the text "note the order, since the vector being rotated is multiplied from the right" is at least ambiguous. Does that mean ${\displaystyle u^{T}A}$ in place of ${\displaystyle Au}$ , and why ?? So we do not even know if the order ${\displaystyle A_{2}A_{1}}$ given is correct, as it depends on the meaning given to the rotation.

Looking now in the same article at the section "Rotation matrix - Euler angles" we should know, but nobody tells us, that the ${\displaystyle A_{X},A_{Y},A_{Z}}$ matrices are coherent with an active rotation, which is also coherent with the given order ${\displaystyle R=A_{X}A_{Y}A_{Z}}$.

Sorry here I made an error ; these matrices represent passive rotations, applied extrinsicly in the order Z,Y,X. Chessfan (talk) 10:50, 26 February 2012 (UTC)

We should of course know, but nobody tells us, that the R matrix shall be multipied from the right by a column matrix representing the vector to be rotated. We should also know why the given order of matrices is different when doing Euler rotations or simply composing rotations in a unique basis. More, it can be very misleading not to mention explicitly that the simple ${\displaystyle A_{X},A_{Y},A_{Z}}$ matrices are initially supposed each written in a different basis. That fact and many others would be obvious if instead of working directly with matrices, we would use tensor algebra, switching if necessary in the end to the equivalent matrix algebra. But even nowadays students seem to learn that very late in their cursus, as a consequence of too theoretically oriented programs ...

Perhaps some anonymus editor will again disdainfully qualify the present text as word salad (see above geometric algebra) ... But at least I have done something to clarify the matter. Readers who would like to have a better understanding of the whole rotation subject will find a detailed analysis of an active rotation (Euler) in User:Chessfan/work1 , and of a passive rotation (Euler) in User talk:Chessfan/work1. Dont hesitate to give me feedback, favorable or not !

Chessfan (talk) 09:14, 24 July 2010 (UTC)

## Direction Cosine Matrix from Vector Observations

A short note that a short discussion linking how to convert vector observations to a DCM would be useful, such as the Triad Method which is commonly used for attitude in multi-antenna GPS and spacecraft dynamics.

Damien d (talk) 00:16, 27 July 2010 (UTC)

I added a new math text in Talk:Triad Method. Chessfan (talk) 19:23, 1 August 2010 (UTC)

## Geometric algebra again.

I suggest that the paragraph concerning Euler angles (with GA) should be suppressed here. It is a difficult subject, as we know, where a lot of ambiguities occur (active, passive, intrinsic, extrinsic, notation conventions, etc ...). A much more detailed corresponding section should be included in the Euler angles article.

I tried to write something like that almost two years ago, but was brutally censored. So I am not willing to try it again, but my work can be found in User talk : Chessfan/work1 . It was formally imperfect, but I think much clearer than the text presently under review.

Some details :

-- γ' and z' should be replaced by γ" and z" ;

-- it is not so obvious that α, β, γ should automatically be attributed to the analog named axes than α, β', γ" ; detailed expressions of all rotors seems necessary, mostly when some axes appear twice ;

-- the incidental fact that Rα and Rγ commute plays no role in the general demonstration.

Perhaps the ambiguities could be dissipated by adopting the following self-explaining names for the rotor :

  R = Z"γ X'β Zα = Zα Xβ Zγ


where the angles, the names of axes, and the indication of which are fixed and moved, appear simultaneously. Another example would be :

  R = Z"γ Y'β Xα = Xα Yβ Zγ


But perhaps that would too much facilitate the task of the layman ... ?

Chessfan (talk) 16:04, 15 January 2012 (UTC)

I also suggest to substitute, or to add to the particular example a more general result which would better illustrate the flexibility of GA for rotations. Let us look at a rotation characterized by :

${\displaystyle (1)\qquad b=R^{\dagger }aR\qquad \qquad R=\cos(\theta /2)+Iu\sin(\theta /2)\qquad \qquad u^{2}=1}$

${\displaystyle (2)\qquad R^{\dagger }aR=R^{\dagger }(a.uu+a\wedge uu)R=a.uuR^{\dagger }R+a\wedge uuR^{2}=a.uu+a\wedge uu(\cos \theta +Iu\sin \theta )}$

where (a.u u) and (a wedge u u) are the projections of (a) on respectively the rotation axis and plane. After some simple algebraic transformations we get the Rodrigues formula :

${\displaystyle (3)\qquad b=a.uu(1-\cos(\theta ))+a\cos(\theta )-a\times u\sin(\theta )}$

which gives the decomposition of (b) in a non-orthonormed frame. If we have to transform it for numeric applications in an orthonormed frame, we have the obvious tensorial formula :

${\displaystyle (4)\qquad b=b^{i}e_{i}=[a^{j}.u_{j}u^{i}(1-\cos(\theta ))+a^{i}\cos(\theta )-a^{k}u^{j}\varepsilon _{kj}^{...i}]e_{i}}$

Chessfan (talk) 16:59, 17 January 2012 (UTC)

## Title

The new title Rotation formalisms in three dimensions is a bit better than Rotation representation (mathematics) but still a bit of a mouthful. How about simply Rotation in three dimensions?

See also discussion at Wikipedia talk:WikiProject Mathematics.-Salix (talk): 19:02, 5 February 2012 (UTC)

That suggests though the mathematical object, i.e. rotation group SO(3) as it now is. Compare e.g. the overlong rotations in 4-dimensional Euclidean space, also known as SO(4). This is about the many different mathematical ways that mathematicians chose to describe rotations in 3D. I think the name is a bit unwieldy but I can't think of anything better.--JohnBlackburnewordsdeeds 19:11, 5 February 2012 (UTC)
Same feeling as JB, slightly clunky but also slightly better than the old one. Rschwieb (talk) 17:53, 6 February 2012 (UTC)
I concur with Salix, not sure I agree with JohnBlackburne. Surely "formalism" adequately suggests choice of a representation of an object, not merely object? But no, Rotation in three dimensions does not work for me. — Quondum 15:50, 8 February 2012 (UTC)
I did think about Rotation in three dimensions. Quite probably, we ought to have an article with that name, as category leader for Category:Rotation in three dimensions. But it seemed to me that, on the basis of the text we have, this is not that article -- this article seems to me to be (usefully) much more about the details of the machinery for expressing such rotations, rather than a general category-leader on the subject itself.
Incidentally, looking through the articles in the category, there would seem to be a lot of overlap and duplication that could use some structure and sorting-out. This article, too, could use some attention; for starters, it currently falls well short of WP:LEAD. Jheald (talk) 20:35, 8 February 2012 (UTC)
BTW, look to Category: Rotational symmetry, which underwent splitting in mid-2011. There are still several articles to re-categorize. Incnis Mrsi (talk) 21:34, 8 February 2012 (UTC)

## Establishing coherence in rotation articles.

I begin feeling like Frankenstein's monster roaming through the fields of rotation articles. Wherever I look I find errors or at least false interpretations, ambiguities, lack of understanding between different professional users. Isn't the idea of establishing coherence between these articles an almost impossible task ? With that question in mind I wrote the section "A general method for interpreting simple and composed rotation matrices" in http://en.wikipedia.org/wiki/Talk:Euler_angles. You might perhaps read it.

Now take a look here at the subsection "Rotation matrix ↔ Euler angles".

First the editor gives us an example ${\displaystyle (\phi ,\theta ,\psi )}$ which is unrelated to the composed matrix given later.

Second we are not told that the detailed matrices correspond to passive rotations.

Third we are not told that ${\displaystyle A_{Z}A_{Y}A_{X}}$ corresponds to the extrinsic interpretation of the global matrix. That ambiguity is reflected in the "Euler rotations" section. What can we understand under "the most external matrix rotates the other two ..." ? Matrices are not intrinsic mathematical objects which can be actively rotated ; they take their signification when attached to a reference system. That is why we must write to fully express intrinsic rotations transformed in extrinsic ones :

${\displaystyle \qquad (R)_{e}=(Z''_{\gamma })_{e}(Y'_{\beta })_{e}(X_{\alpha })_{e}=(X_{\alpha })_{g}(Y_{\beta })_{f}(Z_{\gamma })_{e}=(X_{\alpha })(Y_{\beta })(Z_{\gamma })}$

where Z" and Y' are no more elemental matrices.

Well, I think I have done my job in the Euler angles main article, and I have no more time for that. Who will undertake the coordination task ?

Chessfan (talk) 23:21, 26 February 2012 (UTC)

## Formula to compute Euler angles from matrix uses two different conventions

Rotation formalisms in three dimensions#Conversion_formulae_between_formalisms describes how to compute Euler angles of a rotation from the rotation matrix. The formulas in that section use different conventions for the Euler angles within that same section. Indeed, the article mentions rotating around zxz for the first formula, and rotating around xyz for the second formula. Indeed, you can see the two formulas can't be consistent for the first formula claims that θ = arccos(A33), whereas the last one claims A33 = cos(φ)cos(θ), and A31 = sin(θ).

Could you figure out the correct formulas for a single convention and fix the article? Thanks in advance.

b_jonas 22:47, 11 July 2012 (UTC)

You are right. I already twice draw attention to the fact that this article is partially false , but without success . It should be coordinated with the Euler angles article. That is a lot of work and I will not begin it without support from other editors involved. Chessfan (talk) 18:41, 16 July 2012 (UTC)

Sorry I made an édit error with your title Chessfan (talk) 18:52, 16 July 2012 (UTC)

See http://www.soi.city.ac.uk/~sbbh653/publications/euler.pdf Chessfan (talk) 08:05, 18 July 2012 (UTC)

Have to second this post, since

{\displaystyle {\begin{aligned}\phi &=\operatorname {arctan2} (A_{31},A_{32})\\\theta &=\arccos(A_{33})\\\psi &=-\operatorname {arctan2} (A_{13},A_{23})\end{aligned}}}

and

${\displaystyle {\begin{array}{lcl}\mathbf {A} &=&{\begin{bmatrix}\cos \theta \cos \psi &\cos \phi \sin \psi +\sin \phi \sin \theta \cos \psi &\sin \phi \sin \psi -\cos \phi \sin \theta \cos \psi \\-\cos \theta \sin \psi &\cos \phi \cos \psi -\sin \phi \sin \theta \sin \psi &\sin \phi \cos \psi +\cos \phi \sin \theta \sin \psi \\\sin \theta &-\sin \phi \cos \theta &\cos \phi \cos \theta \\\end{bmatrix}}\end{array}}}$

does not fit together. 212.222.53.78 (talk) 15:45, 10 October 2013 (UTC)

Again, I think the article I mentioned above is OK .Chessfan (talk) 23:47, 25 January 2014 (UTC)

https://wikimedia.org/api/rest_v1/media/math/render/svg/d8f184f89aa7fe5cdbeb6f368c9b7c859abf4802

The matrix above is for left-hand system, not right-hand. — Preceding unsigned comment added by 82.33.216.70 (talk) 11:05, 7 February 2017 (UTC)