Talk:Euler angles/Archive 2

Back to the table of matrices
I have checked the 12 matrix products with a symbolic math package (maxima) and it is correct. If nobody disagrees, I would put it back into the article--Guentherwagner (talk) 07:55, 27 June 2010 (UTC)


 * Working with a math package is fine, but there remains always the problem to be sure having written the right matrices in the right order. If your idea is to calculate a passive rotation with Euler angles in the alphabetic order, then I think you have made a sign error in the elementary matrices (the false signs are in the terms containing an odd number of sinus). With the z,x,z convention I find your result (matrix table) with the sign switched in (1,2) (2,1) (2,3) (3,2), which paradoxically is the result of the original article (section matrix notation) ! I will try to add a detailed passive demonstration in my work talk page. You cannot escape a complete rewriting of the article, but not by me ... :) Chessfan (talk) 08:14, 3 July 2010 (UTC)


 * I agree! This table describes the body rotations (active if you like as the object moves in the frame) in the fixed axis system (aka world axis system, xyz system, or S (static) system).  Not only that, but it describes how to convert to the rotated axis system (aka body axis system, XYZ system, xy'z" system, or R (rotated) system).  With the description, the table is 100% accurate and a very useful reference.  If passive rotations are desired, a single line stating that these should be transposed can be added, also pointing to [].  The whole article is decreasing in quality over time due to editors ignorant of the fact that mathematicians, physicists and engineers all use Euler angles, but all in a slightly different way.  If the article doesn't take this into account - i.e the concepts of how spacecraft move, how an ellipsoid molecule rotates by Brownian diffusion in water, and how to change abstract basis sets, it fails!  --True bugman (talk) 15:31, 6 August 2010 (UTC)


 * In the "euler angles" text (section "matrix notation") as it exists today the global matrix $$R$$ is right if we interprete it as a passive matrix : the result coincides with the one I caculated in User talk:Chessfan/work1. But be careful, we have $$R=C^T B^T A^T$$, and $$A^T$$ is the transpose of the elementary passive A matrix. That is not clear in the text.


 * By Wikipedia's own definition [], $$R$$ is an active matrix. Note that the second matrix in:



\mathbf{R}= \begin{bmatrix} \cos \gamma & \sin \gamma & 0 \\ -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \beta & \sin \beta \\ 0 & -\sin \beta & \cos \beta \\ \end{bmatrix} \begin{bmatrix} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} $$


 * is incorrect and this is an obvious mistake caused by copying http://mathworld.wolfram.com/EulerAngles.html. The transpose of the elementary passive A matrix is the active matrix $$A_a$$ so that $$R=C^T \cdot B^T \cdot A^T = C_a \cdot B_a \cdot A_a$$.  This matches the definition of the table.  --True bugman (talk) 16:01, 9 August 2010 (UTC)


 * The matrices in the "table of matrices" are false. They coincide neither with active nor with passive transformations. That deep rooted error comes probably from the fact that the order $$R(y,\theta_3),R(z,\theta_2),R(x,\theta_1) $$ is false. By the way I recommend to stay with $$\alpha, \beta,\gamma)$$ rather than $$\theta_1, \theta_2, \theta_3 $$ which introduces ambiguity about the order of the rotations. For the $$zxz$$ convention the right matrix is in User:Chessfan/work1


 * These matrices are active in all sense of the word. These are the matrices used universally by physicists, engineers, and even software engineers working on 3D graphics.  A perfect reference is the algorithm of Ken Shoemake in "Euler Angle Conversion. Graphics Gems IV. Paul Heckbert (ed.). Academic Press, 1994, ISBN: 0123361567. pp. 222-229." (http://www.graphicsgems.org/).  This algorithm generates all of the Euler angles conventions listed in this table from an arbitrary rotation matrix R - it assumes the matrices of this table.  To say this table is false is false!  --True bugman (talk) 16:01, 9 August 2010 (UTC)


 * Here is a more convenient link for the Graphical Gems IV book http://books.google.com/books?id=CCqzMm_-WucC&lpg=PA222&dq=euler%20angle%20rotation%20matrix%20xyz%20xzy%20zyz%20zxy%20zyx&pg=PA223#v=onepage&q&f=false. This gives 2 examples which can be compared to the table - xyz and xyx.  Both demonstrate that either the wikipedia table is correct or these books on Euler angles are incorrect.  --True bugman (talk) 17:12, 9 August 2010 (UTC)


 * Here is another reference demonstrating that at least one of the matrices are correct. See Table 1.1 of "Springer Handbook of Robotics" by Bruno Siciliano, Oussama Khatib http://books.google.com/books?id=Xpgi5gSuBxsC&lpg=PA12&ots=lSohU9f39N&dq=euler%20angle%20rotation%20matrices%20zyz%20zxy%20zyx&pg=PA11#v=onepage&q&f=false.  Specifically the second entry of the XYZ fixed angle convention, and compare this to xyz in the old wikipedia table.  --True bugman (talk) 16:40, 9 August 2010 (UTC)


 * Chessfan (talk) 12:47, 9 August 2010 (UTC)


 * What you say about different users of Euler angles is certainly true. But that fact, as it seems to me, speaks precisely in favor of less encyclopaedical writing and insisting more on methods. With precise definitions the users can easily reconstruct the adequate matrix. Chessfan (talk) 13:57, 9 August 2010 (UTC)


 * Certain readers, such as those not having a university degree, may not be able to perform elementary operations, such as dot products, hence giving these matrices is quite useful as a self-contained comprehensive reference on Euler angles. --True bugman (talk) 16:01, 9 August 2010 (UTC)


 * I am very happy to discuss that subject in detail with you. But I am sorry still not to agree with you on the rightness of the matrix table. I think there is between us a difference of notation regarding the $$A^T$$ matrices, but as that is a bit confusing I would suggest you an other very simple method : could you be kind enough to check carefully the active Euler rotations ($$zxz$$) convention I studied in detail in User:Chessfan/work1. There the result does not coincide with the equivalent one in the matrix table. I did the calculation very carefully, and was comforted by the fact that I found in the passive case the exact transposed matrix (see User talk:Chessfan/work1). More : that last passive matrix coincides with the matrix given by the original text in the section "matrix notation". Chessfan (talk) 16:55, 9 August 2010 (UTC)


 * Do you mean in the subsection "The global rotation matrix"? This matrix is correct if you assume the rotated axis rotations.  In this axis system, the xyz rotation matrix notation with the first rotation of $$\alpha$$, the second of $$\beta$$, and the third of $$\gamma$$, the rotation matrix defined using Euler angles is $$R = R_x(\alpha) \cdot R_y(\beta) \cdot R_z(\gamma)$$.  In the fixed axis system - the system used in the deleted table - the rotation matrix is $$R = R_x(\gamma) \cdot R_y(\beta) \cdot R_z(\alpha)$$.  Note that engineers often use the ZXY rotated axis active rotations, physicists often use the zyz fixed axis active rotations, while mathematicians often use the ZXZ rotated axis passive rotations (I know there are many exceptions, but this is most often the case).  --True bugman (talk) 17:31, 9 August 2010 (UTC)


 * Sorry, there was a typo there. The fixed axis rotation is $$R = R_z(\gamma) \cdot R_y(\beta) \cdot R_x(\alpha)$$.  --True bugman (talk) 17:49, 9 August 2010 (UTC)


 * I calculated with my method and with the Springer handbook the case (zyx), assuming of course that we speak about Euler rotations, and found a combined matrix identic with Springer, but different of the matrix table you want to restore. Does that convince you or is there again an interpretation problem ? Chessfan (talk) 19:23, 9 August 2010 (UTC) It seems to me that you do not take in consideration the fact that any Euler rotation can have two interpretations, one intrinsic the other extrinsic; look at the matrix transformations I expose in the zxz convention. Chessfan (talk) 19:47, 9 August 2010 (UTC)


 * Here is my derivation. For the zyx rotation in the fixed axis system, with the first rotation of $$\alpha$$, the second of $$\beta$$, and the third of $$\gamma$$, the rotation is defined as


 * $$R = R_x(\gamma) \cdot R_y(\beta) \cdot R_z(\alpha)$$.


 * Using the definition of right handed rotations of fixed axes given at Rotation_matrix, the matrix form is



\mathbf{R} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \gamma & -\sin \gamma \\ 0 & \sin \gamma & \cos \gamma \end{bmatrix} \cdot  \begin{bmatrix} \cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ -\sin \beta & 0 & \cos \beta \\ \end{bmatrix} \cdot  \begin{bmatrix} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}, $$


 * and after performing the matrix multiplication,



\mathbf{R} = \begin{bmatrix} \cos \alpha \cos \beta & -\sin \alpha \cos \beta  & \sin \beta \\ \cos \alpha \sin \beta \sin \gamma +\sin \alpha \cos \gamma & \cos \alpha \cos \gamma -\sin \alpha \sin \beta \sin \gamma  & -\cos \beta \sin \gamma \\ \sin \alpha \sin \gamma -\cos \alpha \sin \beta \cos \gamma & \cos \alpha \sin \gamma +\sin \alpha \sin \beta \cos \gamma  & \cos \beta \cos \gamma \end{bmatrix}. $$


 * If we replace $$\{\alpha, \beta, \gamma\}$$ with $$\{1, 2, 3\}$$, the matrix then matches that of the deleted table. Hence that matrix is correct.  --True bugman (talk) 20:31, 9 August 2010 (UTC)

Well, that starts being hard work ! Of course your matrix calculation is right, and I suppose you have the right to call it $$(z,y,x)$$. But was not the whole structure of the article very confusing, as it was let's say before I introduced some additional perturbation ...

As one of the editors quickly removed the matrix table and as I was not in encyclopaedical mood, I did no more look at it. So I completely missed, before your intervention, the fact that those matrices where supposed to be interpreted in the extrinsic way.

But that seems to me rather strange, and confusing for readers seeking for help. Having introduced in the "normal" intrinsic interpretation the Euler angles, having -- rather badly -- shown the equivalence with an extrinsic rotation combination, having calculated a global matrix in the intrinsic convention, why suddenly switch to a matrix table established in extrinsic conventions ? Does that even deserve the name "Euler angles" ?

Let us be more precise. I rewrite your matrices with the notations I used in my work. What you call the fixed axis rotations will be noted :

$$(1) \qquad R= (C)_e (B)_f (A)_g$$

where $$(C)_e, (B)_f ,(A)_g$$ will be respectively the simple matrices you associated respectively with $$(\gamma , \beta, \alpha)$$ angles. I redemonstrated a well known fact, which is that :

$$(2) \qquad R= (C)_e (B)_f (A)_g=(A)_e (B)_e (C)_e$$

where $$(A)_e, (B)_e$$ are no more simple matrices, but need not to be evaluated.

The last member of (2) shows that $$R$$ represents the global Euler rotation in the convention $$(X,Y,Z)$$ ( or better $$(x, Y', Z'')$$with the successive angles $$(\gamma, \beta, \alpha)$$ in that order.

So I suggest reintroduce the matrix table if you want but with intrinsic conventions, what i would call the true Euler angles !

Now I will take a break, I am exhausted !

Chessfan (talk) 06:59, 10 August 2010 (UTC)

Indeed the Springer handbook makes the distinction between "Euler angles" and "fixed angles".Chessfan (talk) 18:17, 10 August 2010 (UTC)


 * Yes, the authors there have taken the liberty to call the 2 axis systems by different names. Some will even split this up into 4 categories with different names (e.g. http://www.servinghistory.com/topics/Euler_angles::sub::Conventions).  Nevertheless they are all Euler angles, and there are no Wikipedia articles for these other non-standard naming and categorisation conventions.  Here is a reference for all 24 Euler angles http://www.cgafaq.info/wiki/Euler_angles.  12 of these are in the static axis system (also known as fixed axis, world axis, S, or xyz).  The other 12 are in the rotated axis system (also known as body axis, XYZ, xy'z", or R).  This article also describes all 24 as Euler angles http://www.euclideanspace.com/maths/geometry/rotations/euler/index.htm.  See the sections "Relative to rotating object or absolute coordinates" and "Global Frame of Reference".  --True bugman (talk) 08:32, 11 August 2010 (UTC)

I have no more time now ; I will take that subject up again in two or three weeks. My head spins ... ! Chessfan (talk) 18:42, 11 August 2010 (UTC)

A last try on table of matrices
I will try to explain where I stand now. As I am not at all a specialist in Euler angles, it may be that my conventions do not coïncide with those used by professionnals. But as I feel lost in the jungle of conventions and half explained situations, I must give myself some strict rules.

1/ First I will study active rotations, that is a situation where we determine the new coordinates in the initial reference frame of an object which is rotated (if useful we rotate also a frame to which the object can be referred by fixed coordinates). In operator and matrix notation we write :

$$(1) \qquad \qquad y=R * x \qquad \qquad (y)_e=(R)_e (x)_e$$

where $$(y)_e$$ and $$(x)_e$$ are one-column matrices and all elements are supposed written in the e-frame. When studying Euler angles I will use the successive frames and corresponding vectors:

(e)→(f)→(g)→(h) u → v → w → t

2/ In composing three Euler rotations around $$X$$, then $$Y'$$ and finally $$Z''$$, I must write in operator notation :

$$(2) \qquad \qquad t=Z''*Y'*X*u$$

which I doubtlessly will name the $$(XYZ)$$ convention, and which in matrix algebra gives :

$$(3) \qquad \qquad (t)_e=(X)_e (Y)_f (Z)_g(u)_e=(R)_e(u)_e$$

where the subscripts show in which frame the simple matrices $$X,Y,Z$$ are supposed to work ! The product of these three matrices, in that order gives the global rotation matrix which is of course supposed to work in the e-frame. How could a layman understand what he does when picking a matrix in a table if that construction is not thoroughly explained ? Indeed as we already know User:Chessfan/work1 we can also write the same rotation matrix $$(R)_e$$ as follows :

$$(4) \qquad \qquad (R)_e=(X)_e (Y)_f (Z)_g=(Z)_e (Y)_e (X)_e$$

where we have transformed the simple $$Y,Z$$matrices in their equivalent ones supposed to operate in the e-frame. That explains the change of order between the operator notation and the matrix algebra.

3/ The same$$(R)_e$$ matrix as above can be interpreted as the composition of three rotations around :

-- first the $$z$$ axis [matrix $$(Z)_g$$],

-- second the $$y$$ axis [matrix $$(Y)_f$$],

-- third the $$x$$ axis [matrix $$(X)_e$$].

In operator notation that would be written $$x*y*z$$, so I would like to name it $$(z,y,x)$$. Does anybody disagree ?

As you see we get 12 different matrices ...

4/ But, you have perhaps noted that I still said nothing about the denomination of the angles. Suppose we have chosen in the first interpretation (intrinsic) the angles $$\alpha, \beta, \gamma$$ respectively for the $$X,Y', Z''$$matrices, then in the second interpretation (extrinsic) if we insist on having still an angle named $$\alpha$$ as the first one, etc ..., then of course we must switch $$\alpha$$ and $$\gamma$$ in the global matrix, which is no more the same in both presentations. Thus we get 24 matrices. That seems rather artificial and in any case needs to be explained to the reader.

5/ A similar analysis can be conducted with passive rotations. Roughly speaking that leads to the transposed matrices of those defined above.

6/As a last word I want to say that using transposed vectors (one-row matrices multiplying rotation matrices from the left), to "explain" what happens, seems to be an awfull idea. It more than doubles the confusion and error traps.

7/To be perfectly clear I recall that the matrices used here have the "active" form :


 * $$\mathbf{X}=

\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \alpha & -\sin \alpha \\ 0 & \sin \alpha & \cos \alpha \\ \end{bmatrix}$$

Chessfan (talk) 18:48, 23 August 2010 (UTC)


 * Hi Chessfan.


 * I think the table was right using a) the right-handed rule for the angles and b) active composition of intrinsic rotations. It should enough to state this in a paragraph so that anybody is able to convert it to his needs. Would you agree to reintroduce the table with this remark?


 * About active and passive compositions, in physics they are similar concepts, but mathematically speaking they are very different things. An active rotation is an rotation operator in the space and a passive rotation is a change of basis of the whole space. I think this article should focus only in active rotations (as you have done) and maybe issue a warning about the possibility of how to change for passive rotations used in physics.


 * About your (4) everything you said is true, but the article says that the name "XYZ convention" is ambiguous, and that normally tildes (') are used like in X-Y’-Z’’ to remark that intrinsic composition is meant (See section "naming conventions"). I don't think there is a better way to explain it.


 * I totally agree with not using row vectors. The normal convention is to have it in columns and a row does not contribute with anything.

--Guentherwagner (talk) 09:03, 25 August 2010 (UTC)

Hi Guentherwagner.

I am glad to see that our ideas converge. But there are still problems with the matrix table :

1/If you take a close look you will see that they are in fact active rotations in the extrinsic presentation after the angle name switch I mentioned. The detailed example calculated by True bugman (with a later corrected sign error in the $$\alpha$$ matrix) is in the table under the name (zyx). But the (ZY'Z") example which I calculated in my workpage, which should be in the table under the name (zyz) after angle switch, does not fit. So every matrix must be verified . Anyway the presentation should be intrinsic, that is without angle switch.

2/It seems to me still difficult to reinsert the table of matrices without correcting a lot of other errors or ambiguous definitions in the existing text. It would be much easier to insert the table in a completely rewritten article ; that is what I tried to initiate in my workpage ... I have no "author pride" as we say in French about that article, but I cannot do the work (language, lack of editing skills, nonencyclopaedical writing, etc ...)

What do you think ?

Chessfan (talk) 21:07, 28 August 2010 (UTC)

Sorry, it is so easy to make mistakes here. The matrix I calculated was of course (ZX'Z"), which is in the table under the right extrinsic name (zxz). Thus the table can be put easily in intrinsic form by switching back the first and last angles. Chessfan (talk) 07:05, 29 August 2010 (UTC) Of course by angle switch I mean angle name switch ... Chessfan (talk) 09:49, 29 August 2010 (UTC)


 * Hi again
 * Of course, I find easier to change the names of the entries that to change the whole table. Anyway, as you wrote before, to compose intrinsic rotations about the axes X, Y’ and Z’’ (in this order) you have to left-multiply by its matrices. After that you will get Z.Y.X (this is a product of matrices). Just the opposite order of the rotations.
 * I think this is why the name of the entries are the opposite order that what you expected. This is a new ambiguity that I didn't notice before, but again, it can be just explained in the text.
 * About:
 * It seems to me still difficult to reinsert the table of matrices without correcting a lot of other errors or ambiguous definitions in the existing text.
 * which errors and ambiguities you refer to?
 * --Guentherwagner (talk) 22:29, 29 August 2010 (UTC)

You push me in my last trenches !

1/ First we must speak again about a serious potential misunderstanding. (What bothers me is your text : "After that you will get Z.Y.X ..." ; do you write here in operator notation, or ... ?). Please could you answer the following questions :  Do you agree with the equations (2), (3), (4) I wrote in my contribution above ? If yes, do you agree with the fact that it represents an active rotation, provided of course that the elementary matrices are written in active form, which in its intrinsic interpretation should be named (X,Y,Z) or better (X,Y',Z")?  If yes, do you agree with the fact that after switching the names of the first and last angles in the above R matrix, we obtain an R' matrix, which represents the same rotation in an extrinsic interpretation and should be named (z,y,x).   If yes do you agree that that matrix coincides with the matrix so named in the existing table of matrices ?


 * Z.Y.X is the standard notation in physics/engineering. Hence it can only be operators!  As for active vs. passive, that discussion should be deferred to http://en.wikipedia.org/wiki/Active_and_passive_transformation and not be brought up in detail in this wikipedia page.  True bugman (talk) 15:15, 6 September 2010 (UTC) Well, I suppose you are right, but the notion of operator notation is not in the existing article ! That also needs to be thoroughly explained to the reader. Chessfan (talk) 22:01, 6 September 2010 (UTC) I still have second thoughts on that point ; the operator notation is an abstract notion ; should not the convention, as well for intrinsic and extrinsic interpretation, clearly indicate in what order (axis order) the rotations take place. Then it should be (X,Y',Z") in the intrinsic interpretation and (z,y,x) in the extrinsic one. There remains no ambiguity. Chessfan (talk) 07:45, 7 September 2010 (UTC)

I add a general observation : for me a rotation matrix is not clearly defined if you dont tell in what reference frame it is suppposed to operate. Of course you can always multiply any square table of numbers by a column vector in an arbitrary frame ; that precisely explains the simplicity of the demonstration of the fact that the intrinsic interpretation of any Euler angle system and its extrinsic interpretation give rise to the same global litteral (and numerical) matrices, if of course you do not switch the angle names. As mathematicians and physicists generally neglect to index the matrices by the name of the frame, that idea does not emerge. It is much easier to understand what happens with geometric algebra, because the rotation axes appear explicitly at each stage of the operations.


 * Stating the reference frame is information for a specific implementation. This is often determined by context.  In the context of a wikipedia article, stating this is not necessary as the reader can pick their own frame or perspective.  True bugman (talk) 15:15, 6 September 2010 (UTC)

2/ Referring to something you wrote the 28th april, I supppose that now you agree with the fact that passive/active rotations are different notions than intrinsic/extrinsic interpretations ? By the way I think that physicists and mathematicians have the same definitions for passive/active rotations.


 * Active and passive is described at http://en.wikipedia.org/wiki/Active_and_passive_transformation, whereas intrinsic/extrinsic is talking about which axes (fixed/rotated) you rotate about. They are different.  As for the definition, you are correct, they are the same (http://en.wikipedia.org/wiki/Active_and_passive_transformation).  True bugman (talk) 15:15, 6 September 2010 (UTC)

3/ A general observation is the fact that, due to the history of the article, there is a lack of unity which is, in my view, a serious handicap for the understanding of the subject "Euler angles". I tried to remedy to that, but it will be very difficult without restructuring the article. Well, as you asked I will try to enumerate -- again -- the main problems (there are a lot of smaller, but not negligible details).


 * The lack of unity is due to the fact that there are many different conventions used in different fields (note they are all related), but the authors of the certain parts of this article are oblivious to the fact that there are others using Euler angles with different names/conventions/etc. This article should be general and encompass all of these branches of mathematics/science/engineering.  True bugman (talk) 15:15, 6 September 2010 (UTC)

4/ The section "Euler rotations" is badly defined, badly written, and false from line 7 to the end.


 * I disagree, this is incorrect from the very first sentence. This nutation/precession is actually a very specific example used in certain problems in engineering (and some physics problems).  Many other conventions are used though.  It is a bit of an embarrassment to have this here, pretending to describe all Euler rotations!  It is a specific application and should probably be shifted to the applications section as a subsection. True bugman (talk) 15:15, 6 September 2010 (UTC)


 * To have a better understanding of what motivated me here you could perhaps take a look at the section "Euler rotations" inUser talk:Chessfan/work1 Chessfan (talk) 16:25, 8 September 2010 (UTC)

5/Everything written about intrinsic and extrinsic rotations, including "Matrix expression for Euler rotations" should be regrouped and better explained, in the spirit of my rewriting. You will note that it is not at all necessary to introduce the unfamiliar matrix transformation rules (from frame to frame), to demonstrate the fact that each intrinsic constructed matrix can be interpreted in an extrinsic manner. Alas in the existing article that transformation appears twice in an unexplained, unrelated, manner.


 * Intrinsic vs. extrinsic is an abstract concept which can be explained without the use of a single equation. True bugman (talk) 15:15, 6 September 2010 (UTC) I think nevertheless it is much easier to understand, and to prove, with equations ! Chessfan (talk) 21:40, 6 September 2010 (UTC) I feel it very difficult to visualize the fact that the result of three successive rotations can be explained (with same angle values) by the composition of three other rotations. The animations are nice but demonstrate nothing, in mathematical sense. Could you give us a clue ? Perhaps i do not see something very obvious. Chessfan (talk) 08:51, 8 September 2010 (UTC)


 * Though mathematical proofs are essential for a maths textbook, I don't believe mathematical proofs are necessary for encyclopedic entries. The pictures http://en.wikipedia.org/wiki/File:EulerG.png and http://en.wikipedia.org/wiki/File:EulerX.png demonstrate the fixed vs. rotated axis notation (intrinsic/extrinsic using a slightly different language) rotations in a physical sense in 3D Euclidean space - the way most engineers and physicists see this.  How would you pictorially represent this in a more abstract mathematical sense?  Is the conversion from one notation to the other using operator form (not a formal proof), as in the current "Euler angles as composition of extrinsic rotations" subsection sufficient for mathematically describing these conventions?  True bugman (talk) 11:06, 8 September 2010 (UTC)


 * Sorry, if I insist on that point it is because it seems to me to be central for the understanding of Euler angles. First I think you should not be too dogmatic with the idea that we should introduce as less maths as possible in an encyclopaedical article. You will easily find other articles where theoretical maths are well developped. Second your position is a bit in contradiction with the unexplained matrix transformation rules used in different sections : why not simply suppress them ? Third I have the feeling when reading through all remarks made from the beginning that a lot of people had only a vague notion and even none on the fact that a same Euler combined rotation can have two different interpretations. (That is not a critical remark ; I had myself to learn again almost everything on Euler angles ...). Take a look again at the "Springer handbook" ; both types of rotations are mentioned but as far as I could read, nowhere is it mentioned that the angle triplets shoud be the same for a given global rotation ! Chessfan (talk) 12:37, 8 September 2010 (UTC)


 * Ok, more simply how would you describe S (static) vs. R (rotated) axis system, fixed vs. rotated axis system, world vs. body axis system, zyz vs. ZYZ, zyz vs. ZY'Z", intrinsic vs. extrinsic in a formal mathematical sense. This is not covered in http://en.wikipedia.org/wiki/User:Chessfan/work1.  Note these are all the same.  True bugman (talk) 17:01, 8 September 2010 (UTC)


 * I will answer you soon in a new section, for better readability. Chessfan (talk) 21:44, 8 September 2010 (UTC)

6/ Thus the "convention" sections should also be carefully rewritten in the spirit of the reintroduced table of matrices.


 * The table was originally in the "Matrix notation" section, but the concepts in the paragraph accompanying this table could be introduced in the 'convention' section. The intrinsic/extrinsic, active/passive, rotation axis ordering, frame of reference, left vs. right handed rotation, forwards vs. reverse rotation, etc. should all be brought into this section.  Each concept with its plurality of names from different fields should be concisely covered.  True bugman (talk) 15:15, 6 September 2010 (UTC)

7/The section "Matrix notation" is written in passive form. It must be transformed in active form, in coherence with the table of matrices, whose introduction must also be carefully rewritten.


 * Yes, this was always a problem. I agree with Guentherwagner that one notation should be used throughout (active would be easier for a lay reader), and that the 'convention' section can describe how to switch.  True bugman (talk) 15:15, 6 September 2010 (UTC)

8/ I did not verify the section "Geometric derivation". it should be checked in coherence with the revised article.


 * Is there a purpose for this section? True bugman (talk) 15:15, 6 September 2010 (UTC)

9/My corresponding text should be substituted to the section "Applications".


 * I would suggest that applications should cover all the specific differences between physics/engineering/mathematics as examples. I don't see how the rotation of the moon can be "directly measurable from a gimbal mounted in an aircraft"!  True bugman (talk) 15:15, 6 September 2010 (UTC)

Who will do all that work ?

Chessfan (talk) 08:19, 31 August 2010 (UTC)

Please Guentherwagner you should reread in detail the math demonstration in my textUser:Chessfan/work1, section "Combining successive rotations". When reading it for the first time you made an observation on my initial section title which was not justified. That seems in relation with the misunderstanding we still -- perhaps -- have. Chessfan (talk) 16:11, 1 September 2010 (UTC)


 * Hi Chessfan. I just came from my summer holidays and your text is too long and about too much things. I really cannot understand why you don't want to include the matrix tables or which things you consider wrong in the article. Could you please be more concise? Thanks.--Guentherwagner (talk) 12:01, 5 September 2010 (UTC)

Hi Guentherwagner. I hope you enjoyed yourself. Don't misunderstand me ! I am not against including again a table of matrices. I am just not sure, when I read you carefully, that we have reached an agreement on the definition and the calculation of those matrices. I don't think I now can do more than ask you to take a close look on the math I wrote. Perhaps you could rewrite the matrix table as you understand it, with your own introduction. And rewrite also the section "Matrix rotation" (my point 7). Then I will tell you if I agree with your version, or if I propose changes. Chessfan (talk) 12:49, 5 September 2010 (UTC) Having read the last contribution of True bugman, I suggest that you cooperate with him to rewrite now the whole article. Chessfan (talk) 21:45, 6 September 2010 (UTC)


 * OK. It is nice to have reached an agreement. I will reintroduce the table and I promise to rewrite the rotation section as soon as I have time.

--Guentherwagner (talk) 20:48, 8 September 2010 (UTC)

Final reinsertion of the matrix table
If nobody disagrees, I will reintroduce the matrix table with the following introduction:

The possible Euler angles combinations are shown here. The following matrices assume fixed (world) axes and column vectors, with active composition of intrinsic rotations and the right-handed rule for the positive sign of the angles.

Being an active composition of intrinsic rotations, the names of the entries are such that a matrix like that for yzx is constructed as a product of three matrices, Rot(y,θ3) Rot(z,θ2) Rot(x,θ1), but is the result of performing first an Y rotation, and then Z and X in this order.


 * {| class="wikitable" style="background-color:white;font-weight:bold"

!xzx c_2 & - c_1 s_2 & s_1 s_2 \\ c_3 s_2 & c_1 c_2 c_3 - s_1 s_3 & - c_2 c_3 s_1 - c_1 s_3 \\ s_2 s_3 & c_3 s_1 + c_1 c_2 s_3 & c_1 c_3 - c_2 s_1 s_3 \end{bmatrix}$$ !xzy c_2 c_3 & s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 s_2 + c_1 s_3 \\ s_2 & c_1 c_2 & - c_2 s_1 \\ -c_2 s_3 & c_3 s_1 + c_1 s_2 s_3 & c_1 c_3 - s_1 s_2 s_3 \end{bmatrix}$$ !xyx c_2 & s_1 s_2 & c_1 s_2 \\ s_2 s_3 & c_1 c_3 - c_2 s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 \\ -c_3 s_2 & c_2 c_3 s_1 + c_1 s_3 & c_1 c_2 c_3 - s_1 s_3 \end{bmatrix}$$ !xyz c_2 c_3 & c_3 s_1 s_2 - c_1 s_3 & c_1 c_3 s_2 + s_1 s_3 \\ c_2 s_3 & c_1 c_3 + s_1 s_2 s_3 & c_1 s_2 s_3 - c_3 s_1 \\ -s_2 & c_2 s_1 & c_1 c_2 \end{bmatrix}$$ !yxy c_1 c_3 - c_2 s_1 s_3 & s_2 s_3 & c_3 s_1 + c_1 c_2 s_3 \\ s_1 s_2 & c_2 & - c_1 s_2 \\ -c_2 c_3 s_1 - c_1 s_3 & c_3 s_2 & c_1 c_2 c_3 - s_1 s_3 \end{bmatrix}$$ !yxz c_1 c_3 - s_1 s_2 s_3 & - c_2 s_3 & c_3 s_1 + c_1 s_2 s_3 \\ c_3 s_1 s_2 + c_1 s_3 & c_2 c_3 & s_1 s_3 - c_1 c_3 s_2 \\ -c_2 s_1 & s_2 & c_1 c_2 \end{bmatrix}$$ !yzy c_1 c_2 c_3 - s_1 s_3 & - c_3 s_2 & c_2 c_3 s_1 + c_1 s_3 \\ c_1 s_2 & c_2 & s_1 s_2 \\ -c_3 s_1 - c_1 c_2 s_3 & s_2 s_3 & c_1 c_3 - c_2 s_1 s_3 \end{bmatrix}$$ !yzx c_1 c_2 & - s_2 & c_2 s_1 \\ c_1 c_3 s_2 + s_1 s_3 & c_2 c_3 & c_3 s_1 s_2 - c_1 s_3 \\ c_1 s_2 s_3 - c_3 s_1 & c_2 s_3 & c_1 c_3 + s_1 s_2 s_3 \end{bmatrix}$$ !zyz c_1 c_2 c_3 - s_1 s_3 & - c_2 c_3 s_1 - c_1 s_3 & c_3 s_2 \\ c_3 s_1 + c_1 c_2 s_3 & c_1 c_3 - c_2 s_1 s_3 & s_2 s_3 \\ -c_1 s_2 & s_1 s_2 & c_2 \end{bmatrix}$$ !zyx c_1 c_2 & - c_2 s_1 & s_2 \\ c_3 s_1 + c_1 s_2 s_3 & c_1 c_3 - s_1 s_2 s_3 & -c_2 s_3 \\ s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 s_2 + c_1 s_3 & c_2 c_3 \end{bmatrix}$$ !zxz c_1 c_3 - c_2 s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 & s_2 s_3 \\ c_2 c_3 s_1 + c_1 s_3 & c_1 c_2 c_3 - s_1 s_3 & - c_3 s_2 \\ s_1 s_2 & c_1 s_2 & c_2 \end{bmatrix}$$ !zxy c_1 c_3 + s_1 s_2 s_3 & c_1 s_2 s_3 - c_3 s_1 & c_2 s_3 \\ c_2 s_1 & c_1 c_2 & - s_2 \\ c_3 s_1 s_2 - c_1 s_3 & c_1 c_3 s_2 + s_1 s_3 & c_2 c_3 \end{bmatrix}$$
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * $$\begin{bmatrix}
 * }

Sorry, I think I disagree but I would need some time. See my discussion with True bugman. I think I will give up ; that is only a problem of differences in interpretation of conventions ; as neither you nor True bugman are ready to discuss the math formulas there occur permanently misunderstandings. That plus the language and editing difficulties for me is too exhausting. Anyway it was nice to discuss with you both. Chessfan (talk) 22:49, 8 September 2010 (UTC)

As I promised an answer to True bugman I give here a quick view on my ideas for an unambiguous convention system. I think it would be necessary to give simultaneously the axis name and the associated angle name. The matrix $$R=X_\alpha Y_\beta Z_\gamma$$ corresponds to the convention $$(X_\alpha, Y_\beta, Z_\gamma)$$ in its intrinsic interpretation, and to the convention $$(z_\gamma,y_\beta,x_\alpha)$$ in its extrinsic interpretation. The convention $$(z_\alpha,y_\beta,x_\gamma)$$ would then obviously correspond to the matrix $$R'=X_\gamma Y_\beta Z_\alpha$$. You can easily verify that there subsists then no more ambiguities for matrices like $$R=ZXZ$$. Of course the order in the convention name gives the order in which the rotations occur around the different axis. Chessfan (talk) 07:39, 9 September 2010 (UTC) It is easy to verify that the (zyx) matrix above corresponds to my $$(z_\alpha,y_\beta,x_\gamma)$$ definition and is thus the extrinsic interpretation of the matrix $$R'=X_\gamma Y_\beta Z_\alpha$$, which I said long ago. As an intrinsic interpretation it should thus be named $$(X_\gamma, Y_\beta, Z_\alpha)$$Chessfan (talk) 12:39, 9 September 2010 (UTC) True bugman, you surely noticed that the above example is precisely the one you calculated the 9th of August, but without giving it a new convention name. Chessfan (talk) 11:49, 10 September 2010 (UTC)

I cannot resist to observe again that Geometric Algebra gives by construction unambiguous answers, of course based on a complete definition of each rotation and its order. So it is not surprising that with matrix algebra we must have the same elements to define unambiguously the conventions ! Happily mathematics are coherent. Chessfan (talk) 14:28, 9 September 2010 (UTC)

It might be interesting to note that with the convention naming I propose, we can classify the matrices in two categories, each one composed of 12 matrices.

The first one, which I prefer for active Euler rotations, in the sense given by the Springer handbook (rotated axis system), is characterized by the fact that the rotation order indicated by the axis order in the intrinsic convention name coincides with the normal alphabetic order of the angle names :

$$R=Y_\alpha X_\beta Z_\gamma \qquad (Y_\alpha, X_\beta, Z_\gamma) \qquad (z_\gamma, x_\beta, y_\alpha)$$

The second one, which is adopted in the existing table, privileges the extrinsic interpretation, and thus gives the opposite situation :

$$R'=Y_\gamma X_\beta Z_\alpha \qquad (Y_\gamma, X_\beta, Z_\alpha) \qquad (z_\alpha,x_\beta,y_\gamma)$$

In the beginning of my work I of course spontaneously adopted the first category matrices, which led me to the conclusion that the existing table was completely false. Now I would still prefer that the first category table were introduced. Chessfan (talk) 10:18, 11 September 2010 (UTC)


 * I also think that the active and intrinsic convention should be used. I confused the order the first time. Then it is enough to change the names of the entries to fix the table. Something like xyz->zyx should be enough to fix the matrix, isn't it?.--Guentherwagner (talk) 22:37, 13 September 2010 (UTC)

Hi ! If you follow my suggestions you should agree that your xyz extrinsic convention corresponds to what I named $$(x_\alpha,y_\beta, z_\gamma)$$, thus in intrinsic interpretation to $$(Z_\gamma,Y_\beta,X_\alpha)$$, that is to the matrix $$R'=Z_\gamma Y_\beta X_\alpha$$. You can do that, but if you name it only $$(Z,Y,X)$$ the reader considering the global matrix does not know which angle corresponds to which axis. I suppose that problem does not exist with specific applications, because the order of the rotations is always the same. But the table should be written for everybody ? Chessfan (talk) 09:45, 14 September 2010 (UTC)


 * I think there is no ambiguity because angles are named in the table with numbers. It is normal to assume that 1 is the first, and so on. The introduction saying that we use intrinsic active compositions should be enough to give a single meaning to every angle. --Guentherwagner (talk) 20:40, 19 September 2010 (UTC)

Here is my -- perhaps -- last suggestion :

1/Calculate again with your mathpackage the table of matrices which I called "first category". Be careful to associate in the global matrix formula $$\alpha$$ to the first rotation, $$\beta$$ to the second, $$\gamma$$ to the third.

2/Explain that you give them the convention name (capital letters) in the intrinsic interpretation (of course you must first have explained in detail what extrinsic/intrinsic means -- with the math formulas).

3/Explain that in your table the first matrix (in name and in formula) is always associated with the $$\alpha$$ angle, etc ... .

4/Explain how the same matrices (in litteral writing and numerical value) would be called in their extrinsic interpretation.

5/Explain how and why twelve other matrices can be written. Beware the reader from confusions.

Of course you can do the same with the existing table, but it will be less convincing. I think everybody starting with the intrinsic interpretation would, as I did spontaneously write the first category matrices.

Chessfan (talk) 13:33, 15 September 2010 (UTC)

Final reinsertion of the matrix table. Second try
About conventions:

Before fixing the whole table, please let's agree on this. For the XYZ entry of the table, for example, we should compose Rot(X,θ1) Rot(Y,θ2) Rot(Z,θ3) in this order. This means (calling them A, B and C in this order):

[ 1 0    0   ]                                [             ]   A =                          [ 0  c1  - s1 ] [            ]                                [ 0  s1   c1  ]

[ c2   0  s2 ] [            ]   B =                          [  0    1  0  ] [            ]                                [ - s2  0  c2 ]

[ c3 - s3  0 ] [            ]   C =                          [ s3   c3   0 ] [            ]                                [ 0    0    1 ]

xyz entry = A.B.C = [     c2 c3            - c2 s3         s2    ] [                                            ]                [ c1 s3 + c3 s1 s2  c1 c3 - s1 s2 s3  - c2 s1 ] [                                            ]                [ s1 s3 - c1 c3 s2  c1 s2 s3 + c3 s1   c1 c2  ]

Following Chessfan suggestions the introduction would be something like:

The possible combinations of rotations equivalent to Euler angles are shown here. The following matrices assume fixed (world) axes and column vectors, with intrinsic composition (composition of rotations about body axes) of active rotations and the right-handed rule for the positive sign of the angles.

Being an active composition of intrinsic rotations, the names of the entries are such that a matrix like that for XYZ is constructed as a product of the following three matrices, Rot(X,θ1) Rot(Y,θ2) Rot(Z,θ3) and is the result of performing first an X rotation, followed by an Y and a Z rotations, in the moving axes.

The angles of the rotations in this table are called θ1, θ2 and θ3. They refer to the angles of each of the three rotations in the order they are applied. For example c1 means cos(θ1) and s2 means sin(θ2)

Comments? Mistakes? --Guentherwagner (talk) 20:14, 19 September 2010 (UTC)


 * I suggest to present a less ambiguous example, for example (YXZ).
 * Then, let's confirm the matrices. The entry yxz will be A.B.C, being:

[ c1  0   s1 ] [            ]   A =                          [  0   1   0  ] [            ]                                [ -s1  0   c1 ]

[ 1  0     0 ]                                [             ]   B =                          [ 0   c2  -s2 ] [            ]                                [ 0   s2   c2 ]

[ c3 - s3  0 ] [            ]   C =                          [ s3   c3   0 ] [            ]                                [ 0    0    1 ]
 * Then I would write :


 * The possible combinations of rotations equivalent to Euler angles are shown here. The following global matrices in the table assume fixed (world) axes and column vectors, with intrinsic composition (composition of rotations about body axes) of active rotations and the right-handed rule for the positive sign of the angles.


 * Being an intrinsic composition of active rotations, the convention named (YXZ) is constructed as a product of the following three matrices, Rot(Y,θ1) Rot(X,θ2) Rot(Z,θ3) and is the result of performing first an Y rotation, followed by an X and a Z rotations, in the moving axes. The angles of the rotations in this table are called θ1, θ2 and θ3. They refer to the angles of each of the three rotations in the order they are applied. The notation can be simplified : c1 means cos(θ1) and s2 means sin(θ2).
 * great. I like it.


 * I recall that other rewritings are necessary. Chessfan (talk) 11:47, 20 September 2010 (UTC)
 * Then, what is next? --Guentherwagner (talk) 09:50, 21 September 2010 (UTC)


 * Go ahead ! Perhaps you should speak with True bugman and PAR about the rewriting of the other parts. Chessfan (talk) 10:24, 21 September 2010 (UTC) If you want you can of course use, perhaps with some reformatting, parts of User:Chessfan/work1. Chessfan (talk) 11:08, 21 September 2010 (UTC)
 * The problem with the rest of the text is that it looks like a tutorial about how to compose rotations. I don't know how to include that here. The reader is supposed to know how to do it in advance.--Guentherwagner (talk) 18:25, 21 September 2010 (UTC)
 * Part of it looks like that because I thought a majority of readers would not well understand why the order of matrices in intrinsic interpretation is the opposite of what seems natural. And I think also that our article is the only place where we should explain in detail the mathematical switch between intrinsic and extrinsic interpretation. Please read my preceding detailed observations.
 * I still think that explain rotation composition does not belong here. Maybe we can make a new article called rotation composition. Would you agree with that? Can I just copy your text for that? If you don't want to do it yourself for any reason I can give you credit in the discussion page.--Guentherwagner (talk) 08:34, 22 September 2010 (UTC)


 * There are obvious contradictions and/or errors in the article, with or without reintroduction of the table. Introduce it but dont stop there ! Chessfan (talk) 19:19, 21 September 2010 (UTC)
 * Of course I will fix anything wrong but, could you point it out?


 * Guentherwagner, why dont you adopt my proposal to use capital letters for naming conventions in intrinsic interpretation ? Chessfan (talk) 07:47, 22 September 2010 (UTC)
 * I like the idea. I just didn't realize. I took the old table and I wrote the matrices inside. I have already fixed it.--Guentherwagner (talk) 08:30, 22 September 2010 (UTC)


 * As I see the situation now, the following principal questions should be solved :


 * -- Grouping and rewriting of all sections concerning extrinsic/intrinsic interpretations, including the one named Matrix expression for Euler rotations. One possibility would be to start with an extrinsic example where the order of matrices in the formula is obvious. Then one should explain the necessity of transforming the matrices to adapt them to different frames than the fixed one, write as clearly as possible the resulting formula, then deducing finally the intrinsic order. (Hum, I still think what I wrote in User:Chessfan/work1 is better ...) . I dont think the figures are very helpful.
 * In fact I hate the Matrix expression for Euler rotations section. This is not the proper place to explain how perform a change of basis. Anybody wants to keep it? Can I remove it? --Guentherwagner (talk) 17:23, 22 September 2010 (UTC)


 * -- Correct or suppress the section Euler rotations and Euler angles as composition of Euler rotations. Small Euler rotations are approximatively commutative, but finite ones are not.Chessfan (talk) 11:59, 22 September 2010 (UTC)


 * Do we agree that Euler rotations are precession, nutation and intrinsic rotation? I would say that you are using a different definition.--Guentherwagner (talk) 17:22, 22 September 2010 (UTC)
 * I dont understand; what I critizised were not the names given to the rotations in that particular convention, but as you refuse any discussion on the maths I cannot help ... Chessfan (talk) 20:36, 22 September 2010 (UTC)
 * The word is important because it has a very specific meaning. Please read Precession and Nutation, and verify with a gimbal or something similar that these two movements are commutative always, not just for infinitesimal angles.--Guentherwagner (talk) 21:03, 22 September 2010 (UTC)
 * In case this helps, this is the precession of an arbitrary system (removed)
 * --Juansempere (talk) 21:57, 22 September 2010 (UTC)
 * Sorry Juansempere. I have just removed your precession image because it was getting me nervous. Anyway it is not helping anymore. Thanks.--Guentherwagner (talk) 22:14, 23 September 2010 (UTC)


 * Thanks for the nice animation, but of course I know what is a nutation or a precession. The problem is elsewhere. First, the math quasi-demonstration in the existing text is badly written and incomplete. And you may note that it concerns only infinitesimal rotations.

Second, if we draw on the surface of a sphere a latitude-longitude curvilinear rectangle you may be tempted to say that it proves the commutativity of nutation-precession, but that is an abuse of language. The two latitude segments can be drawn by a unique rotation around the polar axis. But that is not true for the longitude segments which are the trace left on the sphere by equiangular rotations around two different axes lying in the equatorial plane. I do not call that commutativity, even if in mechanics it is a useful propriety. My advice : suppress that section. Chessfan (talk) 09:50, 23 September 2010 (UTC)
 * Parallels are painted with rotations around the polar axis. They are painted with a precession. Painting meridians is a good example of nutation. Just check its equivalence with a the gimbal, whith the external ring movements drawing parallels and the second ring drawing meridians. Do you at least agree in the equivalence of the meridians and the nutations?--Guentherwagner (talk) 20:38, 23 September 2010 (UTC)
 * Of course I do ! But now something must be done about that section. Chessfan (talk) 07:11, 24 September 2010 (UTC)
 * OK. Then you agree that a precession is like a movement on paralels and a nutation on meridians. Then the curvilinear rectangle with finite angles you were using before is really proving that both operations are commutative for finite angles. Don't you agree in this? —Preceding unsigned comment added by Guentherwagner (talk • contribs) 14:29, 24 September 2010 (UTC)
 * I am not a professional mathematician, but I think indeed it would be non-rigorous to call that successive operations commutative. Let's look at it otherwise : If starting from a point P you execute successively a well defined finite precession $$R(u,\alpha)$$ followed by a nutation $$R(v,\beta)$$ you get P' ; if then you alternate these operations with the same angles and axes you get P", which is different from P'. The axis of the nutation depends on the order of the moves. The curvilinear rectangle is closed because we apply two different nutation operators. That has something to do with "parallel transport on a sphere" ..., but looks rather pedantic :) . —Preceding unsigned comment added by Chessfan (talk • contribs) 17:02, 24 September 2010 (UTC)  You can even verify with a program, or with a large precession, that P" is no more on the sphere.Chessfan (talk) 17:08, 24 September 2010 (UTC) One could perhaps describe these alternative moves and conclude that the same final position can be reached by switching them. Could True bugman give his opinion ? Chessfan (talk) 18:08, 24 September 2010 (UTC)
 * Two operations f and g defined over any set of elements are said commutative if f(g(x)) = g(f(x)) for any x. Precessions and nutations (of a given angle) satisfy this definition when acting over reference frames. The reason of course is because the axes are different. Nutation is referred to the line of nodes and therefore is not the same in both cases. Maybe it should be explained more clearly that the commutative property is only valid in this context.--Guentherwagner (talk) 19:30, 26 September 2010 (UTC)
 * I am not quite sure to understand your argument but do what you like ... You forget voluntarily that a rotation is defined by an angle and an axis. The layman will never understand why generally rotations in R3 are not commutative, but here particularly are. Chessfan (talk) 23:03, 26 September 2010 (UTC)
 * A nutation always has an axis, which is the line of nodes. Therefore we can say that nutations are rotations. The only difference with other rotations is that the axis is different for two given nutations. I think the layman will understand this as soon as he takes a look to the gimbal drawing.--Guentherwagner (talk) 06:00, 27 September 2010 (UTC)
 * Please Guentherwagner, do what you like but dont give false interpretations of what I say : I never said that a nutation is not a rotation ! I critizised the "commutation" naming and gave my arguments, thats all.
 * Nop. You said that I was intentionally forgetting that "a rotation is defined by an angle and an axis". I had no other option than explain you that I was not forgetting that.
 * Sorry, I know, but instead of f(g(x))=g(f(x)) you admit that $$f(g_1(x))=g_2((f(x))$$ means commutativity in a strict mathematical sense. Chessfan (talk) 14:34, 27 September 2010 (UTC)
 * Read my equation as operators transforming a frame (x) to another frame (operators in a set of frames). I have the right to define g as an operator that nutates a frame by some amount, for example 20 degrees. Operators defined like this commute. A different thing is how will I express that with matrices. You are thinking probably about rotations that can be expressed by a matrix product. Nutation cannot be expressed by a single matrix product. --Guentherwagner (talk) 19:18, 27 September 2010 (UTC)


 * By the way what will you do with the "infinitesimal" demonstration ?
 * There is no infinitesimal demonstration in the article or here. Where are you seeing such a thing?
 * The math formulas in section "Euler rotations" of course. They are valid only in differential calculus. Chessfan (talk) 14:25, 27 September 2010 (UTC)
 * For sure, we disagree here. The formulas are valid for finite angles, and probably the original author of them thought the same because never said anything about infinitesimals.--Guentherwagner (talk) 19:18, 27 September 2010 (UTC)
 * Why dont you verify ? Calculate one of the formulas. Chessfan (talk) 19:52, 27 September 2010 (UTC) OK the first equations are true with finite rotations, but the last one is parachuted from nowhere and not well defined. Chessfan (talk) 22:18, 27 September 2010 (UTC) If I understand it well it shows precisely that there are two different nutation operators !! Chessfan (talk) 22:31, 27 September 2010 (UTC) The whole math must be verified step by step ; I feel now that the second and third composition relations are false for finite Euler rotations. Chessfan (talk) 06:08, 28 September 2010 (UTC)
 * It would be interesting to hear other advices. I persist in my opinion. Chessfan (talk) 07:27, 27 September 2010 (UTC) Perhaps one could say that the finite movements generated by a gimbal are commutative ? I go as far as I can ... Chessfan (talk) 07:38, 27 September 2010 (UTC)
 * Gimbal movements are the Euler rotations. See the definition (variation of a Euler angles keeping constant the other two). How can you maintain that gimbal movements are commutative and not Euler rotations? --Guentherwagner (talk) 07:54, 27 September 2010 (UTC)
 * That is just a question of rigorous definition of what you call "transformation operator" in maths. Two different nutations, that is with differenr axes, are not the same operator. If you do not admit that we cannot reach an agreement. But happily that does not stop the Earth rotating ! Chessfan (talk) 14:46, 27 September 2010 (UTC)


 * I have found a drawing from function composition article that can help me explain what I mean. Imagine an initial frame like b that is transformed into 1 by a alpha-precession and into 2 by a beta-nutation. Then a beta-nutation over 1 must be the same as an alpha-precession over 2. Precessions and nutations can be defined over any element in an abstract way because the image is defined and is unique, and speaking about abstract operators there is no need to specify an axis of rotation. These operators work with a different rotation axis for each element of their domain, as you said, but is perfectly possible to define them like this. —Preceding unsigned comment added by Guentherwagner (talk • contribs) 20:23, 27 September 2010 (UTC)
 * By throwing away the axes you are able to define on a sphere what I named the "gimbal movement" commutation. But that is only true if you limit your moves to latitude and longitude arcs. Is that perhaps why somebody introduced the notion of partial commutativity ? Our discussion begins to look like Andersen's story "The Princess and the little pea" : do we feel the pain through the triple mattress ? Chessfan (talk) 21:55, 27 September 2010 (UTC)


 * Guentherwagner, hi again. I think you and I, with the help of True bugman, have done perhaps a good job here. But it would be a pity not to push the work as far as we can.
 * I am still in disagreement with you both on the question of explaining or not the math details of the extrinsic/intrinsic interpretations.
 * That is a very specific problem to Euler angles and/or Euler rotations (I have a tendency to mix the two notions).
 * If you write a specific article Rotation composition you will mainly describe rotation composition in a fixed unique reference frame. That might be useful because a lot of people spontaneously write the false $$R=R_1 R_2 R_3$$ for active rotations, instead of the right $$R=R_3 R_2 R_1$$. But the matrix composition for Euler angles precisely gives $$R=R_1 R_2 R_3$$ in intrinsic interpretation, which is far from obvious. In what better place than than the Euler angles article can you explain that ?
 * And, second argument, as you explain the maths of the intrinsic interpretation, you simultaneously explain and demonstrate the existence of an extrinsic interpretation. Why is that so easy ? Precisely because once you have demonstrated the $$R=R_1 R_2 R_3$$, the extrinsic interpretation is under your eyes. Why ? Because the $$R_1$$, $$R_2$$, $$R_3$$ matrices are so simple : in any reference frame the rotation axis is obvious.
 * If that should not be written in the Euler angles article, then I decidely understand nothing about the editing rules in Wikipedia ... Chessfan (talk) 16:54, 22 September 2010 (UTC)
 * For sure rotation composition is a problem strongly related to Euler Angles, but so is Gimbal lock for example. Should we copy that article here? and any other with a relationship? Of course that is impossible. That is why the subjects have to be followed strictly. If you put your text about rotations here and tomorrow somebody else wants to speak about the gimbal lock problem, how could we say not? what will happen when the article is too long to be read?--Guentherwagner (talk) 17:08, 22 September 2010 (UTC)
 * That does not convince me; how can you be satisfied with the existing text on the extrinsic/intrinsic question ? My problem is not to see my text adopted. Chessfan (talk) 17:50, 22 September 2010 (UTC)
 * Well, one of the things that I like about that text is that is not using matrices nor any other mathematical notation. It does not need them, and it does not need to explain in which way matrices would be composed (should this notation be used). We could prove formally the same introducing matrices, quaternions or SO(3) parametrizations, but, would that help the reader?--Guentherwagner (talk) 18:18, 22 September 2010 (UTC)
 * When reading again that text I see that at one side you seem prepared to maintain a rather badly explained mathematical text on extrinsic rotations, but simultaneously you refuse, because of non-encyclopaedical writing, my math formulation which is much clearer ...(yes, I dont hesitate to say that) ! Why ? Chessfan (talk) 14:19, 27 September 2010 (UTC)


 * Please give me advice when you consider that the text of the article is consolidated. For now I dont see where we are ? I hope other editors will contribute or at least give their opinion. Chessfan (talk) 20:43, 22 September 2010 (UTC)

I realize I spent almost half a year on that subject. I will continue to read you, but I take the resolution no more to contribute ! Good Bye ! Chessfan (talk) 15:05, 27 September 2010 (UTC)

Further discussion about Euler rotations
Sorry Guentherwagner I changed my mind. I have still some work to do ; you pushed me too far. Chessfan (talk) 21:16, 29 September 2010 (UTC)

My last word on that subject : Rigorously defined Euler rotations (precessions and nutations) do not commute ! See User talk:Chessfan/work1 Chessfan (talk) 14:54, 1 October 2010 (UTC)


 * Well, then please explain to me what is wrong in the following reasoning:

Could you tell me pleas what is your rigorous definition? --Guentherwagner (talk) 01:19, 3 October 2010 (UTC)
 * I have the right to define a set that contains all the possible rotated frames.
 * I have the right to define some internal operations on this set, which I will call precessions and nutations, as following:
 * Precessions over a frame of this set will be defined as a rotation around the z-axis of the reference frame.
 * Nutations over each frame will be defined as a rotation around the line of nodes of the given frame respect my reference frame
 * Being f and g operators in this set, f an alpha-precession and g a beta-nutation, f(g(x)) = g(f(x)) for any frame x.
 * Therefore we can say that alpha-precessions and beta-nutations as defined before, are commutative for any value of alpha and beta.

What is wrong is the fact you neglect that we are speaking, by the definition of Euler rotations, about four operators : where $$e'_1$$ is $$e_1$$ rotated by $$f_1$$, and $$e''_3$$ is $$e_3$$ rotated by $$g_1$$.
 * $$f_1$$ an alpha-precession around $$e_3$$
 * $$f_2$$ an alpha-precession around $$e''_3$$
 * $$g_1$$ a beta-nutation around $$e_1$$
 * $$g_2$$ a beta-nutation around $$e'_1$$


 * Exactly which point of my reasoning neglects those four rotations? You have claim several times that two nutations have different axes and that is true, but, one thing is the commutativity of the operators and other the commutativity of the rotations used to implement them. I have not used those underlying rotations in anyone of my previous points. --Guentherwagner (talk) 14:54, 3 October 2010 (UTC)
 * Maybe the big misunderstanding between us two is the word rotation. A nutation is a rotation in some geometrical sense, but maybe not in the sense of operators (A rotation operator rotates all the frames in its domain in the same way). Could this be the key to understand each other? Could everything be easier to agree if we consider a nutation as a non-rotation operator?
 * What you write here cannot be true ! Chessfan (talk) 16:11, 3 October 2010 (UTC)

And I demonstrated by three different methods that $$g_2(f_1) \ne f_2(g_1)$$. Do it geometrically for the vector $$e_1$$, and you will be convinced. Chessfan (talk) 09:23, 3 October 2010 (UTC)
 * I cannot perform a nutation over a vector. Such an operation is not defined. My previous definition of nutation is defined only over a complete frame. Again, what defintion of nutation are you using? --Guentherwagner (talk) 14:40, 3 October 2010 (UTC)
 * Nutation is a rotation which you can do over any frame and any vector defined in a frame.Chessfan (talk) 16:11, 3 October 2010 (UTC)
 * Not with my previous definition (rotation about the line of nodes is not defined for a vector) and not with the article definition (increment of the second Euler angle, which is not defined for a vector). Then, for the third time, please define what you call a nutation. I cannot go on speaking about a concept that is only in your mind.--Guentherwagner (talk) 16:22, 3 October 2010 (UTC)

I see now : you take $$f_2=f_1$$. Then indeed $$g_2(f_1)=f_1(g_1)$$, but that I would not call commutativity of Euler rotations ! Chessfan (talk) 09:34, 3 October 2010 (UTC)
 * That's not what I am saying. I am defining nutation as an operator which has a different rotation axis for any frame of its domain. Don't you agree that such a thing can be done? Don't you agree that this is the normal meaning of the word nutation? --Guentherwagner (talk) 14:40, 3 October 2010 (UTC)

No, read again your text. The problem comes from the fact that when beginning with nutation you dont vary the precession axis. Then you get indeed $$g_2(f_1)=f_1(g_1)$$, but that in my opinion does not correspond to a correct definition of Euler rotations. Another way to say that is : a rotation alone cannot be called Euler, it becomes Euler rotation when followed by another rotation, composed in intrinsic way. If you still do not agree with that please try to find help with another experienced mathematician. We go nowhere now. Sorry, I dont want te be rude, but I have the feeling you never read the math texts I write. Chessfan (talk) 16:19, 3 October 2010 (UTC)
 * I have been asking for your definition of Euler rotations several times. Could you please give me one so that we can agree at least about the subject of the discussion? I have the feeling that you don't even read the text I write--Guentherwagner (talk) 16:30, 3 October 2010 (UTC)

Everything is in my user pages, to which I referred since 5 months ! Chessfan (talk) 16:37, 3 October 2010 (UTC)
 * Not the definition that I am asking you for. Or at least I was not able to find it. Would you mind to paste it here?--Guentherwagner (talk) 17:01, 3 October 2010 (UTC)

Everything is here :

''What is wrong is the fact you neglect that we are speaking, by the definition of Euler rotations, about four operators :

* f1 an alpha-precession around e3   * f2 an alpha-precession around e"3    * g1 a beta-nutation around e1    * g2 a beta-nutation around e'1

where e'1 is e1 rotated by f1, and e"3 is e3 rotated by g1.''

To be more explicit, if you begin with a nutation you rotate any vector defined in the e-frame first around the e1 axis with beta, but you also rotate the precession axis from e3 to e"3, then you execute the alpha precession around e"3. Instead if you begin with a precession around e3, you follow with a nutation around e'1 (precessed from e1). The difference with what you do, is that voluntarily or not you forget to rotate the precession axis to e"3 in the first situation. Chessfan (talk) 17:31, 3 October 2010 (UTC)
 * This is really incredible. Please, read slowly. I will repeat the question a fourth time. Can you please give me a definition of what is a precession and what is a nutation for you?--Guentherwagner (talk) 21:06, 3 October 2010 (UTC)

????! Chessfan (talk) 21:20, 3 October 2010 (UTC)
 * Look. I am trying hard to follow your reasonings and there is always a point in which the concepts you are using do not fix with mine. Then I come back here and I ask you to tell me what kind of definitions are you using. As answer I always get from you something that I am not asking. Do you know what a definition is? Is there something in my question that is no clear? If I asked you (for fifth time) to define what a nutation is for you will you do it? —Preceding unsigned comment added by Guentherwagner (talk • contribs) 21:53, 3 October 2010 (UTC)

Well, I think the difficulty comes from the fact that I try, pushed by you, to reconcile three different notions :
 * the fact that here we speak of intrinsic composition of Euler angles ;
 * the fact that as shown in the graph of the article the precession-nutation is essentially a cinematic, dynamic phenomenon, that is the sum of infinitesimal rotations, which of course pose no pb of commutativity ;
 * the fact that you pretend that Euler rotations are commutative for finite rotations.

So I try to realize by successive rotations, well defined in my math texts, successive Euler angles, beginning one with a rotation around e3 (precession axis), the second with a rotation around e1 (nutation axis), and simultaneously follow the rules of Euler angles composition.

Of course you can consider that the precessions by definition take place over a fixed e3 axis. Then I find your commutativity conclusion. But even then the demonstration given in the text seems not convincing. And strictly speaking the successive angles are no more Euler angles. Chessfan (talk) 22:36, 3 October 2010 (UTC)


 * Well, thanks for your response. I am sure it makes sense according to some concept that you have in your mind, but again you refuse to define it. Maybe we are speaking about different definitions since the beginning, but you never reply when I ask you for a definition (why?). Let's do it in other way. Tell me please if you agree with this definition of the Euler rotations: They are the movement obtained modifying one of the Euler angles while leaving the others constant.--Guentherwagner (talk) 23:23, 3 October 2010 (UTC)

That is true, but I start with initial angles equal to zero, and utilize the existing (X,Y,Z) as a, from that moment, staying fixed reference frame (e1, e2, e3). Then I rotate any vector defined in that frame frame around e3 (precession) or e1 (nutation), without forgetting to vary the rotation axis of the rotation executed in second place, to obey (I am very obedient ...) to the rules of intrinsic composition of Euler angles.

That was under your eyes from the beginning, and particularly when I exposed the four operator theory which seems to irritate you. I can also give you the equation references where I exposed that with GA. Of course I cannot ask you to learn GA, but even a superficial lecture would have shown you the definitions of precession and nutation.

To be as clear as possible I add that of course that separation between nutation and precession makes sense only when the nutation amplitude remains small. I learned that fifty years ago, alas...

Another irritating problem with the existing text is the fact that it is quite impossible to well define in matrix algebra an expression like $$A(\alpha+\delta \alpha, \beta+\delta \beta,\gamma)$$ if the variations are not infinitesimal, and if we try to respect the intrinsic composition of Euler angles. I dont see why that should figure in the article.

If again you think my answer is not clear enough, I propose to put an end to a controversy that is now sterile. But I recommend strongly to search for other advices by well chosen editors. True Bugman seems to have some knowledge on the question.

Chessfan (talk) 07:50, 4 October 2010 (UTC)


 * Well, you started your text saying "that is true". I will assume that you agree in the definition that I gave you (finally!!). At least now I can go further. As you pointed out, you cannot express a nutation with a single matrix. Nevertheless, you can speak about nutations, and you can say that they transform a frame into a frame. Do we agree in this?--Juansempere (talk) 08:33, 4 October 2010 (UTC)

Definition agreed
I am happy to see you coming back to that discussion. I would like to check if you agree with me on the following points : Chessfan (talk) 12:03, 4 October 2010 (UTC)
 * Starting with a reference frame, which we now maintain fixed, where e3 is the last precession axis and e1 the last nutation axis, we call precession any rotation around the e3 axis and nutation any (in fact small) rotation around first the e1 axis, later around axes deduced from e1 by well chosen precessions.
 * If we do successive alternate nutation-precession rotations, then precession-nutation ones, we can say and prove that those rotations are commutative (that is what I called latitude-longitude moves).
 * But as we chose to never move the precession axis e3 the combined rotations where nutation precedes precession can no more be called combined Euler rotations (in the intrinsic sense). If instead we had chosen to move also the precession axis the studied combined rotations would no more commute (except in the infinitesimal).
 * The distinction between precession and nutation has a practical interest only when the nutation angles are small (see rotating slightly dissymetrical body).
 * Before replying to your questions, I want to be sure we speak about the same. I will call  to the reference frame,  to the frame after a precession (of course, e3’=e3),  after the nutation.
 * I will try to reword your sentences using these frames. Please, tell me if you agree, and then I will reply to your questions. I have removed the words 'last' because it made no sense assuming only a precession and a nutation. I have changed the part in which you decided what to call precession and nutation by the agreed definition. I anticipate that I don't agree necessarily with the following sentences, but I would like that you assume it or reject it before discussing it further.
 * "Starting with a reference frame  which we will maintain fixed, we perform a precession and a nutation until we reach a frame . e3 is the precession axis and e1' the nutation axis. We call precession (by definition) over a frame to any rotation that keeps its first Euler angle constant and nutation to any rotation that preserves the first and third angles"


 * "If we do successive alternate nutation-precession rotations, then precession-nutation ones, we can say and prove that those rotations are commutative (that is what I called latitude-longitude moves)"


 * "But as we chose to never move the precession axis e3 the combined rotations where nutation precedes precession can no more be called combined Euler rotations (in the intrinsic sense). If instead we had chosen to move also the precession axis the studied combined rotations would no more commute (except in the infinitesimal)"


 * "The distinction between precession and nutation has a practical interest only when the nutation angles are small (see rotating slightly dissymetrical body)"
 * --Guentherwagner (talk) 17:49, 4 October 2010 (UTC)

I am not overenthousiastic with your introduction of a general definition of precessions and nutations with Euler angles. You must at least replace first by second in the precession definition. Go ahead. Chessfan (talk) 22:00, 4 October 2010 (UTC)
 * I know that you must have something different in your mind, but let me remember you that I have been asking you several times to define your thoughts with no reply:


 * Could you tell me pleas what is your rigorous definition? --Guentherwagner (talk) 01:19, 3 October 2010 (UTC)
 * My previous definition of nutation is defined only over a complete frame. Again, what defintion of nutation are you using? --Guentherwagner (talk) 14:40, 3 October 2010 (UTC)
 * Then, for the third time, please define what you call a nutation. I cannot go on speaking about a concept that is only in your mind.--Guentherwagner (talk) 16:22, 3 October 2010 (UTC)
 * I have been asking for your definition of Euler rotations several times. Could you please give me one so that we can agree at least about the subject of the discussion? I have the feeling that you don't even read the text I write--Guentherwagner (talk) 16:30, 3 October 2010 (UTC)
 * This is really incredible. Please, read slowly. I will repeat the question a fourth time. Can you please give me a definition of what is a precession and what is a nutation for you?--Guentherwagner (talk) 21:06, 3 October 2010 (UTC)
 * If I asked you (for fifth time) to define what a nutation is for you will you do it? —Preceding unsigned comment added by Guentherwagner (talk • contribs) 21:53, 3 October 2010 (UTC)
 * And now the only way I have found to agree in a definition:


 * Let's do it in other way. Tell me please if you agree with this definition of the Euler rotations: They are the movement obtained modifying one of the Euler angles while leaving the others constant.--Guentherwagner (talk) 23:23, 3 October 2010 (UTC)
 * That is true, but I start with initial angles equal to zero, and ...


 * What do you mean with "Not overenthousiastic". Aren't you telling me now that you want to change the definition, is it?
 * You are right about the angle. I meant second. I change it and I respond:


 * "Starting with a reference frame  which we will maintain fixed, we perform a precession and a nutation until we reach a frame . e3 is the precession axis and e1' the nutation axis. We call precession (by definition) over a frame to any rotation that keeps its second Euler angle constant and nutation to any rotation that preserves the first and third angles"
 * I agree
 * "If we do successive alternate nutation-precession rotations, then precession-nutation ones, we can say and prove that those rotations are commutative (that is what I called latitude-longitude moves)"
 * I agree
 * "But as we chose to never move the precession axis e3 the combined rotations where nutation precedes precession can no more be called combined Euler rotations (in the intrinsic sense). If instead we had chosen to move also the precession axis the studied combined rotations would no more commute (except in the infinitesimal)"
 * This needs more explanation. I explain my position later.
 * "The distinction between precession and nutation has a practical interest only when the nutation angles are small (see rotating slightly dissymetrical body)"
 * I don't agree, but I think this is irrelevant for the discussion
 * --Guentherwagner (talk) 17:49, 4 October 2010 (UTC)

My position about the third point
Let me start with an example. If I have an operator that converts vectors into vectors I will say that the operator is a rotation when all the vectors rotate around the same axis. In these cases, the operator can be represented by a orthonormal matrix that I will call A. Some other operators are not rotations. For example f(x,y,z)=(0,0,0) is obviously not a rotation.

Let me put another example of an operator that is not a rotation. Let's take two different rotation matrices A and B, and let's define the operator f as following:


 * f(x)=Ax for any vector x on the upper part of the space (z component positive or zero) and
 * f(x)=Bx for x in the lower part (z component negative).

This is a discontinous function and for sure it is not a rotation operator, and for sure it cannot be expresed with a simple matrix product. Nevertheless, we can say for any given x that its image is its rotation around an axis.

Now back to the Euler rotations. The fact is that they are not rotations in the operators meaning. If they were you could write them as a single matrix product, and as you said, a nutation cannot be expresed like that.

Then my position for your third question is that it has still to be rewriten before I can agree or disagree, saying what meaning for rotation and for "combined rotations" we are using. --Guentherwagner (talk) 23:04, 4 October 2010 (UTC)

Guentherwagner I dont know if you have developped a special dislike for me, but the fact is that you never read the math I write, that you pretend I have never given answers to questions which I answered long ago (mainly in the math texts), that you more and more edit your texts in a manner that renders mine unreadable, thus that you adopt systematically an unfair behaviour against me. I know what I have done here, thats enough ! If you want to paint donkeys with wite and black stripes pretending they are zebras, you will do it alone ! Chessfan (talk) 11:10, 5 October 2010 (UTC)


 * Sorry. I have to apologize for my last post. When I read your comment about the definition I got mad. It took me a lot of effort to make you agree. But I understand anyway that you didn't try to upset me, and that all was a misunderstanding. Therefore I apologize for my last comment about the definitions.


 * Anyway, my position for your third question does not change. It has to be rewritten before I can agree or disagree, because the term "combined rotations" is ambiguous. --Guentherwagner (talk) 15:49, 5 October 2010 (UTC)

Apologizes agreed. I will answer you later, but that will be my last contribution, on the subject combined rotations which seems to remain unsolved between us some time ago.

I came to the subject Euler angles, where I had almost everything to learn again, not at all to give lessons to specialists of the question ... , but because it seemed to me to be a good example for showing how efficient it would be to study it with geometric algebra. I was prepared to get a baseball bat reception, because of GA, but not at all to find an article in great disorder, with a lot of ambiguities. Indeed, if you have some time to read even superficially what I wrote with GA,you will see that some misunderstandings between us would not have occured. GA is perhaps not more efficient for our subject than matrix algebra, but it is a lot more precise, and mostly coordinate free.

The struggle for a non-ambiguous reintroduction of a matrix table was hard and time consuming. Now there is a lot more to do -- dont ask me what, because I also would get mad --, and I am intellectually drained by the subject. So dont feel I am unfair when, at least for some time, I stop contributing.

Chessfan (talk) 17:14, 5 October 2010 (UTC)

A last contrib to that question
Let us speak first about your precession definition. You introduce an unusual definition, but I do not oppose it, as long as you dont suppress the mention of the fixed e3 rotation axis. Perhaps instead of precession is a rotation ...  we should write precession is a movement of rotation around a fixed axis ... . One should also consider an observation made by True bugman : ''This nutation/precession is actually a very specific example used in certain problems in engineering (and some physics problems). Many other conventions are used though. It is a bit of an embarrassment to have this here, pretending to describe all Euler rotations! It is a specific application and should probably be shifted to the applications section as a subsection. '' I suggest to follow his advice. I also persist saying that the mention of nutations being generally small would be useful for the reader.

I come now to the general notion of combined rotations. Some time ago -- I did not find exactly when -- you made an observation on the  combined Euler rotations section I wrote in Work1, telling me that in fact I treated there only the general notion of combined rotations. I answered that you were wrong but nevertheless changed the section title. I should not have done that, because there begun our misunderstandings (you obviously had not read my math demonstration, or superficially).

I still think that my text, perhaps somewhat modified, should be incorporated in the article, in a section where also all the definitions and demonstrations concerning intrinsic/extrinsic interpretations would coherently be introduced.

It is most important, but not at all obvious,to help the reader to grasp the following facts which are particular to Euler angles :


 * The position of a mobile frame, and thus of a rigid body associated with it, can be described, starting from the fixed reference frame, by a system of so-called Euler angles constructed by composing three successive rotations.


 * The particularity of those rotations, when compared with rotations composed by successive rotations around fixed axes, is the fact that each rotation acts on the axes of the following rotations, and each rotation axis is perpendicular to the following one.


 * In matrix algebra each rotation is expressed by a simple matrix, acting in the last defined frame around one of the basis vectors. Thus the matrices have all a structure with the number one in the diagonal and zeros in the corresponding row and column. the consequence of that structure is that when applied, without algebraic transformation, in any arbitrary frame, the matrix automatically picks out one of the basis vectors as the rotation axis.

If we write in operator notation the succession of three rotations in the order $$\mathfrak{A},\mathfrak{B},\mathfrak{C}$$, we get :

$$(1) \qquad v=\mathfrak{R}=\mathfrak{C} \mathfrak{B} \mathfrak{A} u$$

If in a fixed reference frame e(e1, e2, e3) we represent these rotations by three matrices we get the same order :

$$(2) \qquad (v)_e=R(u)_e=CBA(u)_e$$

where $$(v)_e, (u)_e$$ are column vectors.

But if we do Euler rotations the order of the simple matrices will be reversed. We get :

$$(3) \qquad (v)_e=R'(u)_e=ABC(u)_e$$

which can also be written :

$$(4) \qquad (v)_e=[ABCB^{-1}A^{-1}][ABA^{-1})]A(u)_e$$

where the expressions in brackets represent the B matrix supposed acting in the f-frame but transformed to e-frame, and the C matrix supposed acting in the g-frame but transformed to e-frame (e,f,g are the successive Euler frames).

That is not too easy to grasp, specialy not how the successive matrices pick out the adequate rotation axes.

Now let us look at the different precession -nutation matrices, which we call P and N if the rotation axes are e3 and e1, and N' if the rotation axis is e'1. A precession folllowed by a nutation N' gives the combined Euler rotation matrix N'P. But as we see by (3) and (4) we have :

$$(5) \qquad N'P=P N$$ In reverse if a nutation N is followed by a precession P (by choice the e3 axis is not moved) we get also the matrix $$P N$$. That justifies in sophisticated matrix algebra the geometrically obvious fact that the latitude-longitude curvilinear rectangle is closed. As defined, precession-nutation operations are commutative !

But we verify also that the defined precession does no more follow the strict Euler rotation composition rules. If it were one should have for the nutation-precession operation a combined matrix equal to P'N. and as we know :

$$(6) \qquad N'P=PN \ne NP=P'N$$

That is the explanation of the commutativity paradox. Why should we hide that ?

Please do not cut my text in pieces !

Chessfan (talk) 14:10, 7 October 2010 (UTC)

Addendum. Forgive me to become a bit pedantic, but the argument is too beautiful to remain hidden. Suppose that you draw on the surface of any sphere, centered on the origin of the axes, such a latitude-longitude rectangle. You have certainly noted that the latitude sides are not grand circles of the sphere. If now you report on the four vertices the position of the rotated frames and follow the contour of the rectangle until you come back to the starting point, will the final frame be identic or not to the initial one ? You know already that the answer is yes.

But what would be the situation if you had sustituted grand circles to the latitude sides ? (Which correponds to the strict application of Euler angles rules ...). Then a theorem first demonstrated by Gauss tells us that there would be an angle between the initial and final frames whose magnitude would be proportional to the surface of the rectangle.

Thus the fact that we adopted the non geodesic latitude sides corrects precisely that angular variation !! (To be more precise the movement of the frames on the latitude sides is not a so-called parallel displacement on the surface of the sphere). Chessfan (talk) 16:50, 7 October 2010 (UTC)


 * I will try to reply not cutting your text, though is really difficult to reply such a big amount of propositions if not one by one.


 * The only way I can reply somehow not cutting your text is focusing on just one thing, and I will take what I think is our big disagreement. I will copy here the assertion I agree with the less and I will discuss it. You said:


 * But if we do Euler rotations the order of the simple matrices will be reversed. We get :
 * $$(3) \qquad (v)_e=R'(u)_e=ABC(u)_e$$


 * You are treating here Euler rotations as rotation operators, and as we agreed long time ago, they cannot be expressed as such. You can find a matrix for a given frame that performs a nutation on it, but you cannot find a rotation operator that nutates frames. I challenge you to give a single matrix that can nutate an arbitrary frame (and we have agreed that nutate here means to change the second Euler angle of the frame while leaving constant first and third). If you want to use a different definition for nutation, please put it here now.

That begins to look like a police audition !!! If you want to discuss with me dont oppose me what I am supposed to have said, but contradict me with precise math reasoning. I am very surprised by the aggressive behaviour I seem to encounter. If you pursue on that tone I will no more read you. Do you really think you are now doing a good job in the spirit of Wikipedia.

Besides you should be surprised that with my supposedly false or impossible definitions I am able to demonstrate by matrix algebra that you are right when you affirm the commutativity of nutations and precessions.

Chessfan (talk) 20:06, 7 October 2010 (UTC)

Sorry, this time was not guentherwagner. And I meant no offense. I was just trying to settle things down.--Juansempere (talk) 20:41, 7 October 2010 (UTC)


 * Please, be careful and sign your contributions, specially if you interrupt the conversation of other people.--Guentherwagner (talk) 20:54, 7 October 2010 (UTC)

OK. Don't worry. Anyway as I said it was my last contribution. What I wrote for you in matrix algebra I wrote also in GA (see my userpages). Everything is coherent. Perhaps you should do some math work on it ? Use it as you like, but I suppose you are aware that the Euler angles article is far from perfect. Good luck. Chessfan (talk) 20:52, 7 October 2010 (UTC)

Juansempere, I forgot to answer your remark on equation (3). If the logical switch from (2) to (3) is not accepted by you, then I suggest you to read the corresponding section in User:Chessfan/work1 Chessfan (talk) 17:10, 9 October 2010 (UTC)


 * As Juansempere has still not replied, I will do it. According to your expression (3) if I want to construct an operator that performs a 0-degree precession, 30-degree nutation and 0-degree intrinsic rotation, its matrix would be:

[ 1  0   0 ] [ 1   0     0   ] [ 1   0   0 ]   [ 1   0     0   ]                 [           ] [               ] [           ]   [               ]   M =           [ 0   1   0 ] [ 0 c(30) -s(30)] [ 0   1   0 ] = [ 0 c(30) -s(30)] [          ] [               ] [           ]   [               ]                 [ 0   0   1 ] [ 0 s(30) c(30) ] [ 0   0   1 ]   [ 0 s(30) c(30) ]


 * This is clearly not the expression of a nutation, and therefore your equation (3) is wrong.--Guentherwagner (talk) 10:03, 10 October 2010 (UTC)

Thank you for your answer which at least has the merit to deal with a concrete example. But it leaves me speechless. The matrix you expose here is exactly the one which represents a rotation around what I named the e'1 axis, which here coincides with e1. That is what I call a nutation, and it is of course easy to verify that we can draw, when varying the angle, a longitude side of the curvilinear rectangle. More generally the relation (3) is a consequence of relation (4) (sorry for the inversion). If you dont agree with that, then we have indeed a much bigger math problem than I thought. Do you be aware of the fact that the relation (3) is the key of the double interpretation of Euler angles, intrinsic and extrinsic ? I think the problem comes mostly from the fact that mathematicians and physicists neglect to specify to and in which frame they apply the matrix. I tried in vain to introduce that idea but was promptly countered by True bugman ! Chessfan (talk) 12:35, 10 October 2010 (UTC)


 * Well, maybe you call this a nutation, but when this matrix is applied over a generic frame (one with its three angles different from zero) it will not move second Euler angle leaving constant first and third. Therefore this operator does not represent a nutation according with the article's definition. --Guentherwagner (talk) 14:08, 10 October 2010 (UTC)

I must think over that, but at least are you aware that by denying the formula (3) you deny your matrix table calculation? Where lies the contradiction ? Chessfan (talk) 16:01, 10 October 2010 (UTC)


 * The table was about intrinsic rotations composition, not about Euler rotations composition. --Guentherwagner (talk) 17:00, 10 October 2010 (UTC)
 * On a second thought you are right here. Strictly speaking maybe we shouldn't treat intrinsic rotations as operators, but just as matrices working on specific frames. In the cases of the table of matrices probably they work as an operator just because they are equivalent to the extrinsic composition, which is in fact the real operator.--Guentherwagner (talk) 18:39, 10 October 2010 (UTC)

Thank you for your fair answer. Chessfan (talk) 19:24, 10 October 2010 (UTC)

proposed end of discussion

I propose a possible end for the discussion. As I have seen, both parts have given some valid arguments about the commutativity and non-commutativity of the operations. It seems that supporting the commutativity requires some weird definitions about operators that are at least uncommon. Of course, only that is not enough for a removal, but I want to propose deletion of the commutativity sentence on other grounds: I want to remove it just because it does not belong here.

This article is about Euler angles and not their associated rotations. It is OK to have a mention of them but for sure to enter in details is outside the scope of the article. If nobody disagrees, I will remove the paragraph of commutativity, not for being untruth, but for being outside of the scope.--Juansempere (talk) 21:32, 7 October 2010 (UTC)


 * I really don't agree. Both operations are commutative and it should be documented. Can anybody here show me an example where the result depends of the order of the operations?--Guentherwagner (talk) 14:14, 8 October 2010 (UTC)


 * I am of course not happy with the last sentence of the new Euler rotation text, but that is a minor problem when compared with other parts of the article which I could not persuade you to modify. But I will no more fight over that ; I have more than once expressed my detailed advices. Mathematically oriented readers can take a look at my User pages ... Chessfan (talk) 22:27, 8 October 2010 (UTC)