Talk:Euler angles

Mistake in expressions to calculate Euler Angles?

Are these actually correct?

   \alpha= \operatorname{arg}(\ -Z_2\ ,\ Z_1\ )
   \beta = \operatorname{arg}(\ Z_3\ ,\ \sqrt{{Z_1}^2 + {Z_2}^2}\ )
   \gamma=-\operatorname{arg}(\ Y_3\ ,\ -X_3\ ) = \operatorname{arg}(\ Y_3\ ,\ X_3\ ). 

I have just coded them up in java and find that a Z rotation never changes any angles which seems odd.

Instead I took some java from http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToEuler/index.htm which does give sensible results. —Preceding unsigned comment added by 24.82.10.226 (talk) 00:49, 21 September 2010 (UTC)

Could you please format the text? I don't understand you. If what you want is to obtain the angles of a given frame, just write the three vectors as columns and make it equal to the matrix corresponding to your desired convention.

--Guentherwagner (talk) 09:44, 21 September 2010 (UTC)

Rewriting the section "Euler angles as composition of extrinsic rotations"

That section is still not satisfying. First of all it is a (ZXZ) convention, but the order of the matrices is false. (I suppose it is written in matrix algebra and not in some abstract operator writing, because then it would be difficult to justify the transformations ). Second those transformations are parachuted without a single word of explanation. And finally the paragraph about left or right multiplication, which at first reading seems to be an astute shortcut demonstration, is false, that is has no relation with the dual representation of Euler angles. If somebody thinks otherwise please explain the math !

I suggest the following new text (Justified amendments are welcome , but I will not fight about it . Please do not cut my text in pieces.)

... This can be shown to be equivalent to the previous statement :

Let us call (e), (f), (g), (h), the successive frames deduced from the initial (e) reference frame by the successive Euler intrinsic rotations described above. We call u, v, w, t, the successive vectors obtained with that rotation. We note $(x)_{e}$ the column matrix representing a vector x in the frame (e). If necessary we add also a lower index to any matrix we wish to operate in a specific frame. We call $Z(\alpha )$, $X(\beta )$, $Z(\gamma )$ the successive "simple" Euler matrices of our example. Thus we can write when describing the intrinsic operations :

$(1)\qquad Z_{e}(\alpha )=Z(\alpha )\qquad X_{f}(\beta )=X(\beta )\qquad Z_{g}(\gamma )=Z(\gamma )$

When describing the intrinsic Euler rotations in the (e) reference frame we must of course transform the matrices used to represent the rotations. Then by the rules of matrix algebra we get :

$(2)\qquad (t)_{e}=Z''_{e}(\gamma )X'_{e}(\beta )Z_{e}(\alpha )(u)_{e}$

$(3)\qquad X'_{e}(\beta )=Z_{e}(\alpha )X_{f}(\beta )Z_{e}(-\alpha )=Z(\alpha )X(\beta )Z(-\alpha )$

$(4)\qquad Z''_{e}(\gamma )=Z_{e}(\alpha )X_{f}(\beta )Z_{g}(\gamma )X_{f}(-\beta )Z_{e}(-\alpha )=Z(\alpha )X(\beta )Z(\gamma )X(-\beta )Z(-\alpha )$

$(5)\qquad (t)_{e}=[Z(\alpha )X(\beta )Z(\gamma )X(-\beta )Z(-\alpha )][Z(\alpha )X(\beta )Z(-\alpha )]Z(\alpha )(u)_{e}=Z(\alpha )X(\beta )Z(\gamma )(u)_{e}$

$(6)\qquad (R)_{e}=Z(\alpha )X(\beta )Z(\gamma )$

The relation (5) can then of course be interpreted in extrinsic manner as a succession of rotations around the (e) axes.

Again, proper Euler angles ...

Chessfan (talk) 09:58, 23 October 2010 (UTC)

Well, I agree, but I would try to use a lighter notation. For example, the (e) basis is the default basis of the space and it does not need to be repeated always. Also things like $(u)_{e}$ can be simplified because $(u)_{e}=I$ using matrices. For me it is OK to make the change anyway. --Guentherwagner (talk) 21:46, 25 October 2010 (UTC)

I tried to modify the article, but did not succeed, and dont understand why. Sorry, I leave it to you. I am not happy with the idea to use a lighter notation. I think the equations (1) and (2) are very important to understand why combining Euler rotations in matrix algebra is different from combining them in standard way (fixed axes). Also I am very in favor to show explicitly the rotation of an arbitrary body point (u), mainly because the different frames play a double role : they are part of the rotating body, and may be utilized as successive reference frames. That would be obvious if I had explained in detail the matrix transformations. I did not do it because I anticipated your reaction :) . Chessfan (talk) 10:11, 26 October 2010 (UTC)

In a second try I succeeded. Chessfan (talk) 12:20, 26 October 2010 (UTC)

Just another proposal about a small change. About the sentence "We call u, v, w, t, the successive vectors obtained with that rotation". Don't you think it would be more accurate to say "We call u, v, w, t, the successive matrices of the vectors obtained with that rotation"?--Guentherwagner (talk) 17:12, 26 October 2010 (UTC)

But they are indeed geometrical invariant (covariant) objects. The vector $x$ is noted $(x)_{e}$, $(x)_{f}$, in its matricial representations in the frames (e), (f), etc ... We must note that :

$(u)_{e}=(v)_{f}=(w)_{g}=(t)_{h}$

Changing the definition I adopted would not be a small change.

Dont you think that Euler angles/rotations is foremost a purely geometrical question, which can be studied with different mathematical tools : matrices, tensors (very efficient ..., see passive transformations in User talk:Chessfan\work1), quaternions, geometric algebra ? After all those discussions I have the feeling that the question of rotations in Wikipedia is seen mostly in a pure algebric manner which privileges matrix algebra and gives sometimes rise to misunderstandings and thus errors. Of course I am myself deeply influenced by the GA tool. Chessfan (talk) 19:20, 26 October 2010 (UTC)

But there is no need to have only one vector in a matrix. You can refer with your matrix $(t)_{e}$ to the three vectors of the frame at the same time, and the same relationships still hold. And in fact you need to refer to the three vectors at the same time, because to speak of the "Euler angles of a vector" has no sense. Euler angles are only defined for frames. (and you should not use simple vectors in your (2) because then it can be true in infinite ways).--Guentherwagner (talk) 22:01, 26 October 2010 (UTC)

About your second question (if I think that Euler angles are a purely geometrical question) I agree. In fact only the first section of the article should be here, because the rest deals with movements, not with angles. Nevertheless I still think that it is important to show the relationship with rotations, even if a rotation is not an angle, and a small section has to be here, but my point is that for going deeper into rotations, a new article should be started. This should be just about angles, not movements.--Guentherwagner (talk) 22:06, 26 October 2010 (UTC)

I do not understand your critic ; I never spoke of the Euler angles of a vector ! My u vector, which is successively rotated to v, w, t is what we call in French le point courant (perhaps in German laufender Punkt ?) of a rigid body, rigidly associated of course with the moving frame which occupies the successive positions (e), (f), (g), (h). That type of notation and equation is (or was ?) classic in cinematics and dynamics of rigid bodies. Perhaps the development of robotics and animation technics has brought in other habits, which I ignore. That could perhaps explain our frequent misunderstandings. Well, I cannot modify my text in a manner which would deeply change its basic ideas, and its pedagogic (? !) value. If you definitely dont like it, revert it and substitute your own text.

Our discussion reminds me of a remark I should have mentioned long ago. I think that one of the major error causes when people write rotation matrices, is the fact that they obey the natural tendency to write the transformation of the frames like that :

$f_{1}=\cos \alpha \quad e_{1}+\sin \alpha \quad e_{2}+0\quad e_{3}$

etc ... Then they jump to the conclusion that THE rotation matrix is the coefficient matrix they have under their eyes. But that is not true, it is the transposed one. They would never make that error if they wrote equations like (2), or better if they employed tensor algebra. Chessfan (talk) 07:35, 27 October 2010 (UTC)

I read in an old comment the ZXZ matrix given in Goldstein's Classical Mechanics . It is the transpose of the matrix you calculated for the table of matrices ! But perhaps Goldstein was speaking about passive rotations ... Could somebody verify ? Chessfan (talk) 11:39, 27 October 2010 (UTC)

I see now. Of course your expression $(t)_{e}=Z''_{e}(\gamma )X'_{e}(\beta )Z_{e}(\alpha )(u)_{e}$ works for any vector (u) of the body that is transformed into its image (t).

What I meant in my first comment is that if you write the three unitary vectors of the frames in square matrices (T) and (U), whith the components of every vector in columns, the same relationship still holds: $(T)_{e}=Z''_{e}(\gamma )X'_{e}(\beta )Z_{e}(\alpha )(U)_{e}$.

In the beginning I thought that your (t) and (u) were these kind of square matrices (T) and (U). That's why I wrote ""We call u, v, w, t, the successive matrices of the vectors obtained with that rotation". Now I have understood what you mean with (2) and of course my comment is no longer valid.

The only problem I see now in your expressions is that they speak about a vector that gets transformed. Therefore it requires to introduce transformations, and that is again outside the scope of the simple euclidean geometry of the article.

About the rest of your text, I really find difficult to argue about a single subject. Let alone two different subjects in parallel. Just one can get difficult enough. Maybe we can discuss the rest later. I prefer to focus first in simplifying the notation --Guentherwagner (talk) 19:08, 27 October 2010 (UTC)

Another possibility for the notation

What do you think about this notation? I think it is equally clear and shorter. (besides, to use parenthesis for the angles could be confused to an operator transforming an object):

$(1)\qquad (Z_{\alpha })_{e}=(Z_{\alpha })\qquad (X_{\beta })_{f}=(X_{\beta })\qquad (Z_{\gamma })_{g}=(Z_{\gamma })$

When describing the intrinsic Euler rotations in the (e) reference frame we must of course transform the matrices used to represent the rotations. Then by the rules of matrix algebra we get in the (e) basis:

$(2)\qquad (t)_{e}=(Z''_{\gamma })_{e}(X'_{\beta })_{e}(Z_{\alpha })_{e}(u)_{e}$

Now using that for these kind of matrices a rotation of a negative angle is the inverse, and in orthonormal matrices also the transposed, we have the equation for change of basis X' and Z:

$(3)\qquad (X'_{\beta })_{e}=(Z_{\alpha })_{e}(X_{\beta })_{e}(Z_{\alpha })_{e}^{t}=(Z_{\alpha })(X_{\beta })(Z_{\alpha })^{t}$

$(4)\qquad (Z''_{\gamma })_{e}=(Z_{\alpha })_{e}(X_{\beta })_{f}(Z_{\gamma })_{g}(X_{\beta })_{f}^{t}(Z_{\alpha })_{e}^{t}=(Z_{\alpha })(X_{\beta })(Z_{\gamma })(X_{\beta })^{t}(Z_{\alpha })^{t}$

Substituting in the former expression (2):

$(5)\qquad (t)_{e}=[(Z_{\alpha })(X_{\beta })(Z_{\gamma })(X_{\beta })^{t}(Z_{\alpha })^{t}][(Z_{\alpha })(X_{\beta })(Z_{\alpha })^{t}](Z_{\alpha })(u)_{e}=(Z_{\alpha })(X_{\beta })(Z_{\gamma })(u)_{e}$

And therefore:

$(6)\qquad (Z''_{\gamma })_{e}(X'_{\beta })_{e}(Z_{\alpha })_{e}=(Z_{\alpha })(X_{\beta })(Z_{\gamma })$

--Guentherwagner (talk) 23:44, 28 October 2010 (UTC)

Good idea ; I made the change. Chessfan (talk) 08:10, 29 October 2010 (UTC)

About the change of basis

About what you said:

Our discussion reminds me of a remark I should have mentioned long ago. I think that one of the major error causes when people write rotation matrices, is the fact that they obey the natural tendency to write the transformation of the frames like that :

$f_{1}=\cos \alpha \quad e_{1}+\sin \alpha \quad e_{2}+0\quad e_{3}$

etc ... Then they jump to the conclusion that THE rotation matrix is the coefficient matrix they have under their eyes. But that is not true, it is the transposed one. They would never make that error if they wrote equations like (2), or better if they employed tensor algebra. Chessfan (talk) 07:35, 27 October 2010 (UTC)

I would like to add just a little remark there. In fact in the general case it is not the transposed matrix but the inverse. Of course both things are the same while speaking about orthonormal frames, but in the general case both things are quite different.

--Guentherwagner (talk) 17:49, 29 October 2010 (UTC)

Referring to Givens rotations ; other ambiguities ?

Sorry Juansempere, I suppressed the reference to Givens rotation you introduced, but I will not argue with you ; do what you want. I think that reference will not be helpful to the reader and even induce him to false ideas. For instance somebody wrote that Givens rotations have no intrinsic equivalence, which seems false to me ...

I also do not quite understand why an Euler angles table is again introduced there with new ambiguities. I think for example the matrix presented as xzy which is equivalent in intrinsic interpretation (after backswitch of angles) to XZY should be named yzx. The names whether in intrinsic or extrinsic mode should be given in the correct physical rotation order.

By the way I still think it is nonsense to switch the angles, doubling the number of matrices. It would be much clearer to give the full names $X_{\alpha }Z_{\beta }Y_{\gamma }=y_{\gamma }z_{\beta }x_{\alpha }$ . A rotation is defined by its axis and its angle !

Chessfan (talk) 16:41, 6 December 2010 (UTC)

Those matrices can rotate single vectors instead of full frames. Does "intrinsic rotation" make a sense when rotating a single vector? I will put at least a reference to former article.

--Juansempere (talk) 02:51, 8 December 2010 (UTC)

The answer is yes because I rotate a body point, that is a whole set of vectors composing a rigid body. You can always make that hypothesis. The structure of the matrix does not matter. The fact they are Givens matrices only permits you to visualize immediately the successive rotation axes.Chessfan (talk) 08:07, 8 December 2010 (UTC)

Rotating a single vector there is not a whole set of vectors. Anyway this matter should be discussed in the Givens rotations article, not here.--Juansempere (talk) 10:18, 8 December 2010 (UTC)

True, but I do not understand how you rotate a single vector in matrix algebra without defining some reference frame ? Then you have your minimal set of vectors ... Chessfan (talk) 12:41, 8 December 2010 (UTC)

I have replied here [1]--Juansempere (talk) 13:54, 13 December 2010 (UTC)

Matrix orientation again.

First, coming back to Euler angles, I revised an obvious typo in equation (3) section " Euler angles as composition of extrinsic rotations " .

Second,I am still not happy with the " Matrix Orientation " text (the upper paragraphs) which I accepted reluctantly several months ago. I think the wording is rather confusing for the reader who hopes to find here a quick answer for the choice of a specific rotation matrix.

One should start from the relations (6) of the above mentioned section :

$(6)\qquad (R)_{e}=(Z''_{\gamma })_{e}(X'_{\beta })_{e}(Z_{\alpha })_{e}=(Z_{\alpha })(X_{\beta })(Z_{\gamma })$

which correspond strictly to the rotation matrices mentioned in the existing text, when replacing the $\gamma ,\beta ,\alpha$ angles respectively with $\theta _{3},\theta _{2},\theta _{1}$ .

Then one should explain that the intermediate member of those relations corresponds to the intrinsic composition of (active) rotations, which is strictly equivalent to the extrinsic composition of rotations in the final member. One should perhaps add that of course the last member , which consists of elemental matrices, is to be used to calculate the global rotation matrix.

Finally one should tell which convention name should be given to the studied example : (YXZ) or (ZXY) ? I am myself still not clear at that. Once I suggested without success something like (YXZ) equivalent to (zxy) in the other interpretation. I leave it to Euler angles specialists to fix that, hopefully unambiguously ! And do not forget to check that everything is in concordance with the matrix table. Chessfan (talk) 15:48, 13 January 2012 (UTC)

I am truely sorry , but the matrix table is still false ! I should have seen that long ago, but I was too tired by endless discussions (please look first at the math and then calculate ...).

First of all one should not introduce the lefthanded matrices (a new ambiguity which I forgot !) : it would be enough to note that lefthanded conventions can be obtained by changing the signs of all sinuses.

Then one should take a close look at the section Euler angles composition of extrinsic rotations where the theoretical formula for :

: $\qquad Z''_{\gamma }X'_{\beta }Z_{\alpha }=Z_{\alpha }X_{\beta }Z_{\gamma }$

is clearly demonstrated. By substituting the $\theta$ angles we see that we get the following $ZXZ$ matrix , which is now falsely indicated as a lefthanded one :

!ZXZ |${\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}\\c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}\\s_{2}s_{3}&c_{3}s_{2}&c_{2}\end{bmatrix}}$

Thus I suppose that the lefthanded matrices are the right(!) ones for righhanded conventions (could somebody check ...).

By the way we have here an obvious example of how confusing a convention simply named ZXZ can be.

Chessfan (talk) 17:11, 19 January 2012 (UTC)

I calculated the above mentioned matrix in User:Chessfan/work1 . I verified it and I am now 100% sure it is right, that is for an active rotation in righthanded frames.

Chessfan (talk) 00:07, 23 January 2012 (UTC)

The ZYZ matrix falsely called lefthanded is in agreement, after transposition, with the same matrix calculated in passive rotation by http://anorganik.uni-tuebingen.de/klaus/nmr/index.php?p=conventions/euler/euler . That gives another coherence example and reinforces my proposal to simplifye the table of matrices by suppressing the now called righthanded matrices and renaming the now falsely lefthanded called matrices. I will not do that unless somebody else checks my affirmation.

Chessfan (talk) 19:59, 29 January 2012 (UTC)

A general method for interpreting simple and composed rotation matrices.

It seems that a vast majority of authors and professors forget, when writing about rotation matrices, to mention a very basic fact : a rotation matrix is or should be a mathematical tool which transforms not vectors but coordinates of vectors. Secondly they often forget to mention if they speak about active rotations or passive ones.

An active rotation is when rotating an e-frame to an f-frame you rotate simultaneously a vector attached to the e-frame (that is a rigid body rotation), and you want to know the coordinates of the rotated vector in the initial frame. Then you get following relations in tensorial notation :

$(1)\qquad f_{j}=\alpha _{j}^{i}e_{i}\qquad x=x^{i}e_{i}\qquad \xi =x^{j}f_{j}=\xi ^{i}e_{i}$

thus :

$(2)\qquad \xi ^{i}=\alpha _{j}^{i}x^{j}$

Now we can write in matricial language :

$(3)\qquad (\xi )=A(x)\qquad A=|\alpha _{j}^{i}|$

where $(i,j)$ are respectively the row and the column indices, and $(x),(\xi )$ are 1-column matrices. It is the relation (3) and not the first relation (1) we must use to define the rotation matrix. One finds out easily by picking out the individual $f_{j}$ vectors that the j-th column of the matrix is constituted by the coordinates of $f_{j}$ in the e-frame. One has the impression that many authors write down horizontally (of course ...) the first relation (1), and then conclude falsely that the transposed matrix $A^{T}$ is the valid rotation matrix !

A simple example is given by an active rotation around the Z-axis by an angle $\alpha$ positively counted in counterclock direction. The rotation matrix is :

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

What about a passive rotation ? We start with the same above mentioned rotation of the frame $(e)$ ⇒$(f)$. Now we do not rotate the vector $x$ attached to $(e)$, but we want to know the coordinates of $x$ in $(f)$. We can write :

$(4)\qquad x=x^{i}e_{i}=\xi ^{j}f_{j}=\xi ^{j}\alpha _{j}^{i}e_{i}$

$(5)\qquad x^{i}=\alpha _{j}^{i}\xi ^{j}\qquad (x)=A(\xi )$

$(5)\qquad (\xi )=A^{-1}(x)=A^{T}(x)$

$A^{T}={\begin{bmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{bmatrix}}$

We are ready now to tacle the confusing question of Euler angles and rotations.

Let us suppose somebody tells us a global active rotation matrix can be represented by the following matrix product :

$(6)\qquad R=ABC$

That means that the coordinates in the initial frame of a vector $(x)$ which we rotate by $R$ will be transformed to $(\xi )$ in the same initial frame by the matricial relation :

$(7)\qquad (\xi )=R(x)=ABC(x)$

That means, or should mean in somebodys mind, as the $A,B,C$ matrices cannot be anything else than elemental matrices, that we have an extrinsic rotation executed by successive rotations around the initial frame basis vectors in the order C , then B , then A . We have no choice, as each rotation axis is predetermined by the structure of the matrix ! And finally that means that the same rotation matrix can be interpreted as composed by intrinsic rotations around moved axis, but in the inverse order A ,B',C". Of course the $B',C''$, matrices are no more elemental, and will not appear explicitly. Matricially that can be written as follows :

$(8)\qquad (\xi )=R(x)=ABC(x)=(ABCB^{T}A^{T})(ABA^{T})A(x)=C''B'A(x)$

That begins to look quite simple. Now we are able to verify what is told in the books !

What if we have to interprete passive Euler rotations ? To be able to compare with the active rotations we choose to use the same A, B, C matrices as before -- that means the rotation (order and angles) of the successive frames are unchanged --, but of course now for each elemental rotation we have a relation of type :

$(9)\qquad (\xi )=A^{T}(x)$

To be sure not to make errors it will be safe to write explicitly the rotation of the frames :

$(10)\qquad h_{m}=\gamma _{m}^{k}\beta _{k}^{j}\alpha _{j}^{i}e_{i}$

$'11)\qquad x^{i}e_{i}=\xi ^{m}h_{m}=\xi ^{m}\gamma _{m}^{k}\beta _{k}^{j}\alpha _{j}^{i}e_{i}$

$(12)\qquad x^{i}=\xi ^{m}\gamma _{m}^{k}\beta _{k}^{j}\alpha _{j}^{i}=\alpha _{j}^{i}\beta _{k}^{j}\gamma _{m}^{k}\xi ^{m}$

The reversing of the order is necessary to comply to matrix multiplication rules (licol ...) :

$(13)\qquad (x)=ABC(\xi )$

$(14)\qquad (\xi )=C^{T}B^{T}A^{T}(x)=(ABC)^{T}(x)=R^{T}(x)$

So we happily find the same transposition relation between the global rotation matrices as between the simple matrices when we interpret the same physical rotation in alternatively active or passive mode.

But we have still not told the end of the story. Indeed it is important to say that in (14) we have in $(\xi )$ the coordinates of the $x$ vector -- attached to (e)-- , observed in the (h) frame, which correspond to a passive rotation . But we could also interprete (14) in a quite different manner. By comparing with (8) we could imagine a composed rotation from (e) to a frame (h')≠ (h) composed by intrinsic rotations with the successive order and values $(-\gamma ),(-\beta ),(-\alpha )$. Then $(\xi )$ would represent the coordinates in (e) of the negatively rotated vector $x$ (active intrinsic rotation). The same vector $\xi$ can also be obtained by rotating extrinsicly the vector $x$ around the basis vectors of (e) successively in the order and with values $(-\alpha ),(-\beta ),(-\gamma )$.

When working with Euler angles be precise, look at the math, and do not spare the indices. Most of the obscurities and misunderstandings arise from the fact that the authors do not follow that advice.

Chessfan (talk) 16:33, 27 January 2012 (UTC)

Contradiction in images.

There is a contradiction in the images illustrating the equivalence of intrinsic and extrinsic rotations. In the second image (extrinsic) the names Φ and Ψ should be switched !

Chessfan (talk) 11:12, 27 January 2012 (UTC)

This is not a contradiction, but conflicting variable substitutions: (φ,θ,ψ)=(−60°, 30°, 45°) for co-moving axes and (φ,θ,ψ)=(45°, 30°, −60°) for fixed axes. Incnis Mrsi (talk) 11:53, 27 January 2012 (UTC)

I think that variable substitutions should be reverted. Chessfan (talk) 13:38, 10 February 2012 (UTC)

Static, Intrinsic, and Extrinsic

The article in its current form uses the adjective "static" twelve times in a manner that is not particularly clear. The meaning of the word in this context appears to be technical, and should be clarified upon first usage. Personally, I've never seen static used in a discussion of Euler angles. The "static" definition is contrasted with definitions based on the composition of three rotations. Similarly, "intrinsic" and "extrinsic" are terms that require precise definitions. I think these terms only have meaning if one describes the coordinate systems involved as either "lab-fixed"/"stationary" or "body-fixed"/"rotating". The primary difference between the two systems is the handedness of the rotation: a counter-clockwise rotation of the body (an extrinsic rotation) will appear to be a clockwise rotation of the room from the perspective of an observer on the rotating body. Thus, an extrinsic definition will define the angles as counter-clockwise rotations from the stationary reference frame to the rotating reference frame, and an intrinsic definition will define the angles as clockwise rotations from the rotating reference frame to the stationary reference frame, in reverse order from the extrinsic definition. 99.11.197.75 (talk) 18:49, 7 February 2012 (UTC)

You are probably right with what you say about the adjective "static" ; but what you say about extrinsic and intrinsic rotations is false. You describe the double interpretation of a same rotation as an active or a passive rotation. The fact that an active - that is a rigid body rotation - can be interpreted as a succession of intrinsic rotations - around axis which move with the body - , but also as a succession of rotations (called extrinsic) around the axis defined by the initial position of the body, is mathematically related to the fact that if you apply a rotation operator $R_{2}$ to an operator $R_{1}$ you get an operator $R_{3}$ whose angle remains the same than $R_{1}$ but whose axis has been rotated by $R_{2}$. You will find in the literature very sophisticated demonstrations based on rotation generators, but that can be done in geometric algebra in one line of calculation ! Chessfan (talk) 10:27, 10 February 2012 (UTC)

Your point about intrinsic and extrinsic is well taken. However, I still think the prose should explain the meaning of extrinsic and intrinsic instead of just assuming that the meaning is clear from the standard dictionary definition. If I am not mistaken, it is the decomposition of a rotation into three successive rotations that can be extrinsic or intrinsic. The composite transformation is neither extrinsic or intrinsic. Similarly, the different conventions are not themselves static, extrinsic, or intrinsic, it is the descriptions of the conventions that are static, extrinsic, or intrinsic. Overall, I think this article spends way too much time focusing on different conventions, which introduces a lot of language that makes the topic sound more complicated than it actually is. Some of the content on the different conventions in use is very useful, but the different conventions should be presented after the fundamental content. 99.11.197.75 (talk) 21:53, 20 February 2012 (UTC)

Move Tait–Bryan angles to simplify discussion of Euler angles

I think Tait-Bryan angles should have their own page. I realize that some of the content would be redundant, but moving Tait-Bryan angles to another page would simplify the discussion of Euler angles. For example, the adjective "proper" could be dropped eleven times. Also, for proper Euler angles I suspect that only the ZXZ and ZYZ conventions are particularly common, while there is much less agreement about the names and signs of the three angles. If this is true, it would be nice factoid to include. The ZYZ convention is presented in Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 198-200, 1985. and is described as the choice "made by workers in the area of group theory and the quantum theory of angular momentum." I would add molecular physics to that list. Also, in what fields are the body-fixed coordinates XYZ and the space-fixed coordinates xyz? In molecular physics, it is usually the other way around. Finally, it would be nice to point out that two of the Euler angles are the spherical coordinates of the Z-axis. 99.11.197.75 (talk) 01:13, 21 February 2012 (UTC)

There is also a slight problem with the static definition. It is insufficient to define <math>\alpha<\alpha> as the angle between x and the line of nodes, since <math>\alpha<\alpha> makes two different angles with the line of nodes, describing two different rotations. Same is true for <math>\gamma<\alpha>. In order to make the line of nodes an axis, as shown in the opposing figure, you need to know the sign of <math>\alpha<\alpha>. This can be done by projecting Z onto xy, and measuring the angle between this projection and -y. This is a little simpler in the ZYZ convention, where \alpha is the angle between the x and the projection of Z on xy. A figure such as the first figure for the ZYZ convention almost always projects Z onto xy.