Talk:Euler angles/Archive 3

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Archive 3

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)

True, the global composite transformation is neither intrinsic nor extrinsic, but when you write $R=ABC$, where $A,B,C$ are elemental matrices you have no choice in the interpretation, as I tried to show above in the sections "Matrix orientation again" and "A general method for interpreting simple and composed rotation matrices". It is extrinsic in the order C,B,A and intrinsic in the order A,B',C". Now about the conventions, I suppose the problems originate mainly in the fact that the professionals have their own rules and are reluctant to adopt unambiguous names for the conventions, which would be understandable for everybody. I have myself worked for months on that Euler angles question, only to see now that the displayed matrices are still falsely interpreted, and nobody seems to care ... Chessfan (talk) 17:19, 23 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 $\alpha$ as the angle between x and the line of nodes, since $\alpha$ makes two different angles with the line of nodes, describing two different rotations. Same is true for $\gamma$. In order to make the line of nodes an axis, as shown in the opposing figure, you need to know the sign of $\alpha$. 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.

(fixed the formatting of your second paragraph). In reply to the first point, they were separate not long ago but it was decided to merge them as there was a lot of overlap and at the time were too many articles on rotational systems. The content was as at yaw, pitch and roll, at e.g. this revision. See also the talk page: Talk:Yaw, pitch, and roll.--JohnBlackburne^words_deeds 01:54, 21 February 2012 (UTC)

Well, I would not recommend that Tait-Bryan angles go back to yaw, pitch and roll, but I still think they are distinct enough to be separated from Euler angles. Many of the applications discussed in the article are applications of Tait-Bryan angles, and proper Euler angles would be noticeably less useful in describing the attitude of an aircraft or the position of a robotic arm. On the other hand, the use of Tait-Bryan angles would unnecessarily complicate the quantum theory of angular momentum. I would argue that these differences in applications underscore the need to treat these systems as separate topics. Anyway, as long as the Euler angles, Tait-Bryan angles, and yaw, pitch, and roll pages are adequately cross-linked, I don't think the separation would create too many articles.

Anyway, there are still too many articles on rotational systems. But, the Euler angles and Tait-Bryan angle pages would be leaves in a hierarchical organization of this tree of articles. I don't really think you can have too many leaf-articles, as long as each article is distinct. The problem right now, in my opinion, is that there are too many grand surveys on the subject, like Rotation (mathematics) and Rotation formalisms in three dimensions. — Preceding unsigned comment added by 99.11.197.75 (talk) 03:39, 21 February 2012 (UTC)

I also find confusing to deal with Euler and Tait-Bryan at the same time. Could be better to have a specific section for Tait-Bryan angles? I propose to move them to the 4th. section. The index would turn into something like this:

1 Definition (Just for proper Euler angles)

2 Geometric derivation (Just for proper Euler angles)

3 Relationship with physical motions (Just for proper Euler angles)

4 TAIT-BRYAN ANGLES -> Here a formal introduction is made for them.

4 Relationship to other representations (both Euler and Tait-Bryan)

5 Properties (both Euler and Tait-Bryan)

6 Higher dimensions (both Euler and Tait-Bryan)

7 Applications

Anybody agrees? Anybody disagrees?

--Juansempere (talk) 15:58, 4 March 2012 (UTC)

On a second thought, this could work better:

1 Proper Euler Angels

- Definition

- Angles signs and ranges

- Geometric derivation

2 TAIT-BRYAN ANGLES -> Here a formal introduction is made for them.

3 Relationship with physical motions (both Euler and Tait-Bryan)

4 Relationship to other representations (both Euler and Tait-Bryan)

5 Properties (both Euler and Tait-Bryan)

6 Higher dimensions (both Euler and Tait-Bryan)

7 Applications

I am going to do it now. If somebody dissagrees, just undo the changes.

--Juansempere (talk) 18:28, 5 March 2012 (UTC)

Matrix orientation (new proposal).

I suggest to substitute the following text to the existing one (do not hesitate to correct my English ...):

Using the equivalence between Euler angles and rotation composition, it is possible to change to and from matrix convention. First, active rotations and the right handed rule for the positive sign of the angles are assumed. As explained above each global rotation matrix can be interpreted as a composition of intrinsic rotations, that is around the axes of the moving body (the moving reference system), or as a composition of extrinsic rotations around the fixed world axes. We choose to name the different conventions by privileging the latter, for a simple and obvious reason : in a matrix composition like

$(1)\qquad R=ABC$

the composing matrices are elemental, and thus pick out automatically successively different fixed world axes in the order C,B,A. We, of course, do not use the complicated intrinsic expression C"B'A to calculate $R$.

To suppress all existing ambiguities and facilitate the understanding between different professional users we introduce a specific denomination principle more detailed than the existing ones. We introduce following rules :

1/ When naming a global rotation we employ the same names in the same order than the matrix formula used to calculate the global matrix. If $R=Z_{\alpha }X_{\beta }Z_{\gamma }$ we name it $Z_{\alpha }X_{\beta }Z_{\gamma }$. The consequence of that is that the name illustrates automatically the extrinsic interpretation.

2/ For each matrix we designate explicitly the axis, which we index with the angles (we abbreviate $\theta _{1},...$ with $1,2,3)$. That may seem redundant but in fact is not, because there is no valid reason to associate an alphabetic or numeric order to the angles, when the order of rotations is already fixed by rule 1 .

3/ We recall something that should be obvious, that is the composing and composed matrices are supposed to act on the coordinates of vectors defined in the initial fixed world axis and give as a result the coordinates in the same system of a rotated vector, rotation primarily interpreted extrinsicly, but to which we can give globally an equivalent intrinsic interpretation.

With those rules we get the following matrix table :

$X_{1}Z_{2}X_{3}$	${\begin{bmatrix}c_{2}&-c_{3}s_{2}&s_{2}s_{3}\\c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}}$	$X_{1}Z_{2}Y_{3}$	${\begin{bmatrix}c_{2}c_{3}&-s_{2}&c_{2}s_{3}\\s_{1}s_{3}+c_{1}c_{3}s_{2}&c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}\\c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}\end{bmatrix}}$
$X_{1}Y_{2}X_{3}$	${\begin{bmatrix}c_{2}&s_{2}s_{3}&c_{3}s_{2}\\s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}\\-c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}}$	$X_{1}Y_{2}Z_{3}$	${\begin{bmatrix}c_{2}c_{3}&-c_{2}s_{3}&s_{2}\\c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}\\s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}\end{bmatrix}}$
$Y_{1}X_{2}Y_{3}$	${\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}\\s_{2}s_{3}&c_{2}&-c_{3}s_{2}\\-c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}}$	$Y_{1}X_{2}Z_{3}$	${\begin{bmatrix}c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}\\c_{2}s_{3}&c_{2}c_{3}&-s_{2}\\c_{1}s_{2}s_{3}-c_{3}s_{1}&s_{1}s_{3}+c_{1}c_{3}s_{2}&c_{1}c_{2}\end{bmatrix}}$
$Y_{1}Z_{2}Y_{3}$	${\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}\\c_{3}s_{2}&c_{2}&s_{2}s_{3}\\-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}}$	$Y_{1}Z_{2}X_{3}$	${\begin{bmatrix}c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}\\s_{2}&c_{2}c_{3}&-c_{2}s_{3}\\-c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}\end{bmatrix}}$
$Z_{1}Y_{2}Z_{3}$	${\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}\\c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}\\-c_{3}s_{2}&s_{2}s_{3}&c_{2}\end{bmatrix}}$	$Z_{1}Y_{2}X_{3}$	${\begin{bmatrix}c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}&s_{1}s_{3}+c_{1}c_{3}s_{2}\\c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}\\-s_{2}&c_{2}s_{3}&c_{2}c_{3}\end{bmatrix}}$
$Z_{1}X_{2}Z_{3}$	${\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}\\c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}\\s_{2}s_{3}&c_{3}s_{2}&c_{2}\end{bmatrix}}$	$Z_{1}X_{2}Y_{3}$	${\begin{bmatrix}c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}\\c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}\\-c_{2}s_{3}&s_{2}&c_{2}c_{3}\end{bmatrix}}$

When we substitute passive rotations to the active ones we have only to transpose the matrix table. Then each matrix transforms the initial coordinates of a vector remaining fixed to the coordinates of the same vector measured in the globally rotated reference system (same rotation axis, same angles).

When we want to establish a matrix table for left-handed sign convention, we need only change the sign of all the sinuses in the tables.

Chessfan (talk) 18:31, 25 February 2012 (UTC)

I strongly think that there is no need to discuss here about rotations composition. Angles are static concepts and they do not require the concept of rotations. Anyway, I agree with your change. Do it if you want.

Apart from that, rotations are for sure related to Euler angles but they are a different thing, and several mathematical tools exist to perform the products. For example check this one: [2]. I propose to remove at least one of the two tables with matrix products. Maybe it would be good to remove both.

--Juansempere (talk) 19:17, 3 March 2012 (UTC)

Thank you for your answer, and the little math program. It is very useful, but I agree only with the extrinsic interpretation. That will become clear if you read my math texts.

I think before moving the matrix table a more detailed discussion is necessary with you and other experts in Euler angles. As I told you two years ago I am not an expert ; I simply tried to understand what Iread, but encountered a lot of difficulties. I tried to solve them. My final conclusions are given in the three sections I initiated above.

I do not claim having found THE best presentation of Euler angles, but I think my approach could have some pedagogical value at least for beginners.

Let me add a few remarks :

-- If somebody tells you $R=ABC$ is the right global matrix, and gives you the detailed expression of the elemental matrices $A,B,C$ , you should have of course no interpretation problem for $R$. But if one tells you : the matrix corresponding to the $ZXZ$ convention is $R$ then you do not know which angle corresponds to which of the unknown composing matrices. More, without some additional rule impossible to memorize, you cannot even tell if the convention $XYZ$ correspond to $R=XYZ$ or $R=ZYX$.

-- Some people might also be confused by the conflict of order between operator language and matricial language. When studying active rotations the opposite intrinsic and extrinsic operator orders are equivalent to the same elemental matrices order, which of course are extrinsic (by construction).

-- My approach would become meaningless if my choices of naming the conventions (with angle indices) were not respected. Then it would be better not to include my proposal in the article. But remember the article as it is now in its section "matrix orientation" is false. Something should be done.

Chessfan (talk) 15:19, 4 March 2012 (UTC)

As I told you, I agree with your names convention, and I also agree with your proposed change. Of course, I cannot speak for others. People normally do not enter in the talk page nor they post their points of view. Maybe you could wait some time and make the change if nobody disagrees. --Juansempere (talk) 16:04, 4 March 2012 (UTC)

If you don't mind, I will introduce your notation into the tables in the meantime. I think is more clear.

--Juansempere (talk) 18:40, 5 March 2012 (UTC)

Proposal for removing the "left-hand" table

As Chessfan says, "When we want to establish a matrix table for left-handed sign convention, we need only change the sign of all the sinuses in the tables".

The second table is therefore redundant. I propose to remove it.

--Juansempere (talk) 19:14, 5 March 2012 (UTC)

I agree but do not forget that in the existing text the two tables were switched. I think you should first introduce my new text, then eventually proceed to other modifications.

Chessfan (talk) 00:10, 6 March 2012 (UTC)

I see, now the tables are right, but it does not fit with the introductory text. Do you have a problem with the text I proposed ? We sit between two chairs ... !

Chessfan (talk) 17:12, 6 March 2012 (UTC)

I have no problem with your text. I was expecting you would do the change. If you want I will do it. Just tell me --Juansempere (talk) 23:28, 6 March 2012 (UTC)

I made the change. Feel free to make adjustments, and perhaps remove the left-handed table. Chessfan (talk) 07:49, 7 March 2012 (UTC)

I checked our $Z_{1}X_{2}Z_{3}$ matrix with the first matrix cited in Wolfram MathWorld. Theirs is the transposed of ours, which is logical because, without telling it ..., they operate passive rotations. I think they follow Goldstein ... Chessfan (talk) 20:30, 7 March 2012 (UTC)

Well, one must be very careful in comparisons ! In fact Wolfram starts with $Z_{1}^{T}X_{2}^{T}Z_{3}^{T}$ which corresponds to our $(Z_{3}X_{2}Z_{1})^{T}$. Thus to compare one must also swich 1 and 3 . Which by the way proves how useful the indices are. Chessfan (talk) 23:20, 9 March 2012 (UTC)

I hadn't realized. Thanks.

--Juansempere (talk) 16:06, 23 March 2012 (UTC)

Please Review References and Contents of this Page

On a subject so well understood and written about, why are we including such dubious references to unpublished personal documents, a series of slides for grammer school children, some online pages at "20kweb.com"?

The section called "Mathematical Description" seems to references some unpublished document by "Paul Verner". The actual reference is some unpublished draft by "Dr. Paul Berner". Either way, its someone's unpublished personal document (unpublished documents are not references for wikipedia), which is unnecessary when there are hundreds of textbooks about this subject.

"Unpublished Draft: Orientation, Rotation, Velocity, and Acceleration and the SRM, Paul Verner" at "http://www.sedris.org/wg8home/Documents/WG80462.pdf"

Later text in the aircraft attitude section uses slides for grammer school children (grades K-12 educational outreach), and used as a argument: "Given this setting, the rotation sequence from xyz to XYZ is specified by and defines the angles yaw, pitch and roll[2][3][4] as follows:". But if you look at the grammer school slides, they do not define any angles in these slides. See http://www.grc.nasa.gov/WWW/K-12/airplane/yaw.html.

24.6.16.141 (talk) 22:21, 21 February 2010 (UTC)

Has been fixed now: as the slides don't show the angles themselves, but respective rotations, the citations have been moved there. --Qniemiec (talk) 23:43, 21 July 2010 (UTC)

Error in the last line of the section Aircraft Attitude

See comparison of versions for the change introduced by me. The details underlying the correction are as follows: the rotation of a point is the inverse (i.e. the transpose) of the rotation of an axis. So, $(R_{x}(\phi )R_{y}(-\theta )R_{z}(-\psi ))^{T}=R_{z}(-\psi )^{T}R_{y}(-\theta )^{T}R_{x}(\phi )^{T}=R_{z}(\psi )R_{y}(\theta )R_{x}(-\phi )$. V madhu (talk) 03:45, 25 September 2009 (UTC)

Aircraft attitude - inconsistency between text and image

The text uses a north-east-down set of space axes, but the image shows something-something-up. Clearly either option is legitimate, but in the interests of clarity would it be better to use a single choice consistently? Some of the rotations in the image seem to be left handed too. Djr32 (talk) 23:13, 7 March 2010 (UTC)

Yes, this should be corrected as fast as possible. The z-axis of the NED world frame must point Down! As far as it concerns the rotations, it's not so easy to draw their direction in a way that can't be misunderstood. The picture should be re-drawn more precisely and with great care not to introduce new ambiguities (or thrown out, as I've added recently a series of 5 pictures, showing amongst others also aircraft attitudes). --Qniemiec (talk) 23:59, 21 July 2010 (UTC)

Pitch axis redirect

Perhaps a redirect from Pitch axis to Aircraft principal axes would be more appropriate than here since the term actually appears in that article, as well as being labeling in an image demonstrating the aircraft principal axes. The term is not mentioned in this article and I assume would come up only in regards to these angles being measured from aircraft principal axes, one of which is pitch. Hyacinth (talk) 19:01, 6 April 2010 (UTC)

"Yaw, Pitch and Roll" are used in abstract mathematical contexts and in more concrete ones such as navigating planes and ships: from where come their names and their preferred ordering. Frankly the mathematics articles are a mess and could do with more rationalisation: there's a lot of overlap between the different articles describing the different ways to do rotations in 3D, so a lot of links between them.--JohnBlackburne^words_deeds 19:25, 6 April 2010 (UTC)

Agreed. I believe the articles should be merged. See below. --Yoderj (talk) 17:25, 21 September 2010 (UTC)

Proposed Merger with Euler Angles

I propose to merge this article Yaw, pitch, and roll, which is about Tait-Bryan angles, into Euler Angles. Tait-Bryan angles are a special case of Euler angles, and both can be understood best when studied together. Futhermore, this article is called "Yaw, pitch, and roll" which applies equally well to Euler angles. --Yoderj (talk) 14:42, 21 September 2010 (UTC)

From a logical point of view it seems the right thing to do. On the other hand, both articles are specialized in a point of view, being "yaw, pitch and roll" more focused in practical applications and there are several of them. If we merge them the result will be huge, and wikipedia policies suggest to split huge articles in two. I am not sure that the merge would be a good idea.--Guentherwagner (talk) 18:29, 21 September 2010 (UTC)

This article should be splitted in two parts, and the part referring to the angles merged into Euler angles. The part referring to the different axis convention should be moved to a new article about axis conventions.--Juansempere (talk) 18:56, 5 October 2010 (UTC)

I have created an article named Axes conventions with the appropiate content from this article and some other frames we use at spacecraft attitude descriptions. I would appreciate if people helps me to gather there more information about possible conventions.--Juansempere (talk) 08:42, 9 October 2010 (UTC)

Bad readability

I agree that the article needs to be rewritten. It contains useful info, but was sloppily written. Too many sentences are not clear or difficult to understand, as they take too much for granted. Some others are lengthy and redundant. For instance, see the difference between these two versions of section Classic definition:

April 13	April 15 (rewritten by me)
The geometrical definition (referred sometimes as static) is based on a reference frame and one whose orientation we want to describe, first we define the line of nodes (N) as the intersection of the xy and the XY coordinate planes (in other words, line of nodes is the line perpendicular to both z and Z axis). Then we define its Euler angles as: α (or $\varphi$) is the angle between the x-axis and the line of nodes. β (or $\theta$) is the angle between the z-axis and the Z-axis. γ (or $\psi$) is the angle between the line of nodes and the X-axis.	The geometrical definition (referred sometimes as static) of the Euler angles is based on the axes of the above-mentioned (original and rotated) reference frames and an additional axis called the line of nodes. The line of nodes (N) is defined as the intersection of the xy and the XY coordinate planes. In other words, it is a line passing through the origin of both frames, and perpendicular to the plane zZ, on which both z and Z lie. The three Euler angles are defined as follows: α (or $\varphi$) is the angle between the x-axis and the N-axis. β (or $\theta$) is the angle between the z-axis and the Z-axis. γ (or $\psi$) is the angle between the N-axis and the X-axis.
Unless otherwise stated, this article will use the convention described in the adjacent drawing, usually named Z-X’-Z’’. To clarify the adjacent drawing, it should be noted that α rotates around the z-axis, β rotates around the N-axis (i.e. X’), and γ rotates around the Z-axis (i.e. Z’’), hence the name Z-X'-Z’’.	This definition implies that: α represents a rotation around the z-axis, β represents a rotation around the N-axis, γ represents a rotation around the Z-axis. As shown in the figure, N can be also described as the orientation of x after the first elemental rotation (by an angle α about the z-axis). This is the reason why N is sometimes also denoted x’, or X’ (see figure). The three axes of rotation can be described as z-N-Z, or z-x’-z’’ (where x’ is x after the first rotation, and z’’ is z after the second rotation), which becomes Z-X’-Z’’ if the axes of the original frame are denoted as X, Y, and Z, rather than x, y, and z.

I am not implying that my version is fully fixed. The content is still incomplete (it gives the impression that z-x’-z’’ is the only "classic" sequence; what about y-z’-y’’, etc.?). However, at least now it is clearly written.
Paolo.dL (talk) 20:30, 15 April 2013 (UTC)

Prime (symbol) may not be U+2019

At some point since I edited the article last time, U+2019 ’ RIGHT SINGLE QUOTATION MARK characters were injected for x′ and x″. Please, do not use this substitute for prime, and use a single character U+2033 ″ DOUBLE PRIME instead of two consequential ones. Incnis Mrsi (talk) 08:59, 23 April 2013 (UTC)

I did it. Thanks. Paolo.dL (talk) 11:32, 23 April 2013 (UTC)

Misdirection

The article starts with the warning "This article is about the Euler angles used in mathematics. For the use of the term in aerospace engineering, see flight dynamics." But that article has no mention of Euler angles. 68.96.49.125 (talk) 05:34, 7 April 2013 (UTC)

Fixed. Redirected to Rigid body dynamics, --Juansempere (talk) 22:25, 2 May 2013 (UTC)

theta of Taitbrianzyx.svg should be negative!

Theta angle of Taitbrianzyx.svg should be negative! — Preceding unsigned comment added by JiewuLu (talk • contribs) 20:18, 7 September 2013 (UTC)

Messy proof

Conversion between intrinsic and extrinsic rotations

Let us call (e), (f), (g), (h), the successive frames deduced from the initial (e) reference frame by the successive intrinsic rotations described above. We call u, v, w, t, the successive vectors obtained with that rotation. We write $(x)_{e}$ for 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 rotations of our example. Thus we can write when describing the intrinsic operations: $(1)\qquad (Z_{\alpha })_{e}=(Z_{\alpha })\qquad (X_{\beta })_{f}=(X_{\beta })\qquad (Z_{\gamma })_{g}=(Z_{\gamma })$ When describing the intrinsic 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: $(2)\qquad (t)_{e}=(Z''_{\gamma })_{e}(X'_{\beta })_{e}(Z_{\alpha })_{e}(u)_{e}$ $(3)\qquad (X'_{\beta })_{e}=(Z_{\alpha })_{e}(X_{\beta })_{f}(Z_{\alpha })_{e}^{t}=(Z_{\alpha })(X_{\beta })(Z_{\alpha })^{t}$ $(4)\qquad (Z''_{\gamma })_{e}=(Z_{\alpha })_{e}(X_{\beta })_{f}(Z_{\gamma })_{g}(X_{\beta })_{f}^{t}(Z_{\alpha })_{e}^{t}=(Z_{\alpha })(X_{\beta })(Z_{\gamma })(X_{\beta })^{t}(Z_{\alpha })^{t}$ $(5)\qquad (t)_{e}=[(Z_{\alpha })(X_{\beta })(Z_{\gamma })(X_{\beta })^{t}(Z_{\alpha })^{t}][(Z_{\alpha })(X_{\beta })(Z_{\alpha })^{t}](Z_{\alpha })(u)_{e}=(Z_{\alpha })(X_{\beta })(Z_{\gamma })(u)_{e}$ $(6)\qquad (R)_{e}=(Z''_{\gamma })_{e}(X'_{\beta })_{e}(Z_{\alpha })_{e}=(Z_{\alpha })(X_{\beta })(Z_{\gamma })$ The relation (5) can then of course be interpreted in extrinsic manner as a succession of rotations around the (e) axes.

This proof is meant to show that the intrinsic rotations x-y’-z’’ by angles α, β, γ are equivalent to the extrinsic rotations z-y-x by angles γ, β, α. It is supposed to be valid for both proper Euler angles and Tait-Bryan angles. However, it deals only with proper Euler angles (z-x’-z’’). The notation is quite messy and unusual. The text is sloppily written.

Proofs are not required in Wikipedia articles, so I removed it from Euler angles#Conversion between intrinsic and extrinsic rotations, but it would be nice if some editor rewrited it with clean notation and cristal clear text.
Paolo.dL (talk) 12:27, 30 April 2013 (UTC)

Compliments ! In a few days you have with full success destroyed the clarification I elaborated with some other editors during more than a year. Well, if you do not understand that by doing that you reintroduced a lot of ambiguities I give up. Editing in Wikipedia is a very disappointing job, but dont worry it was nevertheless an instructive experience :)

Chessfan (talk) 21:50, 25 September 2013 (UTC)

I removed your proof five months ago and explained above the reasons why I did it. Moreover, I copied here the removed text. That's an unusual courtesy. In five months, nobody felt the need to revert my edits or rewrite the proof. That's how Wikipedia works. I am sorry that you decided to give up. Paolo.dL (talk) 11:20, 26 September 2013 (UTC)

Thank you very much for your courtesy ! But reverting the proof is alas not the only mess you and probably others reintroduced, probably faithfully. I recommand strongly to any reader having difficulties in understanding the matricial definition of intrinsic versus extrinsic interpretation to take a look at the article as it was mid 2012. Chessfan (talk) 14:49, 26 September 2013 (UTC)

Please don't misunderstand me ; I will not try to fight with you on particular points, but explain. I am neither a professional mathematician nor a professional user of Euler angles, and additional handicap I am French and sometimes my English is approximative ! I was interested by Euler angles as a possible example for geometric algebra methods. But quickly I became aware that a lot of errors and misunderstandings were undetected in rotation articles. And I began to work without geometric algebra (one should not fight on too many fronts ...). I jump to the conclusion : I introduced one, only one, new or forgotten idea, that is, a matrix means nothing if you do not index it to indicate in which reference system you use it. And that is precisely the idea which you and probably others destroyed. Thus as it is obvious in your introduction of the matrix tables you fall back in the usual ambiguities. Sorry but that is my firm opinion, and i will no more work on the subject. Anyway all possible arguments we could exchange, and the corresponding theoretical work figure already in archive 3 . Chessfan (talk) 22:52, 26 September 2013 (UTC)

Incorrect matrix?

I coded up and was using the rotation matrices for Tait-Bryan angles and the one for Y1 X2 Z3 did not work. I redid the math and found the correct matrix to be:

   c1c3+s1s2s3   c2s1s2-c1s3    c2s1
   c2s3          c2c3           -s2
   c1s2s3-c3s1   c1c3s2+s1s3    c1c2

Anyone else care to verify this? Jimhunt (talk) 18:23, 28 August 2014 (UTC)

I think the proposed matrix is correct. I have generated the YXZ option in http://www.vectoralgebra.info/eulermatrix.html (which I admit could be incorrect) and I have obtained:

c1c3+s1s2s3	-c1s3+s1s2c3	s1c2
c2s3	        c2c3	       -s2
-s1c3+c1s2s3	s1s3+c1s2c3	c1c2

--Juansempere (talk) 21:19, 5 November 2014 (UTC)

I also had trouble with this matrix. I wish I had read the talk page first. Anyway, I independently derived the correct matrix algebraically, and it produces credible answers in MATLAB. (Y1*X2*Z3 gives the same numerical answer for random angles as this matrix.) I went ahead and made the change in the article since people should not be reading a known incorrect matrix. This should have been fixed six moths ago. The previous matrix was not even a proper rotation matrix - its determinant was not equal to 1. I saw no reason to wait to fix it in the main article.

--Rocket Laser Man (talk) 22:31, 16 January 2015 (UTC)

Erroneous diagram

The second last picture, the one with the pale blue aircraft shooting upwards to the left, has the pitch angle labelled as -ve theta for some erroneous reason. That's a positive pitch angle depicted, and theta would be positive.Lathamibird (talk) 07:26, 2 June 2015 (UTC)

Intrinsic Rotations animation

The example/animation shows 3 rotations about z,x', then z again. There's a lack of symmetry in this, and it does not relate well to the way Euler angles are used, in vehicular dynamics for example. The example and animation would be better showing 3 rotations - One rotation for each of the 'axes (z,y',x).

(Mwhincup (talk) 04:59, 3 September 2015 (UTC))

That has to do with the fact that they are Proper Euler (not Gimbal). Check these two videos to see the difference: Euler Angles vs. Euler Angles Cardan

ntg (talk) 11:59, 5 September 2016 (UTC)

Haar Measure

There may be an error in the section where the Haar measure for SO(3) is given. The line says the Haar measure for Euler angles but then gives the formula for Hopf angles. See here for the derivation of each, specifically compare Euler angles volume element in Equation 3 with Hopf angles volume element in Equation 5 AlphaNumeric (talk) 12:13, 21 December 2016 (UTC)

Wrong angle computation ?

I got the feeling that the computation of proper Euler's angle is wrong. Assume we only want to rotate around the Z axis of ${\frac {\pi }{4}}$, so we have

• $Z={\begin{pmatrix}0\\0\\1\end{pmatrix}}$
• $X={\frac {\sqrt {2}}{2}}{\begin{pmatrix}1\\1\\0\end{pmatrix}}$
• $Y={\frac {\sqrt {2}}{2}}{\begin{pmatrix}-1\\1\\0\end{pmatrix}}$

However, using

$\alpha =\operatorname {atan2} (Z_{1},-Z_{2}),$

$\beta =\arccos(Z_{3}),$

$\gamma =\operatorname {atan2} (X_{3},Y_{3}).$

I find

$\alpha =\operatorname {atan2} (0,-0)=0,$

$\beta =\arccos(1)=0,$

$\gamma =\operatorname {atan2} (0,0)=0.$

In other words, this formula does not capture the rotation at all. Looking at this document (page 23), I find a more complicated but more accurate computation of the aforementioned angles. Could you confirm me that it needs some changes or explain me my mistake ?

DrWatson1984 (talk) 09:02, 17 May 2018 (UTC)

The rotation you propose is ambiguous. Maybe that is the source of the problem. Have you tried the arccos expressions? They yield a 0/0 solution that for sure is not the same that the one with atan2.

Matrix to Euler conversion

The article shows how to construct a matrix from three Euler angles. I don't see where it shows the opposite. E.g. derive the Euler angles from the matrix. Is it in the article and I am just not seeing it? I'm not good at math. SharkD  Talk  17:18, 1 September 2017 (UTC)

See this section:

Angles of a given frame: A common problem is to find the Euler angles of a given frame. The fastest way to get them is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix (see later table of matrices). Hence the three Euler Angles can be calculated.

--Juansempere (talk) 20:46, 22 July 2018 (UTC)

A grammar question

I fixed some English here but it was reverted here. The reversion seems incorrect? Neil — Preceding unsigned comment added by 83.219.54.138 (talk) 11:25, 10 June (UTC)

Sorry. There have a lot of dubious IP edits here. Do restore. Xxanthippe (talk) 22:23, 11 June 2019 (UTC).

Thanks. Have restored, and got myself a user account WadoNeil to appear less suspicious on here. — Preceding unsigned comment added by WadoNeil (talk • contribs) 10:54, 4 July 2019 (UTC)

Thanks, I hope you will watch the page to keep it up to scratch. With your computer graphics background you are well qualified to. Xxanthippe (talk) 22:25, 4 July 2019 (UTC).

Tait-Bryan Angles - Distinct Axes

In the section Definition of Extrinsic Rotations a sequence 3-1-3 is mentioned, which then is referred to a common notation of both proper Euler Angles and Tait-Bryan angles. see: "Sets of rotation axes associated with both proper Euler angles and Tait–Bryan angles are commonly named using this notation (see above for details)."

If this last sentence is meant to introduce the notation for rotations in general, it should be moved to the geometrical definition or introduction.

Since in the current context referring to (3-1-3) as an example it violates the definition of Tait-Bryan Angles in the section Tait-Bryan Angles see "The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. x-y-z, or x-y′-z″), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., z-x-z, or z-x′-z″). "

--84.217.196.160 (talk) 09:35, 2 November 2019 (UTC)

Rotation matrix

Can anybody write down all these X, Y, Z explicitly to avoid guesswork about sign conventions? An IP changes signs for s₂ terms and I have few ideas how should it be checked. Incnis Mrsi (talk) 07:09, 7 September 2019 (UTC)

I've noticed this too. Xxanthippe does a good job of cleaning up through the IP edits (including me), but missed the one you talk about which is the only erroneous result as of now. To calculate the matrix you start with

$X_{1}Y_{2}Z_{3}={\begin{bmatrix}1&0&0\\0&c_{1}&-s_{1}\\0&s_{1}&c_{1}\end{bmatrix}}{\begin{bmatrix}c_{2}&0&s_{2}\\0&1&0\\-s_{2}&0&c_{2}\end{bmatrix}}{\begin{bmatrix}c_{3}&-s_{3}&0\\s_{3}&c_{3}&0\\0&0&1\end{bmatrix}},$

which are the three base rotation matrices in X, Y, and Z, as per the Wikipedia page on Rotation Matrix . Mathworld uses the same convention , but the page is misleading because they only state that they do so. The matrices we see explicitly come from the second convention which is coordinate system rotation, using the transpose of the matrices I've written. This is all explained in MathWorld's text leading up to their 3D rotation matrix equations.

Following normal matrix multiplication rules for the last two matrices you get

$X_{1}Y_{2}Z_{3}={\begin{bmatrix}1&0&0\\0&c_{1}&-s_{1}\\0&s_{1}&c_{1}\end{bmatrix}}{\begin{bmatrix}c_{2}c_{3}&-c_{2}s_{3}&s_{2}\\s_{3}&c_{3}&0\\-c_{3}s_{2}&s_{2}s_{3}&c_{2}\end{bmatrix}}.$

Finally, multiplying these last two matrices nets you

$X_{1}Y_{2}Z_{3}={\begin{bmatrix}c_{2}c_{3}&-c_{2}s_{3}&s_{2}\\c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}\\s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}\end{bmatrix}}.$

I've verified these results through the symbolic math software Maple, along with all 11 other conventions. The other results were fine.132.203.102.10 (talk) 14:48, 8 October 2019 (UTC)

This is still WP:Original research (by you). It may be correct but a WP:RS is needed. Xxanthippe (talk) 21:23, 8 October 2019 (UTC).

Correcting an error in the multiplication of three matrices should not require one to cite a source. It is trivial for you to check the correction -- please do so. 74.140.203.84 (talk) 16:49, 30 April 2020 (UTC)

Below I show my work for the correction of the $Z_{1}X_{2}Z_{3}$ Euler angle rotation matrix. $Z_{1}X_{2}Z_{3}={\begin{bmatrix}c_{1}&-s_{1}&0\\s_{1}&c_{1}&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}1&0&0\\0&c_{2}&-s_{2}\\0&s_{2}&c_{2}\end{bmatrix}}{\begin{bmatrix}c_{3}&-s_{3}&0\\s_{3}&c_{3}&0\\0&0&1\end{bmatrix}}$

These are rotation matrices associated with elemental rotations of the three angles about the Z, X, and Z axes in sequence. It follows by basic matrix multiplication that

$Z_{1}X_{2}Z_{3}={\begin{bmatrix}c_{1}&-s_{1}&0\\s_{1}&c_{1}&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}c_{3}&-s_{3}&0\\c_{2}s_{3}&c_{2}c_{3}&-s_{2}\\s_{2}s_{3}&s_{2}c_{3}&c_{2}\end{bmatrix}}={\begin{bmatrix}c_{1}c_{3}-s_{1}c_{2}s_{3}&-c_{1}s_{3}-s_{1}c_{2}c_{3}&s_{1}s_{2}\\s_{1}c_{3}+c_{1}c_{2}s_{3}&-s_{1}s_{3}+c_{1}c_{2}c_{3}&-c_{1}s_{2}\\s_{2}s_{3}&s_{2}c_{3}&c_{2}\end{bmatrix}}$

No source citation should be necessary as it is clear by inspection; see WP:BLUE or WP:CALC.

74.140.203.84 (talk) 23:14, 30 April 2020 (UTC)

You appear to be conducting an WP:Edit war on this subject. Your opinions, as those of any editor, are worthless, unless supported by reliable sources. Please provide such sources. Xxanthippe (talk) 23:20, 30 April 2020 (UTC).

Nonsense. Stop being obtuse. My source is basic linear algebra, more specifically matrix arithmetic. This does not require a citation. Again, see WP:BLUE or WP:CALC. 74.140.203.84 (talk) 23:30, 30 April 2020 (UTC)

Just to be excruciatingly explicit, if you want a source for the elemental rotation matrices that I'm multiplying above, see Rotation_matrix#Basic_rotations. For the rest, see Matrix_multiplication.74.140.203.84 (talk) 00:07, 1 May 2020 (UTC)

Please note that Wikipedia cannot be used as source for itself, nor is WP:OR acceptable. Xxanthippe (talk) 01:46, 2 May 2020 (UTC).

Don't be absurd. There is no original research here. It is neither original nor is it research, by any stretch of the definitions of these words. This is essentially basic arithmetic, though in matrix form. WP:CALC applies. Would you have me cite sources for basic arithmetic on the ring of integers as well?

(By the way, 74.140.203.84 is me). Jmmerchant (talk) 03:45, 2 May 2020 (UTC)

Do you acknowledge that the updated calculation of $Z_{1}X_{2}Z_{3}$ is correct? Jmmerchant (talk) 04:01, 2 May 2020 (UTC)

I found a source that suffices to verify the results of the table of Euler angle rotation matrices and tacked it at the end of the table. I still think this is completely unnecessary, since WP:CALC applies to the sort of simple matrix multiplication needed for someone to verify it for themselves (and even if that is somehow not the case, there are several sources in the bibliography section that likely sufficiently cover the topic). Anyway, if you want to put it someplace else, I'm fine with that. Jmmerchant (talk) 05:51, 2 May 2020 (UTC)

132.203.102.10 here, from above. Just to add to this, this Edit war happened because AGAIN someone modified one of the matrices without any reason, proof or sources. This time, the edit went unnoticed because it provided false info about the changes. This was edit 03:06, 23 March 2020‎ Yulutngu (talk) (cyrillic character meant to be greek phi to match image). No character change was made, only matrix changes. The matrices have been well written in the past. I feel like if someone starts repeatedly trying to make a change using WP:CALC as proof, it's a sign something happened. If that change is identical to the previous time someone checked and corrected ALL the matrices then there's probably a sneaky edit in between. Some sleuthing in the edit history is always good. That said, now that we have a source, this should be the go-to version to be used as comparison because this kind of back-and-forth might just happen again. 70.24.197.137 (talk) 15:42, 20 May 2020 (UTC)

I don't see how my edit provided "false info about the changes." I did not change the matrices. Did you mean another edit? Yulutngu (talk) 18:40, 14 October 2020 (UTC)

You're absolutely right, it was 19:54, 1 April 2020 PlateBowlBottle (talk). The one following yours. My mistake. 70.82.29.61 (talk) 14:45, 27 November 2020 (UTC)

Juanma.jmgg88 (talk) 09:34, 17 July 2020 (UTC) Regarding the statement "1, 2, 3 represent the angles α, β and γ, i.e. the angles corresponding to the first, second and third elemental rotations respectively." I find it a bit misleading that 1 is said to represent the first elemental rotation but it is however applied last (leftmost). The shown matrices are actually applying first the γ rotation. Am I missing something?

This is how chaining intrinsic rotations work. It is counter-intuitive, but that is how they are applied. 70.82.29.61 (talk) 14:52, 27 November 2020 (UTC)

Your claim needs sourcing. Xxanthippe (talk) 21:21, 27 November 2020 (UTC).

References

Distinction of names: cardan, nautical, Tait-Bryan angles

The top of the article sounds like Tait-Bryan, cardan, yaw-pitch-roll and nautical can be used synonymously: Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll

In the Tait-Bryan section there is the comment about only the special case of z-y´-x´´ being called cardan or yaw-pitch-roll or nautical: Tait–Bryan angles, following z-y′-x″ (intrinsic rotations) convention, are also known as nautical angles, (...), or Cardan angles (...)

Unfortunately, I have no good sources on common usage of these terms in english. That's why this Wikipedia weakness bothers me even more.

--2A02:810D:E80:66B1:EA6A:64FF:FED3:1298 (talk) 13:10, 13 April 2021 (UTC)

Added Matrix -> Angles table, reverted for vandalism

Hello, newbie here.

I added a table (with due sourcing) listing formulas for getting angles from rotation matrices.

It got reverted shortly after by Bbb23 for "disruptive editing" and vandalism.

I asked for help over on my talk page and then on the tea house, and it was suggested I should ping the user who reverted my changes again (which I did above I believe), and revert the revert (that is, reinstate my changes).

Thanks for any feedback if this change does break some policy.

Also English isn't my native language so my wording on math stuff may not be the best, feel free to correct or improve it!

Dragorn421 (talk) 06:33, 25 August 2021 (UTC)

Hi, Dragon421, my vandalism revert was wrong, and I apologize. Unfortunately, I misread your initial edits and made some incorrect assumptions. The advice you got on your Talk page and at the Teahouse seems sound, and your reaction to an unpleasant welcome to Wikipedia has been exemplary. The Teahouse is a good resource for the future if you have questions, but I've also left you a Welcome message on your Talk page that has links to various policies/rules/guidelines that you might want to read when you're truly bored. Wikipedia is a tough place to edit when you're new - and even when you're not - and it doesn't help if your first experience is with an administrator who screws up. Best of luck to you!--Bbb23 (talk) 12:40, 25 August 2021 (UTC)

• Comment. It is difficult to maintain this article in good order because of so many redlink, spa and IP edits. Is there a reliable source that it can be compared to, like Mathworld? Xxanthippe (talk) 22:32, 29 August 2021 (UTC).

Fixed error in Rotation Angles, added external link

This previous change introduced an error: the text previously (correctly) stated that the table was for intrinsic rotations, but the change revised it to say that it was primarily for extrinsic rotations. My change restores the original meaning.

I also added an external link to a NASA PDF, which can be used as a reference for the tables in this section. Unfortunately, it does not use the terms "intrinsic" or "extrinsic" explicitly (although the diagrams in it confirm that it uses intrinsic angles).

Bzanks (talk) 19:13, 9 September 2021 (UTC)

Gimbal lock

The article needs to pay more attention to the gimbal lock. AXONOV (talk) ⚑ 21:11, 4 February 2022 (UTC)