# Talk:Euler angles/Archive 1

 Archive 1 Archive 2 Archive 3

## Original discussion

I switched to Z-X-Z since that seems to be everyone's first exposure. For extra emphasis, that's Z as in Zero, X as in Xenon, Z as in Zero. This is not a typo--do not "fix". I hope the fuller descriptions will make it impossible for someone down the road to blindly alter this.

A diagram is certainly needed.

The article has reached a point where most further commentary really belongs on another page. For example, the rigid body description, after expansion, really belongs on the rigid body page.

It's quite possible that the three descriptions include an error, although I tried to doublecheck everything. It's also possible that any error at this point is merely relative to some convention.

The following comments refer to the pre-Dec-2004 version:

For emphasis: Z-Y-Z, not X-Y-Z. This is not a typo--do not "fix".

Euler angles involve a z-y-z rotation description, not x-y-z. Since there is no standard choice of Greek letters, introducing them as part of the definition is misleading, and since the letters are never used, it is spurious noise.

Even as a stub, it seemed to be ridiculously POV, Euler=bad. No mention of their more than two centuries of extensive use in mathematics and physics. A full article should discuss this first, and eventually mention practical limitations.

Computing with Euler angles is quite easy. A dramatic, simple geometric description is in the MTW reference--it is better if explained than if merely referenced. Since Sp(1)=SU(2)->SO(3) is a double cover, issues of interpolation (a local phenomenon) are identical.

The Minkler reference seems to be a vanity press book. It is certainly extremely obscure. I would not be surprised if aerospace engineers do not know the difference between z-y-z and x-y-z. If indeed they go around calling the latter Euler angles, then shame on them, and the article should delineate the difference clearly. My own experience is they simply have no idea what Euler angles are, and use the term rather vaguely, and will concede the terminological issue. That is, I don't think this is a case of a terminological clash between disciplines, but rather one discipline is blithely unconcerned with the actual meaning.

You seem to be saying that an x-y-z rotation does not fit the definition of Euler angles. Is that because Euler only ever used the z-y-z sequence, or what? What should an x-y-z rotation, or z-y-x, or x-y-x, etc. be called, in general? MarcusMaximus (talk) 10:57, 27 August 2008 (UTC)

Your "physicists' description" is simply the already present "moving axes description", with a change of notation. There is absolutely no point in introducing notation (like your Omega=(,,) at the top) that is, if not used within the article, is not at least standardized. Unfortunately, there is no standard notation for Euler angles, which is why I used alpha-beta-gamma. It's the most "neutral", so to speak. So your introduction of new notation is only worse, contradicting the standing notation.

There is no point in introducing section heads for extremely brief mentions, like "Quaternionic description".

Also, your initial description of Euler rotations was too vague. Unfortunately, non-mathematicians and non-physicists have used the "Euler" terminology in a loose way, and the article must be more than clear as a result.

Euler angles are used in several parts of physics. I don't see the point of singling out QM as being somehow special. It is true that rotational questions are inherent in QM "point" particles, but not in CM, which is why angular momentum gets a deeper treatment in QM.--192.35.35.36 17:11, 17 Feb 2005 (UTC)

OK, I have gone back. My edit was a merge - I know that there was an overlap, but by reverting you simply lose all the material posted by someone else. I know that the notation was not compatible, but that was because the original poster on the Euler rotation page was starting from scratch.
You are wrong about section headings. Short sections invite expansion or clarification.

Charles Matthews 17:19, 17 Feb 2005 (UTC)

OK, so it was a merge. SO WHAT? It was a bad merge, where you simply plopped the other page in, cut and paste in a best guess manner, and have made a mess of the result. Instead of a page where people might come away learning something about Euler angles, they are now hit with inconsistent and incompatible commentary and notation, mentioning physics now, dropping mention of physics, bringing it back, and all that. UGH! Saying, well too bad, I had to merge something in, is ridiculous.
Frankly, I get the impression that the former Euler rotation page was almost entirely redundant. A simple redirect and drop the old stuff would be much better. I worked hard turning this page from a somewhat silly and incorrect stub into a serious, accurate, and readable article. It certainly needs more work, but it had reached a stable position. Now it's garbage.
There will be nothing but a revert war now. Please, pick and choose whatever was on the old page that was not here, and rewrite it to conform to the standing notation, and edit it in. If you want, paste a copy of the old page here, or agree to break the redirect, and I'll rewrite it for you.
I don't agree with you about section headings, but I won't object to you putting them in.--192.35.35.35 17:36, 17 Feb 2005 (UTC)

Edit conflict. Here is what I was writing:

I see you have reverted again. You have also used the edit summary in a less than helpful way. It would be much better for you to have answered me on this page. As I have pointed out, I have no personal stake in the material added. As a basic courtesy to the original poster of that, I put it all on the page.
Now, if we could get back to etiquette here, I am quite happy to move the current 'physicist's section' out of the article to this talk page.
I found the Euler rotation page in a project page for neglected articles. It is natural to try to integrate whatever is valuable in it here. Can we have a discussion, please?

I don't do revert wars.

Charles Matthews 17:43, 17 Feb 2005 (UTC)

Could you be consistent regarding indentation?
I don't see why you get special permission to make a merge out of nowhere, without discussion, but I'm on some moral low ground for reverting with the discussion five minutes later. Can you get past being offended, refusing to discuss points, and all that?
I have now checked the history on Euler rotation, and it was simply a brand new article added by someone who did not know the material was here, in much greater depth, on Euler angles. There's nothing new to incorporate. It was just bad luck for the other fellow.
The page did include a reference to a physics text with one particular Euler angle convention. I see no point in keeping that reference. In particular, Goldstein has a long footnote on the horrible prolification of Euler angle conventions, how it has sometimes subdivided by field and/or nationality. Sakurai does not.--192.35.35.36 18:04, 17 Feb 2005 (UTC)

OK, to get down to basics. Let's assume the Sakurai reference is not at issue.

Do we agree that these are all matters that can contribute to the page?

Charles Matthews 18:47, 17 Feb 2005 (UTC)

• I don't see the need for an initial "In geometry". It's common WP style, but it's redundant so far as the rest of the first sentence goes. If you really like it, include it.
• It is absolutely standard to include an orienting topic in the first phrase or so. Charles Matthews
• Yes, I know. As I said, go ahead.
• There is no point. Rigid bodies are one particular application, and should not be singled out in the introduction. Do not include it. Similarly with your one sentence paragraph regarding physics and QM in particular. Euler angles are also used in classical mechanics, and in CM applications/simulations: robotics, flight control, 3D gaming. Please don't include it.
• Well, the inclusion of something less abstract in the introduction is considered good practice. I'm a mathematician, you don't need it - most of the rest of the world finds rotations of rigid bodies more hands-on than an abstract transformation from one orthonormal frame to another. Charles Matthews
• And as a mathematician, you mistakenly consider Rigid Bodies more concrete than 3D Rotations, just because physicists, those real-world concrete down-to-earth people use both phrases. I agree with your regard for the value of a starting concrete example, but from the point of your more typical real world engineers, say, the two are equally airheaded smartypants.
• The real concrete item needed is a diagram. (A good diagram.) I would do it, but I don't have the time to research what software to use and learn.
• I'm mildly opposed. It seems better to me that the mention of Lie groups should be made inside the particular Lie groups, or inside the Haar measure article. Here, it strikes me as unnecessary terminological overload. That, too, is WP style, so if you really like it, include it.
• We aim for a concentric development, also called 'news style'. Say it once in the most concrete way, again at one technical level, again from a more abstract point of view. Obviously this ends up with rotations as a manifold - if we get that far. Charles Matthews
• Which I think sometimes plays better when split between articles. Euler angles are a parametrization of SO(3). Treating it as a waystation on the way to Real Topic Of Interest, "rotations as a manifold", seems like rushing things. I wouldn't go so far as to say the standard mathematical POV that "coordinates are bad, functoriality is good" is a violation of NPOV, but I would urge you to resist it when possible. 'News style' may be good for the rest of WP. In mathematics, the advanced view has a tendency to obliterate outsiders' interest in a subject. Thus, I would reserve the Lie group mention for the reader who clicks through. In other words, in a WP article about Euler angles, Lie groups are not the Real Topic Of Interest, no matter how much your mathematical training says otherwise.
• If you would like to expand on the question of interpolating Euler angles vs quaternions, I would be very appreciative. The existing documents I've found out there are about Tait-Bryan angles, but call them Euler angles. I certainly don't care if you wikify the word.
I said a few weeks back:
The article has reached a point where most further commentary really belongs on another page. For example, the rigid body description, after expansion, really belongs on the rigid body page.
This is my real objection to short short sections here. You envision expansion on this page, I envision expansion on different pages. That was my gut feeling after I got the article up to par and was thinking of where to take the article next. As such, I would not interfere with you sectioning the article.
Can I just say that WP is a collaborative project? Your opinion is noted, but it is rare that pages do attain any degree of closure. Charles Matthews
Of course! I am not opposing enhancements or improvements, and I will ignore most things that look like judgment calls. Sectioning is a judgment call, ergo, go for it. If I give you pause, that too is good. As it is, you are confusing my no-prisoners-taken approach to the downside of your merge as if that was my philosophy to any edit someone makes here.
If you compare my last Dec 31 edit with the Jan 1 edit, you'll see I was improved upon immensely. I'd like to see more of that. But you are still responding as if that merge you did had been a good thing, if only I'd look past the 95% of it that was a priori totally unacceptable, and it was darned uncharitable of me not to sort through the sludge. Frankly, it was darned uncharitable, but I think it was well-deserved.
As for section headings, "Definition" is the only one I have been able to think of that I like, so I did not add it.
I was interested in the history/Gruppenpest aspect - partly having been 'called in' to do something about the Hermann Weyl page, and here we have the other side of the story, i.e. the resistance to what Weyl was preaching. Charles Matthews
I can dig that up. Just as Weyl was making group theory important in atomic structure, a chemist figured out how to redo Weyl's work in concrete terms, and someone chortled that the Gruppenpest was slain, and as a result group theory disappeared from physics for about twenty years, and from chemistry for about forty years.
For the record, "Euler rotation" is sometimes used by people who know the difference between Euler angles and Tait-Bryan angles, but wish to talk about both together. I'm not conversant enough with the literature, though. See Jack B. Kuipers Quaternions and Rotation Sequences.

To sum up - we seem not to be able to reconcile some of the points where you wish to close down mentions of matters, where I would see at least some room for growth. On the specific matter of the merge, you can perhaps observe that my second edit noted the duplication caused by it. This would normally signal to an editor that I considered that cuts would be an advantage. Now, what I get from you here is the shutting of a few doors in my face. You know, most people like section headings. I'm just going to say this once, and not in a spirit of antagonism - but the edits you have made here do not make the article yours. Charles Matthews 20:14, 17 Feb 2005 (UTC)

Your merge wasn't just a merge with a few edits to fix. Completely different notation, duplicated definition, simply horrible. That was a loud and clear signal that someone was not reading the article, not noticing the most basic content, just thoughtlessly plumping in. Hence the double revert.
Are you aware of how awful the convention disagreements are regarding Euler angles? Read the Goldstein footnote, and then multiply his census by three. Good grief, the first two angles (in the alpha-beta-gamma convention of the article) correspond to spherical coordinates, so that means even if you find people agreeing on right-vs-left-handed coordinates, clockwise-vs-counter rotation senses, X-vs-Y for the middle axis, active-vs-passive description, you still find people who disagree about naming these angles phi-theta vs theta-phi, because mathematicians and physicists are split over what to name the two spherical coordinate angles. AARGH!
I mention this to make one fact clear: it's essentially impossible to write a coherent, consistent article about Euler angles by merging. Having read through various papers (where the authors don't reveal their conventions) and debugged code (where the original programmers were even more clueless) several times involving Euler angles, I feel qualified to judge aversely anyone who attempts such a merge.
When I comment on something as simple as section headings, I am doing so because I would appreciate if you do something more than "everybody does this on WP", pick section headings, and say mission accomplished. For example, I too like section headings. Good section headings. I've added some. I've rewritten articles a bit to enable good section headings. But just sticking them in every couple of lines doesn't sound like a plan or a plus.
I don't think the article is mine at all. But I've put a good deal of effort in converting it from a weak and inaccurate stub into a factually accurate medium grade article that walked carefully over the fundamental terminological dispute itself, dodged past the convention multiplicity, and to boot integrated all three definitions, something I've not seen in print. Enhancements are welcome.

--192.35.35.34 22:00, 17 Feb 2005 (UTC)

I have explained how I got here - housekeeping. I have explained that my edit comments flagged the duplication. I have explained that certain things are 'normal for Wikipedia'. You don't seem very interested in my points about improving the introduction. You are quite entitled to object to my edits, though if you use language designed to communicate your points you would get much further with others, not just me. I am not questioning your qualifications to judge aversely anyone who attempts such a merge. I am pointing out that that 'judging adversely' is not what we come here for. I happen to have a quite adequate background to understand most points made about most mathematical topics, if the conversation is about that.

Charles Matthews 22:22, 17 Feb 2005 (UTC)

Yes, you thought you were merely housekeeping. I think you did a really bad job of it, for the reasons I explained. You have not refuted my basic contentions about your merge. That is the issue, not the path that brought you here.
Your edit comments did say "some redundancy". Is this British humour? Get real.* You plopped in a redundant definition, with a different convention, leaving an inconsistent and incoherent mess. That was about 95% of your added content, and I felt totally justified reverting, rather than figuring out what might actually be an improvement. Nothing you've said has changed my opinion. I see someone picking nits to assuage wounded pride, spelling out hypothetical scenarios about how I could have saved the 5% that might have some value. Well, the 5% didn't look wrong, but didn't look they were worth the effort either.
• Am I sounding irked? Well, yes. You're defending the indefensible, with decidedly feeble excuses, blaming me for not giving due weight to your warning that there was some redundancy. Well, I did give it due weight (0), so can you drop this line?
Yes, certain things are WP normal. I've acknowledged that, and I have pointed out I don't think they work here the way they would elsewhere. In some of the cases, I've further explained why I don't think they would work as well. I've also spelled out that I will not revert such changes. Why are you still discussing this as if it were an issue?
I don't think your points regarding the beginning of the article would constitute an improvement. They reflect a grab bag "what can we toss in" philosophy, not a "think it through first".
What have I failed to communicate? You seem to continually dodge the issue that your merge was 95% duplication with inconsistent notation, and as such, made the article worse. You're still responding to me with some lurking resentment over my justifiable reversion of your botch. See above about my attitude regarding the 5% that was not actually inconsistent.
You're right, I'm not here to "adversely judge" you. My apologies, I meant your edits.
We're both mathematicians by training, so there is no reason we can't talk mathematics. Meanwhile, you have wandered into a topic that's far enough outside your past expertise that you had no idea of what you were actually doing. I remain genuinely astonished that any mathematician would fail to notice that the content of "Rotation sequences" was already here in a different notation. I assume you did a rush job.--192.35.35.36 00:00, 18 Feb 2005 (UTC)

What you are really telling me here seems to be that conventions in this area are fraught with pitfalls. OK, why not make that explicit in the article? I won't be the last person coming here. If you flag up the conventions that are being applied, and supply a reference when you do so, it is not only an enhancement but an actual indication that you have a view on presentation based on research.

Charles Matthews 22:22, 17 Feb 2005 (UTC)

I'm telling you many things, but many of them boil down to this: your initial merge was horrible. Not just mistaken, not just in need of adjustment, not just a little goof. Just plain horrible. And that you have been replying to me with the impression that it was merely a trifle worse than a typo, and until you catch on that it was seriously doubleplusungood, this conversation is going to be unnecessarily unpleasant all the way through.
Could you read the article? I include a warning about the many conventions. As I have said repeatedly, I got the article up to medium grade quality and have stopped. Tracking down the various conventions in print is extremely tedious, nor do I see the point. I gave numerous references. I don't think any of them use the same convention I did, nor do I think it matters.
I have long range plans for enhancements, but the above detail is just a waste of time.
Yes, there will probably be more people visiting and messing things up. But there is one less name out there for someone to create a duplicate article to then get badly merged in with this one. Really, you laid an egg. If you read my original comments above, or check the history edit summaries, you'll see that one of my goals was to write the article in a way that would make one particular error close to impossible.
Frankly, your suggestion that I might indicate somehow that I have a view based on research is quite insulting. I think it's pretty clear that I know what I am talking about. And that I have been fluently communicating.
Please, stick to the issues, and not throw in condescending comments.--192.35.35.36 00:00, 18 February 2005 (UTC)

## Friday

Can I just point out that elaborating the conventions which you use would be in line with definite WP policy? [...]

Charles Matthews 06:05, 18 Feb 2005 (UTC)

The changes you made are now something I'll have to think about. For one thing, I don't see any point in mentioning Euler's rotation theorem at the beginning. You are, in essense, starting off the article by referring to a different chart. What for? This is an article about the Euler angles chart. Similarly, I don't see why you introduced Cardan angles at the beginning. That, too, is a different chart. And if you read down the article, you'll see they are already linked, under the synonym Tait-Bryan angles. Similarly, there's a link down later to charts on SO(3), where the connection with Euler's rotation theorem is discussed.
Now, it certainly makes perfect sense in mathematics to write the way you did. It may even add to the reader's motivation, something like, ah, Euler's rotation theorem, yes, I get it, three parameters, now what are the simplest three, and oops, here's a misleading choice, now on to the Real Conclusion. But that's inappropriate for an encyclopedia article on Euler's angles. Why not get more abstract, talk about linear algebra in the first paragraph and diagonalization? How about if we begin by mentioning that SO() is a functor, and work our way down?
In other words, this is the game mathematicians play, and it is to be encouraged for the sake of mathematics. I am fully aware of what the answer to my "What for?" question above is. But that kind of answer is not useful on WP. What's useful is to start by discussing Euler angles. Full stop. An engineer or programmer who comes here should not be discouraged by mathematicians writing for the greater glory of the bigger picture, the mathematical point-of-view that you and I take for granted as the Right Way To Really Understand. I won't say MPOV is a violation of NPOV, but I will say that on WP, MPOV should be treated with caution and skepticism.
To give a related example, I don't like an "In geometry, ..." beginning (in this article) because Euler angles are a widely used subject. By rights, it should begin "In geometry, physics, flight dynamics, graphics programming, ...". All perfectly logical, but atrocious.
I have the impression that while you know what a good mathematics article is, and what a good WP article is, you've not noticed the two are subtly different. You are writing (not deliberately, a side effect of your mathematical past) in a way that says to the expert reader, there goes that Charles, he now really knows his math, whereas I am writing (deliberately, refactoring all my mathematical past) in a way that says to the general reader, you now really know Euler angles.--192.35.35.35 16:15, 18 Feb 2005 (UTC)

[...] And that belaboring me is certainly not?

Charles Matthews 06:05, 18 Feb 2005 (UTC)

I have absolutely no idea what you mean when you say belaborment on my part is against WP policy, and frankly I find this repeated tack of yours, a general purpose zing, petty, pointless, and insulting, nothing but an extended sulk on your part. As long as you keep carping about how my reversions were wrong-headed or thoughtless or suboptimal, I will carp back. Frankly, the more I've scrutinized your initial merge, the worse it looks, and I've been deliberately understating things.
As for spelling out my philosophy of editing WP math, how is that not acceptable?--192.35.35.35 16:15, 18 Feb 2005 (UTC)

Because of policy: see Wikipedia:Be bold. Under most circumstances, people are encouraged to edit within their entitlements. It is not the case that, aside from obvious political minefields, people need worry so much. WP has grown on the basis that tiptoeing around the site is not required.

Charles Matthews 16:55, 18 Feb 2005 (UTC) [and so on down the page]

And it also says at the bottom of every edit page that "bad edits to articles are watched for and will be quickly removed". You are still responding in this imaginary world where you made a pretty good merge, needs some work, and I rudely trudged over it.--192.35.35.35 19:00, 18 Feb 2005 (UTC) [and so on down the page]

What really is required is a mature attitude to discussion, in any condflict that arises.

You are deliberately, continually insulting me. I see little but long term sulking on your part.

You reverted with a message 'some good, some bad'; I went back asking you to use the talk page.

Would you check the timeline? I reverted at 16:49, discussed at 17:11, and you went back asking me at 17:14. Can you drop the unhelpful implications that you, being bold without any discussion for two days, are on a moral high ground, while me, taking 22 minutes to finish typing my response, had to be prodded three minutes backwards in time to use the talk page, revealing what an immature cad I am?

You reverted once more and started using upper case [...]

Yes. I emphasize a small portion of my words. What of it?
The first "not" was important.
Next, you were defending your bad edit--which I explained in my response why it was a bad edit--on the grounds that it was a merge. I said SO WHAT? I feel the same now, just as bold and majuscule as before. Because all that matters is the content, not the path. This is obvious, right? Right from the beginning, instead of discussing the relevant issue, to wit, the page's actual content, you brought up irrelevancies like you were only trying to be helpful, and I had to explain in slow motion that, sad to say, it was not helpful. Right from the start you ducked the mathematical problems I pointed out in my discussion (you know, the one posted three minutes before you asked me to, but which you are still taking credit for prodding me to do) and have been defending Charles, not your edits.

Unhelpful implications? Where? I really have no idea what you are talking about.
I find nothing helpful in your repeated insults in my general direction.

I am entitled to ignore all that shrill stuff from you.

You have been long, tiresome, and shrill in your sulking resentment over getting reverted.

I am entitled to ask for specifics.

That's what "specifics" is all about.

I am entitled to ask you to consider accessibility and 'house style' here. Charles Matthews 16:43, 18 Feb 2005 (UTC)

"Accessibility"? Sigh. What does that refer to??
House style is ambiguous enough that your cannot claim you are in the WP right and I am in the WP wrong.

Specifically, I quote this from the policy page:

With the vast majority of articles, feel free to dive right in and make broad changes as you see fit.

The difference between you and me is ...?
This isn't specifics. Specifics is like the question of why or why not the article should begin with "In geometry", or mention or not mention in the first paragraph two different non-Euler charts.

I did. You give me a hard time for that alone,

That is false. It's an outright lie. I reverted your article--which is not the same thing as giving you a hard time--for making a horribly bad edit, as I've only said over and over and over again. As per policy, bad edits get removed. When it comes to discussing specifics over this edit, I started off immediately with specifics. Most of your response has been avoiding the issues I raised.
You are getting a hard time because you keep insulting me, you continue on as if your initial merge was a bold push that ruffled my sensitive feelings of article ownership that, by opposing and taking 22 whole minutes to finish my first round of discussion, reveal me to be a thoughtless clod. The reality is you botched, badly, very badly, very very badly, introducing inconsistent notation, making a factually wrong distinction between mathematician and physicist in their conventions, and mostly said nothing new anyway, and I reverted the article back and began the discussion so that we can find a better Euler article down the road.
Since then, you've been defending Charles more than your edits. I repeat: what difference did it make, way back when, that it was a merge? Why mention it?

and you are pushing the envelope of policy here. Strange but true - the success of WP is largely due to people taking a different line from you.

You are just being deliberately rude. The success of WP is largely due to people who know their stuff putting in more work collectively than people who don't. Normally, this is because the latter voluntarily get out of the way of the former, often by knowing their stuff somewhere else and going there.

Reverts take seconds only: no rush.

So this is what it boils down to. You got reverted and you resent it.
Here we go again. You made a horrible edit. You've tried to justify it on the grounds that you were "only housekeeping" (so? it's still horrible), and that you admitted upfront that there was "some redundancy" (some=95%). It was a horrible edit, and it got properly reverted, as per WP policy. Reversions take mere seconds for good reason.
Let me guess: you're trying to make an unhelpful implication here? That since the reversion took but a few seconds, there was no thought involved? Piffle. I spent a good part of a month thinking about this article, and I looked very closely over your edits and found there was too much redundancy to bother salvaging, reverted, and immediately started a discussion. But you started accusing me falsely from the beginning, starting with the nonsense that you had to prod me to do the right thing and discuss matters. An understandable and obvious mistake when you put it in your edit summary, what with timing issues, but that you keep raising it repeatedly after I pointed out that no such thing happened is simply a blatant lie.

You meet eople with other takes: could be interesting. Community customs and norms: there for a reason. House style: not one size fits all, but based on 10 million plus edits. Specialist blinkers: leave at door, please. Wikipedians: on the whole get along. Charles Matthews 16:55, 18 Feb 2005 (UTC)

I also meet people who are rude, insulting, and will sulk forever over having their mistakes identified. Hiding behind "house style" justifications for your goofs is bad enough, but then using it as grounds for insulting me is simply garbage. I see no effort on your part to get along. I do see you making an unhelpful implication that I refuse to do so.
I have pointed out why I think your latest changes are not for the good. You have done nothing more than assert it's simply "house style", without explaining why "house style" means the opening paragraph should refer to two other charts. Since you point blank refuse to justify them, I can only assume you are running on resentment. I will revert your latest changes, and then re-introduce (sometime next week) the parts that were good. Cut-and-paste, unfortunately, does not work, since your good parts depend on the prior do-not-belong-in-paragraph-one parts.--192.35.35.35 19:00, 18 February 2005 (UTC)

## Progress?

You are quite wrong to revert in that way. Your beef against Cardan angles would not be shared by most readers. I'm not refusing anything. Your put-downs on the basis of my nationality - alone - would justify me in writing you off as someone to discuss with. Charles Matthews 19:46, 18 Feb 2005 (UTC)

My beef against Cardan angles is that this is the Euler angles page.
I talked about my mathematical reasons for objecting to your second round of edits. You did not defend them. I call that a "refusal", since you certainly had the time to do so and instead used it for this rather pointless flamewar. Had you tried, I'd have probably given it positive consideration. This isn't "reward/punishment" type thinking, or anything remotely close. This is just that having enumerated the negatives, I had absolutely no idea of what could have been a positive in this context. Clue me in and I'll start weighing plus-and-minus, and definitely slow down.
There was no put-down based on nationality. "British humour" is simply a pejorative-free synonym for extreme understatement, especially when made by one of your nationality. As in your defense of "some redundancy" when talking about "95% redundancy" in your original edit. If I wanted to put you down based on nationality, I'd have throw in some crack about "bleedin' pommy bastards" or whatever the phrase is my Australian friends keep saying. (I'll doublecheck, if you like.) I will reveal that I am an American, and you have my permission to throw in one crack at me based on my nationality.
As it is, your various suggestions have included some good points, and I've incorporated them as best I can. You were right that the introduction was too stark and abstract, and needed something concrete. You were also right that the article needed sectioning, and my wait for the perfect scheme first was impractical and ignorant of Wiki article evolution.
Nevertheless, my objections to your particular versions remain.--192.35.35.34 22:04, 21 Feb 2005 (UTC)

I admit to not reading all of the above commentary (it's a little long winded). However, I believe that both of you (Charles and anon) are interested in writing a good article without trying to own the page. So can we agree not to revert edits without discussing it here first?

I consider the first one justifiable without discussion first--the edit was that bad.
The second revert followed discussion of why I thought it was bad on my part, with no counterreply on those issues.

My suggestions for improvement:

• Can we discuss what material from the proposed merge needs to be included?
• It was nearly all redundant. A different convention from Sakurai's text was the main body of the merge. If we're going to expand this page to an enumeration of texts and their conventions it would of course be worth reconsidering. (To see the now hidden page, go

here and check the history. (Side query: is there an internal way to link without redirection?)

• Some emphasis needs to placed on the different conventions used to define Euler angles. I think I have seen as many conventions as I have seen authors. Emphasizing this will help prevent future errors.
• It is at least in a separate section now, with an added warning. Do you want more?
1. Definition
2. Conventions
3. Applications for rigid bodies, quantum mechanics, Lie groups, etc.
It's not clear to me if "applications" is a good word to use to encompass both physics and mathematics here.
1. Alternate methods of representing rotations to include mention of SO(3), quaternions, etc.

There is a separate page for this: charts on SO(3). I have left the quaternions down on the bottom, since the existing description refers to applications.

• It wouldn't hurt to include some mention the applications in the introductory paragraph (before the Definition section).

-- Fropuff 21:24, 2005 Feb 18 (UTC)

• How about now, with the expanded pre-definition section? A second sentence listing areas of applications in the first paragraph seems reasonable to me.--192.35.35.34 22:04, 21 Feb 2005 (UTC)

• Since I expect some fluidity at the moment, I am leaving in redundant links until it looks like a consensus has emerged.--192.35.35.34 22:13, 21 February 2005 (UTC)

## New stuff

I added the explicit calculation that the moving and fixed axes descriptions are equivalent. I believe the phrase "To be explicit ..." lets us off the hook from explaining too much, and there ought to be a link to something here about rotations and conjugation. A link of conjugation in the previous paragraph is useless to the non-mathematician.

Be careful about modifying the equations. You might think you can combine some of them to make a shorter display, but that will mess up anyone reading at one or two levels of magnification above the default.--192.35.35.34 18:40, 8 March 2005 (UTC)

## applications of Euler's angles

Can anyone tell me how Eulers angles can be used to represent head and neck movement? i know about the use of goniometers but i'm still a bit confused. it's for a project i am doing.

— Preceding unsigned comment added by 84.65.68.14 (talk) 17:51, 30 January 2006 (UTC)

Hi, someone else here, I think the article should give more info on what the 3 angles actually mean about the orientation of an object in some concrete examples. Someone just mentioned head movement (nodding, shaking, ... ) , I would suggest a camera's orientation ( azimuth, elevation, tilt ) or an aircraft's orientation ( yaw or heading , pitch or attitude, roll or bank ). With pictures indicating what effect each of the 3 parameters has. I will do it when I have time, if nobody objects...

—Preceding unsigned comment added by SpaceDreamer (talkcontribs) 14:11, 5 May 2006 (UTC)

## Why only SO(3) ?

All special orthogonal groups have an Euler angle decomposition...

— Preceding unsigned comment added by 132.187.63.106 (talk) 09:24, 28 February 2006 (UTC)

## Meaning of the 3 angles

If I understand correctly, the angles α, β and γ (in the definition given in the article) indicate azimuth, elevation, and tilt , respectively. I think it should be said in the article.

Also, the article completely omits the fact that θ, Φ and ψ are usual notations for these angles, (although there there are inconsistencies to determine which is which). The reader might just think "so α is θ, β is Φ, and γ is ψ", for example, which is hazardous. The article just doesn't warn him about it. I think the paragraph "Conventions" should explicitly name the culprits, θ, Φ and ψ, to make sure the reader understands where the danger is. The article as it is warns the reader of a potential danger, but doesn't name that danger!

I used to think that "θ = azimuth, Φ = elevation, ψ = tilt" was the most widespread convention, I'm not so sure about it now, but shouldn't the most widespread convention be mentioned in the article, with the appropriate warning ?

—Preceding unsigned comment added by SpaceDreamer (talkcontribs) 14:34, 5 May 2006 (UTC)

In my experience in aeronautics and aerospace, we use $\psi$ = azimuth/yaw, $\theta$ = elevation/pitch, and $\phi$ = roll/bank. I am fairly sure this is not universal, especially considering you, me, and the article use at least three different conventions. MarcusMaximus (talk) 09:07, 27 August 2008 (UTC)

## Tait Bryan - Euler

Could anyone please answer a little question I have regarding the Tait-Bryan angles and the Euler angles? In the Tait-Bryan article, it says

"They are also called Cardano angles or nautical angles. For a craft moving in the positive x direction, with the right side corresponding to the positive y direction, and the vertical underside corresponding to the positive z direction, these three angles are individually called roll, pitch and yaw.

In aeronautical and aerospace engineering they are often called Euler angles, but this conflicts with existing usage elsewhere."

If I look at the definition given in the Euler angle article, it says

"Fixed axes of rotation Start with the XYZ system equalling the xyz system. Rotate the XYZ-system about the x-axis by α; the xyz-system does not move, now or later. Rotate it again about the y-axis by β. Rotate it a third time about the z-axis by γ. (Note that the first and third axes are identical.)"

To me, this sounds equivalent, doesn't it? So Tait Bryan angles are Euler angles, too?

Second, I have problems understanding the Euler definiton on its own. It says "rotate around x, then y, then z.". But, "note that the first and third axes are identical". I conclude x=z?

May someone please put this together for me? If I understood something wrong here, then maybe this is a good occasion to see where the problems in understanding the article lie.

Thanks again. 172.180.237.98 19:41, 17 July 2006 (UTC)

It is not equivalent at all. If you think of euler angles as defining a rotation with three matrixes, you will see that the inner one represents a intrinsecal rotation and the external one a rotation in the observer's frame. Tait-Brian angles are intrinsecal all the three. —The preceding unsigned comment was added by Juansempere (talkcontribs) 19:05, 7 February 2007 (UTC).

I just got done teaching rigid body motion in a graduate course in physics on Classical Mechanics, and I thought I would grab a picture from Wikipedia, for my class notes the next time around. So I started to read the article and found it massively confusing, and put in my own two cents by rewriting the Preview and noting in the Applications that the moment of inertia is really a tensor, and simplifies only in the principal axis frame. I left the rest of the article alone, but I think it indeed merits the phrase "confusing". As noted in the discussion, we just have to define our angles every single time, or others will go nuts. I like the definitions θ, φ, and ψ, but that's because I'm used to them. Note that their time-derivatives give nutation, precession, and spin. However, applied to the earth, we find that the earth precesses daily, but takes hundreds of days to "spin" (and back-spin at that, unlike a football). So the technical definitions don't necessarily agree with the conventional language. Cheers, everyone. Wayne Saslow, wsaslow"at"tamu.edu. —The preceding unsigned comment was added by 165.91.181.49 (talk) 00:48, 8 December 2006 (UTC).

## Fixed and Moving are wrong

I'm quite sure the descriptions in terms of "fixed axes of rotation" and "moving axes of rotation" descriptions are mixed up. Under "moving axes", one should have "rotate angle α around Z (=z)", then "rotate β around (the new) X", then "rotate γ around the (new) Z". As it is, the angles are reversed. "Fixed axes" is similarly mixed up. I will change these soon unless someone shows me to be in error, unimaginable as that may seem :) PAR 00:08, 24 January 2007 (UTC)

I'm in agreement that things are messed up. If we take the diagram as "correct", then the descriptions of "Static" "Fixed Axes" and "Moving Axes" all seem to have capital and lower case axes swapped (e.g., X instead of x, Y instead of y, Z instead of z, and vice versa). Please go ahead and fix this.

tbo 169.229.144.67 00:41, 24 January 2007 (UTC)

Don't take the image too seriously yet, I just uploaded it, and I see now that it is wrong. The XYZ axes are labelled xyz and vice versa. I will fix that right now. Even with this fix, though, the descriptions remain wrong. PAR 02:09, 24 January 2007 (UTC)
Image is now fixed. PAR 02:35, 24 January 2007 (UTC)

## Figure and narrative don't match the "proof of equivalence"

(This is my first attempt at contributing to the WP resources, so bear with me if I'm not observing proper "etiquette." I'm sure others won't be shy about correcting me.)

I'm reasonably sure that the "proof" in the "Equivalence of definitions" section is backwards. It is still correct as a proof, but the symbols &alpha, &beta, and &gamma appear in the reverse order to their use in the figure and narrative in the "Definition" section (this assumes that the order of composition is read right-to-left as is customary in mathematics). To be explicit, in order to agree with the "definition", the "proof" should be proving:

Z″(γ)X′(β)Z(α) = z(α)x(β)z(γ)

(I don't know how one gets the attention of the page's original author; perhaps (s)he is automatically sent a copy of this posting. If I don't hear back soon (say, a week) then I gather that I'm empowered to just go ahead and edit the page directly. Please let me know if this is not the case. Thanks.)

Hello - The history is this - the original explanation was wrong, I fixed it, but did not fix the "proof of equivalence". Please go ahead and fix it. If you don't do it in a day or two, I will. Thanks - PAR 21:06, 8 February 2007 (UTC)

Please go ahead. You're more expert at this than I. (BTW, a niggle directed toward the artist of the figure on this page: It would be clearer to represent the directions of the rotations (alpha, beta, gamma) with single-headed arrow arcs, rather than double-headed arrows. This comment notwithstanding, I very much appreciate the inclusion of such a concise geometrical illustration. If ever there were a concept where "a picture is worth a thousand words", this is it!)

On the angles, I agree and will fix it. If you know enough to know the proof is wrong, then you know enough to fix it, right? Go ahead. PAR 03:11, 9 February 2007 (UTC)

## I have an applet for this subject

Hi.

I have developed an interactive applet in which the user can move the three angles and see how the moving reference frame behaves. It is made with a free tool and based on free libraries, and I suppose that it allows me to upload it. The problem is that I need also to upload the libraries. Somebody knows how can be done? --Juansempere 18:48, 23 February 2007 (UTC)

• It'd probably be better to post the link here, and then if someone likes it, they can post a link to it in the article. !jim 21:47, 9 May 2007 (UTC)
• And how can I post it here? If I try to insert html I get this

<applet code="euler.eulerApplet.class"

       codebase="." archive="_library/ejsBasic.jar,euler.jar"
name="euler"  id="euler"
width="300" height="376">


</applet>

by John Lewis May 21, 2007

An explanation missing in this article is that the formulation of the Euler angles actually consists of two different problems: A “forward” problem concerned with obtaining the final orientation (“direction cosines”) of a rigid body for given values of the Euler angles, and an “inverse” problem, where the orientation of the body is given, and the question is, What are the Euler angles needed to accomplish that orientation?

Angle ranges: The article states “With β, 0 and π give the same 3D rotation”. Not sure I understand this. If I take a spherical ball marked with both a North and a South Pole and choose to rotate it so that α= γ=0, then choosing β= π places the North Pole where the South Pole used to be. Thus, β=0 and β=π can’t possibly be quite the same. Indeed, even non-zero values for α and γ with β=π will leave the North Pole at the bottom.

Based on Goldstein's Classical Mechanics, pg 153,the rotation matrix R should be
$\begin{bmatrix} c1c3-s1c2s3 & s1c3+c1c2s3 & s2s3 \\-c1s3-s1c2c3 & -s1s3+c1c2c3 & s2c3 \\s1s2 & -c1s2 & c2 \end{bmatrix}$
The r11,r12,r21,r22 terms are different from your calculation.
For the angle ranges, when $\beta=0$, R matrix is
$\begin{bmatrix} c1c3-s1s3 & s1c3+c1s3 & 0 \\-c1s3-s1c3 & -s1s3+c1c3 & 0 \\0 & 0 & 1 \end{bmatrix}$
However, when $\beta=\pi$, R matrix is
$\begin{bmatrix} c1c3+s1s3 & s1c3-c1s3 & 0 \\-c1s3+s1c3 & -s1s3-c1c3 & 0 \\0 & 0 & -1 \end{bmatrix}$
The two matrices are not same, so, there must be a misstatement in the article.
About a rotation $\delta$ around y-axis, the rotation $\alpha=\pi/2,\beta=\delta,and\ \gamma=3\pi/2$ should work. the rotation matrix turns out to be
$\begin{bmatrix} \cos\delta & 0 & -\sin\delta \\0 & 1 & 0 \\\sin\delta & 0 & \cos\delta \end{bmatrix}$

Thurth 23:55, 1 September 2007 (UTC)

Addendum by John Lewis, Sept. 23, 2007 Thanks to Thurth for the clarification. I had indeed an error in my equations, which actually originated from a problem in the article. To avoid confusing other readers, I have now removed my equations as well as the now superseded parts from my earlier comments.

## Comparsion with spherical coordinates

Conventional spherical coordinates as described in this article are a combination of rotations along the z and y axes. However, Euler angles are most commonly combinations of z and y rotations. Maybe someone could mention a few words about this issue? I was a bit confused by this difference. I posted this at Talk:spherical coordinate system too; I propose to discuss the issue over here. Han-Kwang (t) 10:53, 4 December 2007 (UTC)

## Relative Euler angles

I think this section is not quite correct, in the proof check "First we perform a rotation of θ1'-θ1 on Si. When we do it, the first intermediate frame of Si will reach the first intermediate frame of Sf." This is in question since the first rotation axis of Si is not the same as that of the initial frame, besides, if it was correct, we should be able to get Sf by rotating by θ2 and θ3 after that step. I am not sure if this way of combining is correct at all. I've tried the method visually with 3DSMAX and it doesnt work. And as far as I know combining Euler angles are supposed to be harder than that. Since maybe that is a well proven fact, and I am making a mistake in reasoning, I am leaving this part there, but if you share my views, please remove it. —Preceding unsigned comment added by 130.226.26.158 (talk) 10:44, 19 December 2007 (UTC)

Yes, I find that section rather dubious too. I'm moving it here until it's sorted out:

### Relative Euler angles

We can deal now with the following problem. We have two frames Si and Sf with angles Si=<θ123> and Sf=<θ1',θ2',θ3'>. and we want to know the Euler angles of Sf when they are measured from Si.
Theorem: The Euler angles of Sf measured from Si are <θ1'-θ12'-θ23'-θ3>
Proof: We know that Euler angles are equivalent to three rotations composed from the initial frame. If we perform three elemental rotations with the given values, starting from Si, and we reach Sf, we will have proved that these really were the relative Euler angles. First we perform a rotation of θ1'-θ1 on Si. When we do it, the first intermediate frame of Si will reach the first intermediate frame of Sf. In this situation, the second-intermediate frames of Si and Sf will be both in the same plane. We perform now the second rotation, with value θ2'-θ2 and then, the second intermediate frame of Si will reach the second intermediate frame of Sf, and the given frames Si and Sf will be then in the same plane. Doing the same with the third rotation, the initial frame will reach the target frame and this proves the former theorem.

Mathematically, a vector u is transformed into the i and s frames as

$u_i = R_z^a R_x^b R_z^c u,$
$u_f = R_z^A R_x^B R_z^C u,$

where R_x^a is the rotation operator along the x axis over angle a. Hence,

$u_i = R_z^a R_x^b R_z^c R_z^{-C} R_x^{-B} R_z^{-A} u_f = R_z^a R_x^b R_z^{c-C} R_x^{-B} R_z^{-A} u_f.$

Since rotations are not commutative, I'm pretty sure that

$u_i \neq R_z^{a-A} R_x^{b-B} R_z^{c-C} u_f.$

Han-Kwang (t) 11:22, 19 December 2007 (UTC)

### I have removed that paragraph

Hi. I am the one who added that paragraph. Now I have removed it because it was wrong.

What I wanted to say is that the three rotations are independent, but for sure, it does not imply that they are relative angles as I wrote. Sorry. I have added now a paragraph and an image speaking about gimbals which show much better what I meant.

Regards,
--Juansempere (talk) 23:54, 22 December 2007 (UTC)

## Typo?

In section "Table of matrices" it says "use the matrix for yzx with θ1 and θ2 swapped". Shouldn't that read "use the matrix for yzx with θ1 and θ3 swapped"?

I don't have the temerity to actually edit the page, but the following text might be helpful:

The following matrices assume fixed (world) axes, with rotations acting on objects rather than on reference frames. They are used to pre-multiply column vectors. For example, matrix xzy is constructed as a product of three matrices, Rot(y,θ3)Rot(z,θ2)Rot(x,θ1). To obtain a matrix for the same axis order but with rotating (body) axes, use the matrix for yzx, but with θ1 and θ3 swapped. The result is the product Rot(x,θ1)Rot(z,θ2)Rot(y,θ3), which is used to post-multiply row vectors. In the matrices, c1 represents cos(θ1), s1 represents sin(θ1), and similarly for the other subscripts.

SpinCoupling (talk) 15:48, 23 February 2008 (UTC)

## The matrices of the 12 conventions are wrong?

I strongly believe that at least the ZXZ and ZYZ matrices in the picture/table of all the 12 different conventions are wrong.

I assume (because it is not mentioned) that all matrices are in rigth handed (anti-clockwise) notations. It is claimed that those matrices rotate the objects (and not the axes). BUT, looking at mathworld website http://mathworld.wolfram.com/EulerAngles.html the two matrices (ZXZ and ZYZ) are different. And in mathworld, they are anticlockwise and rotate object. —Preceding unsigned comment added by Cofcof oz (talkcontribs) 04:13, 26 March 2008 (UTC)

## Possible reason why 12 matrices are wrong

It seemed the evaluation of those matrices were performed in the static frame by rotating the objects. That means three consecutive rotations have to be arranged in reverse order, because:

  $Z''(\gamma)X'(\beta)z(\alpha) = z(\alpha)x(\beta)z(\gamma)$


However, this is not done properly in the article. The mistake can be traced back to the section "Matrix notation", where the product of three consecutive rotational matrices is in wrong order already. The correct form should be:

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

The consequence of this is angle 1 and angle 3 are swapped in position in the final matrix. Matrix obtained from this procedure should be the coordinate transfer matrix from the body frame to world frame. After swapping angle 1 and 3 , we can see it is consistent with Goldstein's and Mathworld.

On the other hand, we can also take a look at the derivation of the rotation matrix in Mathworld, where they followed a scheme of rotating the body frame instead of objects.

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

Note each matrix here is the transpose of its counterpart in this article. And the order of three matrices is reversed as well, which leads to:

  $R' = z^t(\gamma)x^t(\beta)z^t(\alpha) = (z(\alpha)x(\beta)z(\gamma)) ^t = R^t$


They arrived at a matrix that transfers coordinates from world frame to body frame, which is essentially a transpose of zxz matrix here. It can be seen as well that the formula in this article was wrong.

Jiwuliu (talk) 21:29, 6 June 2008 (UTC)

### I will change the Matrix notation section

You are right. The matrix expression is clearly wrong if we take the angles as defined in the main picture. I will correct the section Matrix notation, and I will check the other. Regards, --Juansempere (talk) 12:06, 19 July 2008 (UTC)

It seems that the example just before the table also contains an error. The rotations in the matrix are around the x, then around the y and finally around the z axis, but in the explanation it is stated that they are around z, x, and z 134.225.48.60 (talk) 11:53, 2 December 2009 (UTC)

### The final word!

Right:

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

Wrong:

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

To define what is meant with the matrix one must also say:

$\begin{bmatrix} \hat{x}\ &\ \hat{y}\ &\ \hat{z} \end{bmatrix} [\mathbf{R}]= \begin{bmatrix} \hat{X}\ &\ \hat{Y}\ &\ \hat{Z} \end{bmatrix}$

At this moment this is all OK!

Stamcose (talk) 11:41, 23 July 2008 (UTC)

This is not usual or common practice. Vectors are usually column vectors and all conventional text books for the tranformation matrix is shown as \bold{A'}\=\bold{A}\•\bold{x}\. No need to add unnecessary confusion on a wiki page.- Ck.mitra (talk) 04:10, 22 September 2008 (UTC)

Ck.mitra, I think you made errors in your math formatting and in the equation itself. What you wrote, if formatted properly, is rendered as
$\bold{A'}=\bold{A}\cdot\bold{x}.$
I believe what you meant to write was
$\mathbf{x'}=A\cdot\mathbf{x}.$
I posed this question in a new topic at the bottom of this talk page. I agree with your answer. MarcusMaximus (talk) 04:54, 22 September 2008 (UTC)

Sorry, I was wrong and you are right about what I meant. Thanks- Ck.mitra (talk) 10:26, 22 September 2008 (UTC)

## Angle ranges and exceptions

I agree with the angle range definitions:

* α and γ range from 0 to 2π radians. * β ranges from 0 to π radians.

but i think what follows is wrong:

These angles are uniquely determined, with certain exceptions.

* With α and γ, 0 and 2π radians give the same 3D rotation. * With β, 0 and π give the same 3D rotation.

This corresponds to the xy and the XY planes being identical, so the rotation is just a rotation of α+γ about the z-axis. (This last ambiguity is known as gimbal lock in applications.)

for α and γ, I think it is better to say that there are defined in [0, 2π[, as rotations of 0 and 2π are always identical. for β, 0 and π just do not give the same rotation: (0, 0, 0) and (0, π, 0) are not identical. β is defined in [0, π].

The thing is that when β equals 0 or π, α and γ are not uniquely determined: this is the "Euler space degeneracy". At β = 0, only (α+γ) matters and at β = π, only (α-γ) matters.

Niac2 (talk) 14:34, 29 April 2008 (UTC)

### Clarification

α and γ are uniquely defined modulo 2π! Or in other words (cos α , sin α) and (cos γ , sin γ) are uniquely defined but not α and γ!

Stamcose (talk) 11:48, 23 July 2008 (UTC)

I have added some more information on the ambiguities: the value of β and the ways α and γ are not uniquely determined. I have also added an external link to a free software in which all the stuff is implemented. I did that months ago without including explainations (no much time), and so the link was removed, but there are few such software, and so I think it would be a pity not to include it.

Niac2 (talk) 10:10, 2 September 2008 (UTC)

### Error in range definition

There is something else. In the article it says:

• β range covers π radians (but can't be said to be modulo π). For example could be [0, π] or [-π/2, π/2].

If we have Z-X-Z convention, how would you represent 0°-180°-0° with β ranging from [-π/2, π/2]?

--217.229.6.31 (talk) 18:21, 3 May 2010 (UTC)

## Use of row vectors vs. column vectors

Is it commonplace to use a 1x3 row matrix containing unit vectors premultiplying the direction cosine matrix, as opposed to a direction cosine matrix premultiplying a 3x1 column matrix containing unit vectors? All of my engineering experience involves the latter, yet this article uses only the former. MarcusMaximus (talk) 09:18, 27 August 2008 (UTC)

I have never seen the rotation transformation being shown as I see here. I do not know where students are taught to use row vectors in rotation transformations. Of course this is basically the transpose of the equation x'=A.x and nothing is lost (or gained) by using the transpose. Also the angles are more commonly called phi, theta and psi. See Goldstein (classical mechanics; particularly the footnote) for a clearer picture. I think in the last 50 years more than a million students have read or referred Goldstein - why break the convention unless you gain something new? Ck.mitra (talk) 06:54, 23 September 2008 (UTC)
Either way, can we at least have an indication in the article as to which convention is being used? For example, by showing the entire transformation equation
$x' = xR = [x_1, x_2, x_3] \begin{bmatrix}...\\...\\...\end{bmatrix}$
we can clearly see what convention is used. The issue of which convention to use is less important, because switching between the two is trivial. However, for the record, I do support the column vector convention. Ksimek (talk) 03:58, 21 November 2008 (UTC)

I agree about the column vectors. The Greek symbols are arguable, because I think both conventions are in wide use. MarcusMaximus (talk) 07:03, 27 September 2008 (UTC)

I am for column vectors. In all of my engineering experience, I have never seen a vector represented as a row. In dealing with rotation matrices, I have always seen the notation as

$\begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix} = R \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix}$

Most references I am familiar with use the above notation. [1][2][3]

I also noticed that the rotation matrix presented in this section is the transposes of the same matrices represented else where. In order resolve these, the matrices should be consistent with the rest of the documentation. As [4] states, zxz is the product of the respective matrices, $A_z \, A_x \, A_z$, to obtain the rotation matrix. This, after Matlab's expansion, results in the transpose of the matrix mentioned in this section Euler_angles#Matrix_notation. To keep consistency, I believe these need to be changed. If there are no objections, I will change these within the week.

There should also be a discussion of rotational differences of a reference frame, a point between reference frames and a vector rotation somewhere.

[1] Stevens, B and Lewis, F. Aircraft Control and Simulation, 2nd Edition, 2003

[2] Zipfel, P. Modeling and Simulation of Aerospace Vehicle Dynamics, 2nd Edition, AIAA, 2007

[3] Unit_vector

--This is Life? (talk) 19:32, 2 April 2009 (UTC)

## Rotation convention in second moving graphic

I am questioning the label of the second moving graphic which shows a primary rotation about the z Axis Secondary about the (New) Y Tertiary about the (New) X... shouldn't the notation be Zyx??? Spock 1010 (talk) 13:26, 15 October 2008 (UTC) --(the preceding post was placed here by me after it was previously inserted as a an edit description and not actually on the talk page) MarcusMaximus (talk) 03:11, 16 October 2008 (UTC)

I agree with your assessment. It's also strange that after the initial three simple rotations, there seems to be another very large rotation about fixed Z (which is then undone), followed by a negative rotation about body x and then negative about body y to return to the original orientation. That is very confusing because it departs from the first convention to take a different route back to zero. The first moving picture right above also suffers from this. MarcusMaximus (talk) 03:20, 16 October 2008 (UTC)

## Duplicate image

Is it just me are the two images with captions: "Behavior of a reference frame embedded in a body, under modification of its Euler angles. Using z-x-z convention." showing exactly the same thing (but with the top cut off in one image)? - David Forster —Preceding unsigned comment added by 91.177.221.208 (talk) 10:43, 2 January 2009 (UTC)

## Other conventions

• The "Derivation of the Euler angles of a given frame" is a useful result. However...
• How is this result affected when a different convention (i.e. not zxz) is used (e.g. when rotating about x, y, then z) ?
• Z-bar should be Z-hat in the first definition of beta.
• In the following two sections, the convention used is unclear (presumably zxz, which tells us what alpha, beta and gamma mean):
• Relationship with physical motions
• Derivation of the Euler angles of a given frame
• The caption of the first diagram should identify which convention it is describing (presumably zxz).
• Unfortunately, by itself \scriptstyle{\hat{z}} is unreadable!

Warwick.fifield (talk) 07:06, 16 February 2009 (UTC)

## "Common Misconception"

I am quite sure that the following part was wrong so I removed it: "There is a common misconception that replacing the use of rotation matrices with quaternions will avoid gimbal lock, but this is not correct. Merely changing the representation of rotation transformation matrices with equivalent quaternion forms does not help with gimbal lock, because the problem lies in the logic and not the representation. However, the logic of interpolating between rotations is expressed with quaternions in such a way that a gimbal lock that might have occurred if straightforward Euler angles were used, is avoided." Correct me if everything I've learnt is wrong, but:

• Neither rotation Matrices nor Quaternions know anything comparable to gimbal locks.
• I believe whoever wrote the above section in fact meant to say this: A common implementation error when trying to avoid gimbal locks using Quaternions or Matrices is replacing each of the 3 elementary rotations with Quaternions or rotation Matrices. This - of course - does not avoid gimbal locks. It does not even avoid euler angles. But I'd still like to question the relevance of this implementation error. Sbstn (talk) 09:45, 17 February 2009 (UTC)

## Why partial commutativity?

The article says that the following properties hold true for Precesion, Nutation and intrinsic rotation:

$A(\delta\phi + \phi, \theta, \psi) = P(\delta\phi)A(\phi,\theta,\psi)$
$A(\phi,\delta\theta + \theta, \psi) = N(\phi, \delta\theta)A(\phi,\theta,\psi)$
$A(\phi,\delta,\delta\psi + \psi) = R(\phi,\theta,\delta\psi)A(\phi,\theta,\psi)$

Then it says:

"This partial commutability can be easily seen for γ =0 using the analogy of the gimbal. The same applies for any combination of all three rotations".

It can be shown from the former equations that any combination of these rotations is commutative. For example:

$P(\delta\phi)N(\delta\theta)A(\phi,\theta,\psi) = A(\delta\phi + \phi, \delta\theta + \theta, \psi) = N(\delta\theta)P(\delta\phi)A(\phi,\theta,\psi)$

Somebody added the word partial in the past. I would say that these rotations are whole commutative. I would like to know why the partial word was added. --Juansempere (talk) 07:36, 16 June 2009 (UTC)

Sorry, I don't understand what about we speak in the section Euler rotations ; I would be happy if anybody explained me. As I see it we speak here about infinitesimal rotations, which indeed are commutative. But of course the finite Euler rotations are not commutative. Is this section usefull ?? Chessfan (talk) 15:58, 24 April 2010 (UTC)

## Featured article?

I want to propose this article as a Featured article. Anybody disagrees?--Juansempere (talk) 21:26, 11 November 2009 (UTC)

## Matrix notation- Am I missing something here?

IN matrix rotation the three rotations are described as

\begin{align} \mathbf{p}' &= \mathbf{p}\mathbf{R} \\ &= [z,x,z]\begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ -\sin \alpha & -\cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \cos \beta & 0 & \sin \beta \\ 0 & -1 & 0 \\ \sin \beta & 0 & -\cos \beta \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & -\cos \gamma & \sin \gamma \\ 0 & -\sin \gamma & -\cos \gamma \\ \end{bmatrix} \end{align}

OK, But should the element 2,2 in Beta be 1 not -1 ? If I apply a,b,y as 0 0 0. the object should stay still so the results should be [1 0 0; 0 1 0; 0 0 1]; At the moment it comes out as [1 0 0; 0 -1 0; 0 0 1]; Which turns the object upside down, or am I missing something? —Preceding unsigned comment added by 130.123.128.114 (talk) 22:25, 21 December 2009 (UTC)

Yeah, this is wrong. Just try plugging in 0 for any of the angles, this would effectively be no rotation, and should yield the identity matrix; however, with the negative cosine terms, this does not happen. —Preceding unsigned comment added by Belegorn (talkcontribs) 23:26, 2 February 2010 (UTC)

## Should we rewrite the whole article ? !

What I have to say here may not be well accepted by earlier contributors of that article, but I must do it. The more I read it the more I discovered uncorrected errors, a lack of definitions and explanations, and in my view some confusing and unnecessary sections.

Of course there are excellent diagrams and animations, but that is not enough. If we take the viewpoint of a young student or a seasoned amateur trying to learn, for later applications, Euler angles, he or she will be very disappointed.

First of all let us distinguish between active and passive transformations, which I believe is not exactly the same thing than intrinsic and extrinsic rotations (perhaps I am wrong here). The first misconception is the fact that mostly the article speaks about rotation of frames and not of some fixed body point in a mobile frame ( and sometimes there are notational confusions between mobile and fixed frames ...). So we often do not know to what we want to apply the combined matrices -- representing the change of frames, or ready to multiply a column vector, or a row vector ?

Being more precise, the relations in section Matrix notation are double wrong: First the minus sign in all matrices is misplaced (it should be in position (1,2)or (2,3)), second the order of the matrices should be reversed. By the way they have of course to be multiplied from the right by column vectors.

I am pretty sure of that, having done the theoretical work with three different methods : geometric algebra, tensor algebra, matrix algebra. Let us call e, f, g, h the successive reference frames in the z,x,z convention, A(alpha, three), B(beta,one), C(gamma, three)the matrices (sign corrected), u, v, w, t, the successive position vectors of a body point remaining fixed in the mobile frames. Then we get, using tensor notations :

 (1)          t(i)= a(i,j)b(j,k)c(k,m)u(m)


that is in matrix notation :

 (2)          t= A B C u


There is no choice : the order of the terms is fixed by tensorial and matrix multiplication rules.

An important remark is that the A,B,C matrixes are , after sign correction, the simple matrices determining rotations in respectively the e,f,g reference frames. But as the only sign marking the rotation axis in those matrices is the position of the number 1 [respectively (3,3), (1,1), (3,3)], we can immediately reinterprete relation (2) as the successive rotations in the following order , gamma around e3, beta around e1, alpha around e3.

I must insist : we can easily transform relation (2) to obtain

  (3)         t= C'B'A'u


where A'=A , B'= B evaluated in (e) , C'= C evaluated in (e). Of course those matrices are no more simple, and we have no reason to calculate them. Now (3) is in the right order for the main interpretation.

As a conclusion I would suggest a complete rewriting of the article, reenforcing in just one convention the theoretical calculus, suppressing and/or simplifying some other sections, suppressing encyclopedical results (matrix calculation in the 12 conventions). Chessfan (talk) 15:19, 23 April 2010 (UTC)

I made some minor corrections. Chessfan (talk) 20:52, 23 April 2010 (UTC) Chessfan (talk) 22:39, 24 April 2010 (UTC)

I agree, the article seems rather scattered, with many concepts undefined. If you are willing to undertake this task, can I suggest that you create a work page, perhaps named User:Chessfan/Rewrite or something. In it, you can make a rough outline of what you propose, let other editors know the name of the page, allowing edits to your page, and if all goes well, you can then replace the present page with the new page. I would be in favor of a single convention, starting out rather simply for the new reader, then getting more involved, clearly defining concepts as they are needed. Once the fundamental concepts are clear, later sections could deal with different conventions. We should be very clear about the difference between a coordinate frame being rotated and a vector being rotated. I am guessing that the minus sign problem you made note of results from this kind of confusion, whether on your part or the original editors part. PAR (talk) 00:16, 25 April 2010 (UTC)

Thank you very much for your quick answer and proposal. It is tempting, but I must think it over. I should at least do a teamwork with somebody having experience in editing wiki articles, and writing latex. Is anybody interested ? Chessfan (talk) 13:07, 25 April 2010 (UTC)

Chessfan, on the question of "active" and "passive" rotations, am I right to guess that you are talking about rotating a body and expressing it in the same basis, in one case, and holding a body fixed but expressing it in a different basis, in the second case? I have been frustrated by Matlab, for example, because their documentation doesn't specify which they are doing. MarcusMaximus (talk) 15:18, 26 April 2010 (UTC)

That is true. I prefer active transformations, because I am used to work with geometric algebra -- which is mostly coordinate free. With tensorial algebra of course a physical law which you find covariant in an active transform will also be in a passive transform ; the transformation elements (matrices) are similar, but not identical. Thus you must be careful and define precisely what you are doing. Chessfan (talk) 20:53, 26 April 2010 (UTC)

I mostly disagree. I consider specially important to keep "encyclopedical results" in an encyclopedia. I also disagree with rewriting the whole article, because I like the structure. The matrix section clearly should be rewritten anyway. Appart of that, I like the active and passive transformations instead of the intrinsic/extrinsic. Maybe this change could be directly in the article with no more discusion, instead of using a sandbox. --Guentherwagner (talk) 08:41, 28 April 2010 (UTC)

I wish you good luck. Restoring coherence in that article without starting from grassroots seems to me to be an almost impossible task. One or two examples : -- The matrix of " matrix notation " is the transpose of the equivalent one in " Table of matrices " ; -- The extrinsic rotations are poorly explained and seem unrelated to the section " Matrix expressions for Euler rotations " ; -- The order of the matrices as explained in "Table of matrices " is poorly explained and probably false. Without a solid theoretic explanation and coherent definitions through the whole article you will go nowhere. That does not mean that there are not excellent things in the present article to be preserved. Chessfan (talk) 17:23, 28 April 2010 (UTC)

## It is urgent to fix the matrix section

All the mistakes pointed out by Chessfan) about the matrix section are quite important. I propose to correct them urgently to have the article while he does his new version. Anybody disagrees?--Guentherwagner (talk) 07:37, 29 April 2010 (UTC)

I agree - Also, to Chessfan, if you can enter equations in readable form, I will rewrite them in proper form. Also, regarding the intrinsic/extrinsic or active/passive issue, being a physicist, I view a vector as a coordinate-free entity which is fixed, while the coordinate system rotates, yielding different triplets of numbers which represent the same vector. PAR (talk) 10:58, 29 April 2010 (UTC)

OK. I need a few days time, then I will begin a user page. I will let you know. Chessfan (talk) 11:57, 29 April 2010 (UTC)

PAR, I agree with your sense of vectors. MarcusMaximus (talk) 15:23, 29 April 2010 (UTC)
Good. One of the things I really dislike is editors who assume that their field of expertise is what everything is about. While editing the Fourier transform page, there was one editor who, being an electronic engineer, insisted that the Fourier transform be dealt with in terms of time and frequency, and power in decibels, and that anything else would be just the scribbles of some holier-than-thou mathematician who was trying to snow everybody with a lot of gibberish. With that in mind, I think that this page should be written for the person most likely to use it. Am I being prejudiced to assume that the people most likely to use it will be people who are dealing with "real" situations, in physical space, rather than a mathematically defined space of ordered triplets of real numbers? PAR (talk) 18:55, 29 April 2010 (UTC)

Well, I see a difficulty here. When describing Euler angles everybody spontaneously displays an active rotation, but often forgets to associate a rigid body with the successive positions of the frames. That explains some of the errors. And also it seems that very few people know that the same final position of the body, that is the final frame, can be reached through : 1/the rotation about the Euler axes, 2/the rotation, reversed in order,about the fixed initial axes. Now if you want me to write mathematically the passive transformation, I must confess that I do not even know what that means precisely! You must tell me. The editor of the extrinsic section seems to try a passive transformation but finally he finds the second interpretation without recognizing it -- as I understand it. Anyway I want only to help you to present a sufficiently rigorous justification of the results you give, and of course to eliminate the errors. I do not want to interfere with other aspects of the article. By the way I write scientific articles on Internet, using Texmacs. So it would be very easy for me to communicate with you through postscript or pdf articles. Is that posssible, as I know it on some French forums, to send you personal messages with joined notes in .pdf or .ps ? Chessfan (talk) 14:51, 30 April 2010 (UTC)

Lets deal in two dimensions for simplicity. Suppose we have a vector and a cartesian coordinate system such that the vector is represented as [1,0], i.e. a unit vector pointing in the +x direction. If we rotate the coordinate system by 45 degrees, that vector is represented as [a,-a] where a=sqrt(1/2). On the other hand, if we rotate the vector by 45 degrees, it is represented in the original coordinate system as [a,a]. From the way people are using the words "active", "passive", I would say the first case in which the coordinate system was rotated is the "passive" case, the second is the "active" case. I'm not fully clear on the distinction between "intrinsic" and "extrinsic" however, but the sections that compare the two will no doubt contain clues. Apparently it has to do with a case of combined rotations, and whether or not the second rotation is relative to the original coordinate system or the once-rotated coordinate system. PAR (talk) 15:55, 30 April 2010 (UTC)
Should this page even have a "table of matrices"? It seems bound just to cause more confusion and be misinterpreted by everyone. Maybe this article should describe Euler angles and provide one or two specific examples. If somebody wants to use Euler angles they should get hold of a textbook. MarcusMaximus (talk) 18:11, 30 April 2010 (UTC)

MarcusMaximus I agree. There are so many possibilities, interpretation ambiguities, that Wjkipedia should concentrate on showing : 1/What one should clearly define , 2/how to construct the three matrices , 3/how to put them in the right order , 4/how to use the global matrix , 5/how to interprete the results. Chessfan (talk) 19:33, 30 April 2010 (UTC)

I still think that the existence of a "table of matrices" is appropiate, but for sure, it should be quickly fixed or quickly removed, to prevent confusion in the readers. While we speak there are people consulting the article and maybe using those matrix. Maybe we should move that table here until it gets fixed. --Guentherwagner (talk) 23:34, 30 April 2010 (UTC)
Guentherwagner, the trouble with a table is that in order to be complete, it must include active/passive, intrinsic/extrinsic, and every combination of x-y-x, x-z-x, x-z-y, x-y-z, y-x-y, y-z-y, y-x-z, y-z-x, z-x-z, z-y-z, z-y-x, z-x-y, which results in 2x2x12=48 entries. And based on experience, it's very difficult for the average person to identify which entry from which table they are actually using. What I think this article should do is teach the theory and how to build a matrix, rather than display information that is normally tabulated in the appendix of a textbook. MarcusMaximus (talk) 07:35, 1 May 2010 (UTC)
Anyway it does not harm to leave it in the article saying something like "This is the table for active intrinsic rotations compositions. Other similar tables can be obtained multiplying matrices in the required order". What I find the bigger problem is that it is ambiguous and possibly incorrect. I move it here, and maybe we can paste it back to the article if an agreement is made.--Guentherwagner (talk) 08:07, 1 May 2010 (UTC)
What is the encyclopedic reason for including any kind of table, as opposed to just presenting the theory and method? I go back to my previous statement that a table of matrices really belongs in the appendix of a textbook. I think the article should present an example of each type with an explanation of how to derive it, but not a huge eye-jarring table. MarcusMaximus (talk) 00:07, 2 May 2010 (UTC)
Of course, the only reason to keep the table would be to help the reader, saving his time and preventing him from making mistakes. That's why I think is so dangerous to leave an incorrect or ambiguous table. Its effect would be just the opposite. I still think it makes sense to leave it, but as it seems that I am the only one with this point of view, I will not insist on that.--Guentherwagner (talk) 08:36, 2 May 2010 (UTC)
Assuming the table is useful, I agree with Guentherwagner, lets include it. Next best idea is to have an off-Wikipedia link to an accurate set of tables, and hope the link does not change or disappear in the future. Next is to explain how to calculate a matrix and trust the user not to make an error. Worst is to tell a reader to shell out \$100 for a book, or get in their car and drive to the nearest library. PAR (talk) 11:36, 2 May 2010 (UTC)
I just realized that the matrices can be multi-listed, so the size of the table might be relatively small. For example, xyz intrinsic is the same as zyx extrinsic with the same angles on each axis. I don't know why I didn't realize that before. I guess the challenge is going to be explaining the concepts clearly enough to make the table useful. If you guys think that can be done, then I'll join you in support of including the table. MarcusMaximus (talk) 04:16, 3 May 2010 (UTC)

I am beginning to work at a theoretical text, but slowly ... As I don't know how to create subpages I write on my userpage which is perhaps not the best idea. Could anybody give me some tips, on my usertalkpage ? Chessfan (talk) 09:48, 3 May 2010 (UTC)

My work is on User:Chessfan/Work1 Chessfan (talk) 16:13, 3 May 2010 (UTC)

### Table of matrices

The following matrices assume fixed (world) axes and column vectors, with rotations acting on objects rather than on reference frames. A matrix like that for xzy is constructed as a product of three matrices, Rot(y3)Rot(z2)Rot(x1). To obtain a matrix for the same axis order but with referred frame (body) axes, use the matrix for yzx with θ1 and θ3 swapped. In the matrices, c1 represents cos(θ1), s1 represents sin(θ1), and similarly for the other subscripts.

xzx xzy $\begin{bmatrix} c_2 & - c_1 s_2 & s_1 s_2 \\ c_3 s_2 & c_1 c_2 c_3 - s_1 s_3 & - c_2 c_3 s_1 - c_1 s_3 \\ s_2 s_3 & c_3 s_1 + c_1 c_2 s_3 & c_1 c_3 - c_2 s_1 s_3 \end{bmatrix}$ $\begin{bmatrix} c_2 c_3 & s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 s_2 + c_1 s_3 \\ s_2 & c_1 c_2 & - c_2 s_1 \\ -c_2 s_3 & c_3 s_1 + c_1 s_2 s_3 & c_1 c_3 - s_1 s_2 s_3 \end{bmatrix}$ $\begin{bmatrix} c_2 & s_1 s_2 & c_1 s_2 \\ s_2 s_3 & c_1 c_3 - c_2 s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 \\ -c_3 s_2 & c_2 c_3 s_1 + c_1 s_3 & c_1 c_2 c_3 - s_1 s_3 \end{bmatrix}$ $\begin{bmatrix} c_2 c_3 & c_3 s_1 s_2 - c_1 s_3 & c_1 c_3 s_2 + s_1 s_3 \\ c_2 s_3 & c_1 c_3 + s_1 s_2 s_3 & c_1 s_2 s_3 - c_3 s_1 \\ -s_2 & c_2 s_1 & c_1 c_2 \end{bmatrix}$ $\begin{bmatrix} c_1 c_3 - c_2 s_1 s_3 & s_2 s_3 & c_3 s_1 + c_1 c_2 s_3 \\ s_1 s_2 & c_2 & - c_1 s_2 \\ -c_2 c_3 s_1 - c_1 s_3 & c_3 s_2 & c_1 c_2 c_3 - s_1 s_3 \end{bmatrix}$ $\begin{bmatrix} c_1 c_3 - s_1 s_2 s_3 & - c_2 s_3 & c_3 s_1 + c_1 s_2 s_3 \\ c_3 s_1 s_2 + c_1 s_3 & c_2 c_3 & s_1 s_3 - c_1 c_3 s_2 \\ -c_2 s_1 & s_2 & c_1 c_2 \end{bmatrix}$ $\begin{bmatrix} c_1 c_2 c_3 - s_1 s_3 & - c_3 s_2 & c_2 c_3 s_1 + c_1 s_3 \\ c_1 s_2 & c_2 & s_1 s_2 \\ -c_3 s_1 - c_1 c_2 s_3 & s_2 s_3 & c_1 c_3 - c_2 s_1 s_3 \end{bmatrix}$ $\begin{bmatrix} c_1 c_2 & - s_2 & c_2 s_1 \\ c_1 c_3 s_2 + s_1 s_3 & c_2 c_3 & c_3 s_1 s_2 - c_1 s_3 \\ c_1 s_2 s_3 - c_3 s_1 & c_2 s_3 & c_1 c_3 + s_1 s_2 s_3 \end{bmatrix}$ $\begin{bmatrix} c_1 c_2 c_3 - s_1 s_3 & - c_2 c_3 s_1 - c_1 s_3 & c_3 s_2 \\ c_3 s_1 + c_1 c_2 s_3 & c_1 c_3 - c_2 s_1 s_3 & s_2 s_3 \\ -c_1 s_2 & s_1 s_2 & c_2 \end{bmatrix}$ $\begin{bmatrix} c_1 c_2 & - c_2 s_1 & s_2 \\ c_3 s_1 + c_1 s_2 s_3 & c_1 c_3 - s_1 s_2 s_3 & -c_2 s_3 \\ s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 s_2 + c_1 s_3 & c_2 c_3 \end{bmatrix}$ $\begin{bmatrix} c_1 c_3 - c_2 s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 & s_2 s_3 \\ c_2 c_3 s_1 + c_1 s_3 & c_1 c_2 c_3 - s_1 s_3 & - c_3 s_2 \\ s_1 s_2 & c_1 s_2 & c_2 \end{bmatrix}$ $\begin{bmatrix} c_1 c_3 + s_1 s_2 s_3 & c_1 s_2 s_3 - c_3 s_1 & c_2 s_3 \\ c_2 s_1 & c_1 c_2 & - s_2 \\ c_3 s_1 s_2 - c_1 s_3 & c_1 c_3 s_2 + s_1 s_3 & c_2 c_3 \end{bmatrix}$

## its like spherical coordinates - ish

The best way that i could explain euler angles to someone would be by relating it to something they may already know - spherical coordinates. Define a vector on the object. Two of the coordinates (say, theta and phi, the classic spherical coordinates) give a direction, which is where the vector on the object points. The third coordinate gives the rotation of the object around that vector.

For example, an aeroplane. The vector on the object is down the length of the plane, pointing forwards. Pretty easy to understand. Theta and phi give the direction the plane is pointing. The third angle gives the rotation of the plane about that direction. All the business with XZX or YZY and all the others is doing is defining from which axes theta and phi are measured.

For example, for YZY, theta is measured from the y axis and phi measured left handedly around the y axis from the x axis.

I recognise that this only works for euler angles where the first and last axis are the same - the proper euler angles in the article. The second set, the tait-bryan angles, can instead be explained via yaw, pitch and roll, as mentioned. Maybe the article could be edited so that euler angles and tait-bryan angles are explained simply first, as spherical coords + an extra angle and as roll-pitch-yaw respectively before then confusing people by talking about rotating about rotated axes and things.

So i would support explaining it extrinsically first before explaining it intrinsically (and i hope i got that the right way around).

I'm not making and edits though, because the article is already a mess, and it seems people are already working on fixing it. Nyb.Thering (talk) 20:34, 8 May 2010 (UTC)

In my opinion, it's more confusing to say they are "like spherical coordinates only different because instead of a displacement you do another rotation about that axis, but only when the first and third Euler rotations are about the same axis". MarcusMaximus (talk) 22:54, 8 May 2010 (UTC)
I don't think that introduce the angles with rotations would be a good idea. That would force us to prove two things in advance: a) the three rotations are unique for any position of the target frame and b) for any distinct set of three rotations there is a different result frame. It seems simpler to introduce the angles in the static way, and then prove their equivalence with the rotations.--Guentherwagner (talk) 11:15, 9 May 2010 (UTC)

## Chessfan draft article

Finally I put in my workpage User:Chessfan/Work1 a more detailed reworking than I first thought technically possible for me. If you still think it is urgent to correct the existing article, some more experienced editors should read in detail my proposal, complete and perhaps correct it and then substitute it to the present text. If we instead pursue the search for new ideas we will never get it right. It's up to you. Chessfan (talk) 11:53, 9 May 2010 (UTC)

To make accesss faster, the direct link to the page of Chessfan is this one: User:Chessfan/work1, and the discussion page for the draft is here: User_talk:Chessfan/work1. Post your comments there to keep apart the discussion over the draft --Guentherwagner (talk) 05:59, 12 May 2010 (UTC)
I have made an introduction to Chessfan article about Geometric Algebra, which is in Rotation representation (mathematics)--Guentherwagner (talk) 08:24, 29 May 2010 (UTC)

Sorry, I found nowhere your introduction. I will not try to go further in that work. There are too much people on Wikipedia whose main job seems to be destroying what others and perhaps specially newcomers try to build. I am too old now for such stupid games. Thanks for your contribs. Chessfan (talk) 10:23, 31 May 2010 (UTC)

I have put the final touch on my draft article User:Chessfan/work1. The principal errors in the Euler Angles article are still there. I hope the draft article is correct. I did my best for that. Chessfan (talk) 11:42, 13 June 2010 (UTC)

## Figure on XYZ Euler angles

It looks like the 1st and 3rd are right handed rotations but 2nd (gamma) rotation is left handed. Is this intentional or a mistake? Thanks —Preceding unsigned comment added by 207.47.77.18 (talk) 19:38, 15 July 2010 (UTC)

gamma rotation in the main drawing is counter-clock-wise when looking at it from the top of the Z axis, as all of the others are. Do you referred to that? —Preceding unsigned comment added by Guentherwagner (talkcontribs) 19:06, 21 July 2010 (UTC)

## Reference frame of reference

Hi Juan, I agree the phrase 'reference frame of reference' sounds awkward, but it is technically correct:

Euler angles are a means of representing the spatial orientation of any frame of reference (coordinate system) as a composition of rotations from a reference frame of reference (coordinate system).

Vectors can be defined by coefficients relative to a frame of reference. Similarly, frames of reference can be described by other coefficients (Euler angles) relative to another frame of reference, a reference frame of reference so to speak. It doesn't sound so confusing if you use the word coordinate system instead:

Euler angles are a means of representing the spatial orientation of any (coordinate system) as a composition of rotations from a reference (coordinate system).

Technically, the coordinate systems need to be orthonormal and of the same handedness too.

The formulation needs to be improved, but I don't think simply dropping one of the two consecutive instances of the word 'reference' does the trick. Replacing 'frame of reference' with 'coordinate system' would be better. Any suggestions for a better solution? Martijn Meijering (talk) 16:54, 26 July 2010 (UTC)

Well. I admit that it could make sense somehow, but anyway it sounds really weird, and it can be replaced by equivalent expressions. I would really prefer "coordinate system", "set of axes" or just "frame" for the referred frame.
Something more important is what you have just pointed out. Euler angles are not defined when the referred frame is of the opposite handedness. Don't you agree that such a remark should be added to the article? --Juansempere (talk) 17:31, 26 July 2010 (UTC)
We are talking about "the orientation of a rigid body", and about *rotations*, so opposite handedness doesn't come into the question. David Biddulph (talk) 19:23, 26 July 2010 (UTC)
It would if we replace 'frame of reference' with 'coordinate system', but I think you've shown us a good solution: use 'orientation' instead of 'frame of reference'. Martijn Meijering (talk) 20:05, 26 July 2010 (UTC)