Talk:Quaternions and spatial rotation: Difference between revisions

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== Compared to rotation matrices ==

Author: M. Romero Schmidtke.

Translator: August Pieres.

The article says

Please can the authors confirm that this is original work, donated under the GFDL?

"Compared to rotation matrices they are more compact, more numerically stable, and more efficient"

:The article appears to be a translation from [http://enciclopedia.us.es/wiki.phtml?title=Cuaterniones+y+rotaci%F3n+en+el+espacio this Enciclopedia Libre article]; M.Romero Schmidtke is a regular contributor there. [[User:AxelBoldt|AxelBoldt]] 01:26 Apr 7, 2003 (UTC)

However, quaternion rotation requires 24 add/mul operations but a 3x3 matrix requires only 15 add/mul operations. Also, the "more numerically stable" claim is unjustified and I cannot find a reference. <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.47.47.163|80.47.47.163]] ([[User talk:80.47.47.163#top|talk]]) 15:22, 19 May 2020 (UTC)</small> <!--Autosigned by SineBot-->

:I'm not sure where are those numbers from. As far as the number of operation goes, quaternions require less operations to compose rotations and more operations to apply a rotation to a vector. What ends up being more efficient depends on the application.

Thank you, August Piers, for this translation of my article. I would surely not have done better. And yes, it is an original work, writen specifically for ''enciclopedia libre'' and the spanish ''wikipedia''.

:Besides, efficiency is not only the number of mul/adds. Quaternions take less than half the memory and, accordingly, bandwidth. Which is, again, important in some applications (e.g. tangent space calculations on a GPU).

:Applying and composing rotations aren't the only operations to consider either. Quaternions are easier to interpolate (again, useful for tangent space calculations on a GPU for example, but also for animation, modeling, etc...).

:"more numerically stable" -- this is un-doubtfuly true. When repeatedly composing rotations (eg in rigid body simulations) rotation matrices will inevitably become non-orthogonal. There are different ways to re-orthogonalize them, trading off precision and performance. In contrast to those, quaternions don't suffer from that issue at all. [[User:Ybungalobill|bungalo]] ([[User talk:Ybungalobill|talk]]) 20:09, 19 May 2020 (UTC)

::"compact, efficient, and numerically stable" better are sourced or reasoned indeed IMO.

::The rotation matrix in the tooltip of the linked words "rotation matrices" in the sentence of above quote alone for example requires only Theta, making it more compact than a quaternion. Here reasoning is missing.

::"efficient" in what respect: efficiency is a kind of output per input; are quaternions less time consuming to use for a high research effort? This would make them inefficient. So the respect is missing.

::"numerically stable": comparing rotation matrices sin and cos range 0...1 being multiplied with each other multiple times with unit quaternions being multiplied with each other multiple times, I don't get an idea of a significant different numerical stableness. Non-othogonal is a semantic issue, stableness is rather a syntactic issue. [[Special:Contributions/2A02:2455:30C:7D00:3931:FDDB:9442:9713|2A02:2455:30C:7D00:3931:FDDB:9442:9713]] ([[User talk:2A02:2455:30C:7D00:3931:FDDB:9442:9713|talk]]) 23:55, 29 August 2025 (UTC)

== Labelling the formulas in the Alternative Convention section ==

M. Romero Schmidtke, April 26 , 2003.

The paragraph argued that usage of the Shuster convention is discouraged, as did the cited article "Why and How to Avoid the Flipped Quaternion Multiplication" by Sommer et. al. But the formulas in the paragraph are not clearly distinguished (except by red minus signs). These formulae are beginning to show up on Google Images out of context, creating a lot of confusion to students. I tried to add labelling "\qquad \text{alternative Convention, usage discouraged} to the right of the formulas but the edit got reverted by a anti-vandalism bot. If anyone (especially registered users) agree, please help with the edit.[[Special:Contributions/184.147.40.19|184.147.40.19]] ([[User talk:184.147.40.19|talk]]) 04:04, 25 January 2022 (UTC)

The following text was cut from the main page. -- [[User:Fropuff|Fropuff]] 03:48, 2004 Aug 2 (UTC)

:I already reversed the bot edit a few minutes ago. [https://en.wikipedia.org/w/index.php?title=Quaternions_and_spatial_rotation&diff=1067782018&oldid=1067781032] I'll just nod and pretend I knew for sure it wasn't vandalism. Cheers. <small>_''signed'',</small>[[User:Willondon|Willondon]] ([[User Talk:Willondon|talk]]) 04:14, 25 January 2022 (UTC)

== a thought ==

== Recover Axis Angle ==

One can specify a rotation in n dimensions by specifying two unit vectors '''A''' and '''B'''. The specified rotation is that which maps '''A''' onto '''B'''. The ''axis'' of rotation in n dmensions is a surface of (n-2) dimensions. Only in three dimensions is this axis itself one-dimensional.

Since quaternions are rotations about arbitrary axis, it makes sense as a matter of convention that the angle is ''always'' a non-negative number. Moreover, the angle must always be in the 1st and 2nd quadrant as all other quadrants have equivalent negative angles. If for example we want to represent a rotation of <math>-\frac{\pi}{6}</math> about the <math>\bigl( \begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix} \bigr)</math> axis, this is entirely equivalent to a rotation of <math>\frac{\pi}{6}</math> about the <math>\bigl( \begin{smallmatrix} -1 \\ 0 \\ 0 \end{smallmatrix} \bigr)</math> axis.

I would guess, then, that a quaternion of rotation is [[equivalent]] to the [[cross product]] of the two unit vectors '''A''' and '''B''', which is also a vector only in three-space, and whose [[magnitude]] also varies as the sin of the angle.

So indeed the formula to recover the rotation angle presented <math>\theta = 2\, {\rm arctan2} \left(\sqrt{x^2+y^2+z^2},\,w\right)</math> will indeed produce the correct answer, in the 1st and 2nd quadrants because thee first argument is always going to be positive, but it is un-necessarily complicated. I propose to recover the angle we simply use

I'm not sure what you mean by "The specified rotation is that which maps '''A''' onto '''B'''." In 3 dimensions, there can be many rotations that map '''A''' onto '''B''', not just the ones with the cross product as axis. [[User:Jwwalker|Jwwalker]]

<math display="block">\theta =2\, \arccos \left(w \right)</math>

which also produces results in the 1st and 2nd quadrants, Once the angle is recovered, then the axis is simply the vector part divided by the sine of the half angle

29 November 2005

<math display="block">\vec{\rm z} = \tfrac{1}{\sin \left( \frac{\theta}{2} \right) }\,\vec{v}</math>

I think perhaps the original suggestion is well defined, but the contributor is confusing the point with incorrect vocabulary and muddled concepts. What is probably meant is as follows:

<!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Jalexiou|Jalexiou]] ([[User talk:Jalexiou#top|talk]] • [[Special:Contributions/Jalexiou|contribs]]) 16:26, 11 June 2022 (UTC)</small>

One can specify a rotation in n dimensions by specifying two ''reflections''. Indeed, this is the definition of a rotation. Each reflection can then in turn be specified by a corresponding unit vector, orthogonal to the (n-1) dimensional subspace which is invariant under the reflection. There are two such vectors: '''A''' and '''-A'''. If we choose positive notation by convention, '''A''' is mapped to '''-A''' under the reflection represented by '''A'''.

== rotations and matrices ==

Composing with a second reflection '''B''' moves '''-A''' to its final desination = 2['''A'''|'''B'''] '''B''' - '''A''' , where [x|y] specifies the finite dimensional inner product, following the notation of quantum mechanics for infinite dimensional inner products (<x|y>). ['''A'''|'''B'''] is of course the cosine of the angle between '''A''' and '''B''', call it ''t'', and defines this angle unambiguously. Therefore, '''A''' is rotated by an angle 2''t ''through '''B''', in the two dimensional subspace containing both '''A''' and '''B'''. The 'axis' of rotation is the n-2 dimensional subspace fixed by the composition of the two reflections '''A''' and '''B''' and is orthogonal to the plane containing '''A''' and '''B'''. I say 'axis' because its use in this context is an unforgivable corruption of terminology.

:{{xt|A geometric fact independent of quaternions is the existence of a two-to-one mapping from physical rotations to rotational transformation matrices.}}

Returning to the topic at hand, namely quaternions, one may form the rotation which takes the three dimensional vector '''A''' to '''B''' in the plane of '''A''' and '''B''' by calculating sqrt('''AB'''). As the second contributor correctly notes, there are infinitely many other rotations which move '''A''' to '''B'''. Of course, if '''A''' and '''B''' are not of unit magnitude, the resulting square root needs to be normalized.

Shouldn't that be the other way around? That is, one rotation has two representations – not that one matrix can represent two rotations. —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 05:41, 2 August 2023 (UTC)

The square root of a quaternion rotation operator is the quaternion operator which, when applied twice to all vectors in three space, yields the same movement as the orginal operator applied once. The square root rotates about the same axis, but by only half the angle. So given '''q'''=-cos(''t '')+'''u'''sin(''t ''), which rotates 3-vectors about the axis '''u''' by an angle 2''t'', sqrt('''q''')=-cos(''t ''/2)+'''u'''sin(''t ''/2). '''u''' is of course a unit pure quaternion specifying the axis of rotation (i.e. a three dimensional vector), and the resulting operator moves vectors by an angle ''t'' about '''u'''.

== xyzw order? ==

An interesting alternative axis of rotation for moving '''A''' to '''B''' is the unit vector bisecting the angle between them. In this case, the rotation operator is a pure quaternion (''t ''= 90 degrees) formed by computing ('''A'''/|'''A'''|+'''B'''/|'''B'''|)/|('''A'''/|'''A'''|)+('''B'''/|'''B'''|)|, where |x| denotes the norm or magnitude of x.

I believe when written as four numbers, it is relatively common to write them with the cosine term last, ie it is a 3-vector containing the axis multiplied by sin/2, followed by cos/2. This lines up the vector with 3-space vectors written in homogenous coordinates, making it very popular in code that already has to deal with those. Anybody know if there is any real convention to do it this way, and/or the way shown in the article with the cos term first? [[User:Spitzak|Spitzak]] ([[User talk:Spitzak|talk]]) 17:55, 8 October 2024 (UTC)

P.S. As an after thought, the main article on quaternions and spatial rotation needs to be completely rewritten. It is confusing and misleading in places and was clearly authored by individuals who rarely use quaternions in practice, and therefore lack intuitive understanding.

:In many contexts, quaternions are written as the sum of four parts in a specific order: the real part, the i part, the j part, and the k part. That form carries over to having the cosine be first, and is the one that I am most familiar with. If you've seen it the other way ... then there are (at least) two ways to do it. If this is a vote, I vote for having the {{math|cos}} term first. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 18:19, 8 October 2024 (UTC)

14 December 2005

::Not suggesting it be changed, just wondering if it should be noted in case somebody is looking at code that works that way. Looking at some existing code such as Pixar's math library, it looks like they are pretty careful to make the api return the real and vector parts as two different pieces, and indexing only works on the vector (ie you write quat.vector[2] to get the z, not quat[3] or quat[2]). I think though there are some tiny advantages on GPUs to storing the quaternions as xyzw so this is often done. [[User:Spitzak|Spitzak]] ([[User talk:Spitzak|talk]]) 23:56, 8 October 2024 (UTC)

:::Looking at Pixar's `GfQuatd` it does appear to lay out the memory as I said, but they definitely hide this in the api. There is a 4-number constructor but it has the real part first, then the imaginary vector. In the private part of the data structure they store 4 doubles and they put the real part last, this may have been done so the memory can be sent to the GPU. Some other libraries I looked at seem to store the real part first. [[User:Spitzak|Spitzak]] ([[User talk:Spitzak|talk]]) 00:08, 9 October 2024 (UTC)

== Reasoning for #Performance_comparisons missing, unspecific ==

Well, it seems either no one is tracking this article's discussion, or no one is critically reading it, since the egregious error in my equation for the final destination of '''A''' above went unnoticed and uncorrected by others. It is now correct. It seems I muddled the forms for orthogonal bases and self referential bases in my head, which just goes to show extemporaneous online mathematical discourse is error prone. The correct equation is completely self referential.

[22] and [23] are currently present. [22] is unspecific: is it a replier in some forum thread? [23] needs a qualifier such as "as in code published at ...".<br>

Ironically, I had the presence of mind to correct the omission of the sqrt in ['''A'''|'''B'''] '''B''' - sqrt(1-['''A'''|'''B''']^2)'''A''' the first time I revisited the page, but didn't notice the orthogonal basis adjustment factor was unnecessary, and incorrect in this context.

<br>

Method in table 1 under Storage requirements is linked to articles but does not specify the look further which makes "Rotation matrix" at least ambigious (2D? 3D? 4D?) and the numbers in the storage column are not referenced in the paragraph below the table which is why they are not reasoned.<br>

In spite of these foibles, per wiki etiquette, I am announcing here in the discussion that I have decided to rewrite this article in stages over the next few months, whenever I have a few hours to kill. This is a heads up, in case someone else has adopted this article, or may wish to collaborate on the revision. While I am evidently not the most competent person for the job, apparently no one in the current time frame has greater competence.

<br>

"Performance comparison of rotation chaining operations" does not include angle-axis after angle-axis which is 2 x angle-axis of "Performance comparison of vector rotating operations".<br>

When I am finished, the article will be completely new, and I will therefore delete the copyright issue noted above, as it will become irrelevent.

<br>

"Performance comparison of vector rotating operations" does not explain intermediate matrix, does not consider transforming vector to quaternion first which requires C and S of Theta/2 which is 2 sin/cos and 1 mul/div.<br>

:Hi. Just a note that you are being watched :) . I don't know much about quaternions though and I never used them, so don't expect any help on the maths from me. Please go ahead and improve the article as you see fit. I hope you'll create an [[Wikipedia:Account|account]] first; that makes it easier to communicate and it enables you to tell us your background; alternatively, if that's what you want, it gives you more anonymity because it hides your IP address. Oh, and don't use this [ | ] notation for inner products unless it really is standard in this context; it most certainly is not standard in the general area of mathematics. I use ''x'' &middot; ''y'' or ''x''^T''y'' or &lang;''x'',''y''&rang;; you may also use the physicists' &lang;''x''|''y''&rang; notation if you like it; it certainly is not restricted to infinite dimensions. Good luck! -- [[User:Jitse Niesen|Jitse Niesen]] ([[User talk:Jitse Niesen|talk]]) 23:02, 14 December 2005 (UTC)

<br>

"Rotation matrix" is likely a rotation around a 3D base vector as axis which is not equal an arbitrary angle-axis or a quaternion and thus should be marked with an asterisk.<br>

3 January 2006

<br>

Performance comparisons can consider time ___domain when parallel computing if this whole section is original work: a dot product would be 1 op instead of 3 mul and 2 add for example. [[Special:Contributions/2A02:2455:30C:7D00:3931:FDDB:9442:9713|2A02:2455:30C:7D00:3931:FDDB:9442:9713]] ([[User talk:2A02:2455:30C:7D00:3931:FDDB:9442:9713|talk]]) 23:41, 29 August 2025 (UTC)

Well, it is good to know ''some'' discussion takes place before revisions. Regarding anonymity, I have no personal online access and use public terminals exclusively. For the purposes of this effort, I've been making a point of using the same public terminal so that the URL in the history consistently identifies me. Therefore, contributing without an account gives me the greatest anonymity, for I can change terminals in a trice.

Regarding notation, I will stick with what I've used above for reasons which will become apparent. I've nearly completed the revision in my head, and will begin data entry later this week. As far as unconventional notation goes, it is worth noting that Conway, in his recent book on octonians, implicitly defines the norm of a vector as the square of what is usually considered to be a norm, as a contradistinction to the magnitude. It is a good idea, and it sets a useful precedent, something which someone of Conway's stature can easily get away with.

14 Jan 2006

A word of encouragement: The author of the article is doing a better job of explaining a somewhat tricky subject than is usual - rather than simply displaying erudition, as some authors do, he/she seems to be genuinely aware of the pitfalls that can trap (and even dishearten) the interested but non-expert reader; if the Wikipedia is to be useful, it is to this level that I believe it should be targetted. Thanks. PGE.

17 January 2006

Since I'm still quite a wiki-neophyte, there are some editing details that need attention

* The citation mechanism I'm using is ad hoc, created merely to enter the data. What is needed is a footnote placing the data at the end of the article, with a numbered link in situ. I don't know how to do this.

* As things progress, the addition of several figures would be illuminating. I don't know how to make or insert such graphics.

* The math mode creates very awkward font size changes. My predisposition thus far is to make it look right in an ad hoc way. This will become increasingly burdensome as things progress. On the other hand, the formatting choices of the math system are usually positively repulsive aesthetically.

I trust there are more experienced individuals who can help with this. Thus far, I've mostly simply imitated notation existing in the page, and gleaned a few techniques from the "cheat" link on the edit help page. Therefore, if someone would fix a citation, I can figure out what needs to be done with the rest of them.

: Adding an image to this wikipedia is a simple matter - I did the images on the [[Mobius Transformation]] page. Creating them is quite a bit trickier. I did the 3d images with a tool called "PovRay", but that is not an interactive graphical tool: you have to greate a description file and tell your machine to render it. [[User:Pmurray bigpond.com|Pmurray bigpond.com]] 23:56, 20 February 2006 (UTC)

==Versors==

Versors are described in [http://members.aol.com/jeff570/t.html Earliest Known Uses of Some of the Words of Mathematics] (under ''Tensor'') and in [http://www.itk.org/CourseWare/Training/QuaternionsI.pdf this tutorial], but I found neither clear or relevant enough to be included in the References. -- [[User:Jitse Niesen|Jitse Niesen]] ([[User talk:Jitse Niesen|talk]]) 11:29, 10 September 2005 (UTC)

== In Summary ==

The "Quaternion rotation" has a summary where it gives the cookbook formula for making a quaternion given an angle and a unit vector.

Could someone add the converse - the formula for getting the vector and the angle from an arbitrary quaternion? Yes, I suppose the reader can work it out from what's given but it would be nice and encyclopedic to include it.

Actually - why not make this summary a separate section? Name it "this is how you actually do it", or something.

Oh - and why use <''u'', ''v''> (rather than the dot operator) for the dot product when the cross operator is used for the cross product?

As an afterthought: please don't remove these equations. There's an annoying trend on the math pages for hard-core math guys to remove the cookbook ("analytic") stuff. I imagine they feel that people should make the effort to understand the deeper beauties and derive the equations themselves, because the actual resulting equations are uninteresting: but it really imposes a learning curve and makes it difficult for people like me who just want to write code.

== Too technical ==

This article certainly needs many cleanup, accessibility change, tone and style change, and even an rewrite. Personally, I think this article should be moved to wikibooks, but it is also true that wikipedia could use an article that explain how to do spatial rotation using quaternion in detail, through '''not excessively'''(translation: not too long that scare away all people).

I suggest that it be moved to wikibooks, and either rewrite this article here, or take the following measures to take it up to encyclopedia article standard:

* The introduction on certain special properties of quaternion can be summarized and moved to the main quaternion article.(And yes, the non-commutativity of rotation can be better explained by using a diagram)

* While useful for an indepth understanding, I don't think doing rotation would need understanding projective space, operator space, and all the like. The concepts section cannot stay, I'm afraid

* Historical terminology(!) can be moved to their respective article.(PS: does this section imply that sometimes historical terms will be used?!)

* Modern terminology can be moved to the last part of this article, sort of like an appendix, until we know what to do with them.

* I know there is difference in notation in computer science and mathematics, even for the same concepts, but I think for the part that shows the mathematics, derive how to do rotation, prove that it is correct, etc, they should use standard mathematics notation(We have LaTeX, sir) , so that it is accessible to mathematician.

* And then, if there's even an algorithm section, that can use the computer science group's notation, if there's a difference.

* Instead, the algebraic rules section should be moved to the beginning, as a summary/revision.

* The rest is ok as it stands. Through the derivation of quaternion for rotation(That is, the main point of this article, afterall) need quite some expansion.

* And as a sidenote, the citation can be done in a footnote style, rather than brackets everywhere(check any featured article and notice some small number that occassionally appear at the end of a sentence.)

That's all for now, thanks. --[[User:Lemontea|Lemontea]] 02:34, 23 April 2006 (UTC)

:Agree. The article is not too technical; its too verbose. The mathematics behind this is actually quite easy, but you would never guess that from this article. The sheer heft of this article is daunting. This whole thing should be moved to wikibooks, and re-written from scratch. [[User:Linas|linas]] 14:56, 18 August 2006 (UTC)

:Yes. FAR too technical. The person talking about rotating vectors makes much more sense to me than talking about reflections. The person saying reflections are the basis of the definition of rotation may be right for 4 or higher dimensions, but I really don't think this needs to be mentioned in the article at all, it only confuses me.

== Copied? ==

Has this article been copied from some other site? For one thing, there's this:

''...which will not equal the absolute position implied by the sum produced in another representation. This problem arises when simply translating the origin some small distance from the original representation. '''(illustrative figure here)''' The positions are unique absolute physical positions, whereas the n-tuple is merely a symbolic abstraction....''

Which looks as if a hyperlink was lost in the conversion to text, and nobody noticed.

For another thing, this is not Wikipedia style at all. It reads like a very, very boring book. Worse still, it doesn't actually make sense to anyone who doesn't already know what the article is about. I'm trying to make sense of it, but it's just frustrating the hell out of me.

My brother once said, "I've looked on Wikipedia, but I didn't understand it." This is my first go at that experience. [[User:86.136.82.105|86.136.82.105]] 15:31, 9 May 2006 (UTC)

::The source is mentioned at the top of this talk page. Actually this is a small corner of wikipedia that I'm afraid are rarely visited, as shown by how my comment and suggestions(see above) recieve no reply, not even acknowledgement or objection. And it's true that this topic is a quite specific and difficult one. Anyway, I tried to understand [[quaternion]] by starting from the matrix representation of [[complex number]]. ''Why'' rotation was done ''that'' way is still unknown to me. --[[User:Lemontea|Lemontea]] 11:25, 10 May 2006 (UTC)

== no pictures ==

That's pretty extreme to discuss that whole article without a single picture.

Re. ^^^^:

People's heads probably started aching before they got up to there.

== Could we have something a bit more to the specific topic? ==

Could we have something a bit more to the specific topic? The topic is not [[quaternions]], which already exists (elsewhere), it is quaternions and spatial rotation. I would be expecting something to tell me

#how to represent rotation in terms of quaternions

#why quaternions are better than [[matrix]]es.

It seems that [[Blender (software) | Blender]] uses quaternions to represent rotations; perhaps that would provide some assistance from the [http://www.blender3d.org/cms/Development.81.0.html Blender documentation]. [[User:M.e|m.e.]] 05:38, 16 October 2006 (UTC)

Ya, *unfortuneately* (from a user standpoint), Blender does use Quats for "bone" rotation representation in the "IPO" (graph) editor.

== Book Description and zy rotation ==

I changed some of the description for the initial "book" example, it appeared to me that the "spine direction" description was incorrect. If I have changed it incorrectly, please advise .. (and I'll be confused :) if so).

Also in the description of "In fact, the composition zy represents a single rotation of 180 degrees about the [-1 1 0] axis", does that imply from [0 0 0] to [-1 1 0] ? I assume so, as otherwise I don't understand how an axis can be defined by a single point:) .. Unless it's more "understood" math terminology that I'm not familiar with :)

[[User:Mike Stramba|Mike Stramba]] 17:19, 2 December 2006 (UTC)

:Conventionally, an axis is represented by a unit vector (magnitude 1), such as the above mentioned [-1 1 0], so your assumption is correct. Btw, in two-dimensional space, a rotation axis is actually defined by a point. – [[User:AdrianLozano|Adrian Lozano]] 08:02, 4 December 2006 (UTC)