Talk:Quaternions and spatial rotation
This article is marked:
Author: M. Romero Schmidtke. Translator: August Pieres.
Please can the authors confirm that this is original work, donated under the GFDL?
- The article appears to be a translation from this Enciclopedia Libre article; M.Romero Schmidtke is a regular contributor there. AxelBoldt 01:26 Apr 7, 2003 (UTC)
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.
M. Romero Schmidtke, April 26 , 2003.
The following text was cut from the main page. -- Fropuff 03:48, 2004 Aug 2 (UTC)
a thought
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.
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.
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. Jwwalker
29 November 2005
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:
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.
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 2t 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.
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.
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 )+usin(t ), which rotates 3-vectors about the axis u by an angle 2t, sqrt(q)=-cos(t /2)+usin(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.
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.
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.
14 December 2005
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.
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.
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.
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.
- 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 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 · y or xTy or ⟨x,y⟩; you may also use the physicists' ⟨x|y⟩ notation if you like it; it certainly is not restricted to infinite dimensions. Good luck! -- Jitse Niesen (talk) 23:02, 14 December 2005 (UTC)
3 January 2006
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.
Versors
Versors are described in Earliest Known Uses of Some of the Words of Mathematics (under Tensor) and in this tutorial, but I found neither clear or relevant enough to be included in the References. -- Jitse Niesen (talk) 11:29, 10 September 2005 (UTC)