Talk:Quaternions and spatial rotation: Difference between revisions

Revision as of 22:47, 6 December 2005

Revision as of 20:50, 14 December 2005

Line 30:

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 = ['''A'''|'''B'''] '''B''' - sqrt(1-['''A'''|'''B''']^2) '''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.

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.

Line 39:

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.

==Versors==