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 = [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 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.
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)