Dear Fred, Sorry -- I seem unable to consistently remember to change the recipient from math-fun to an individual I'm writing to. So I'll follow suit and continue in public. Since conjugation of the unit quaternions S^3 by any quaternion implements only arbitrary rotations of the equatorial S^2, fixing the real axis and preserving all the S^2's of various radii given by Re(q) = constant, there are actually a lot of measures on S^3 preserved by just conjugation. Left multiplication of S^3 by an arbitrary unit quaternion implements the (left) isoclinic rotations -- (half of all the) rotations that move everything by the same angle, with all of R^4 as the common eigenspace. I'm almost sure that only (scaled) Haar measure on S^3 is invariant by this S^3 action, but I don't have a proof at the moment. --Dan On Jul 6, 2014, at 4:41 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
You apparently intended this for me personally; but since it was addressed to the list, I feel it only polite to others to reply publicly.
On 7/6/14, Dan Asimov <dasimov@earthlink.net> wrote:
Dear Fred,
Where you say "uniformly distributed unit quaternions" I am confused. Uniformly distributed in what?
The 3-sphere: presumably the Euclidean measure there is essentially the only one to remain invariant under conjugation by quaternion --- or perhaps composition --- I guess that makes no difference [vide Haar measure ...]
I also don't know what "transitive in all four components" means.
The random generator has obvious symmetries involving transposing variables and sin,cos, which permute the vector components transitively.
Clicked on the link to Stack Exchange and confronted a section of . . . I'm not sure what language of something by Ken Shoemake, but the question was not written in a way I can understand easily, so I don't care to read it.
Try instead http://planning.cs.uiuc.edu/node198.html
I certainly am reasonably familiar with quaternions and with probabilityt distributions, so if you give me just a few hints I can probably understand.