Dear Dan, 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.
Thanks,
Dan
P.S. Did you watch any of Wimbledon (on the telly) ? I don't care for spectator sports in general, but top-notch tennis I like a lot.
Don't have a telly. Nor time to watch it. Regards, Fred
On Jul 6, 2014, at 2:09 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A feature of quaternions that is even more remarkable has not so far been explicitly discussed at all: the fact that uniformly distributed unit quaternions (transitive in all 4 components) correspond to "random" 3-space rotations (scalar angle unrelated to 3 axis vector components)!
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