Yes, this observation is very helpful. I now recall that this mapping -- at least applied to the quaternions -- was also noticed by Hamilton himself: If "u" is what Hamilton called a "pure" unit quaternion (no real part), then Hamilton noticed that x+y*u (real x,y) is isomorphic to the ordinary complex numbers, so exp(x+y*u) is well-defined. So, exp(u*t) will "explore" a circle in this space, in the same way that exp(i*t) explores the circle group. --- So now if the choice of directions "u" are made uniformly -- i.e., we can uniformly randomly pick a point on the ordinary sphere located in ordinary 3D -- and subsequently and *independently* pick a random point on this circle -- then the overall result should be uniformly random. Is this correct? At 05:42 PM 3/15/2018, Jean Gallier wrote:
The exponential map from the Lie algebra su(2) of SU(2) to SU(2) is surjective,
(for example, see my manuscript http://www.cis.upenn.edu/~jean/gbooks/manif.html <http://www.cis.upenn.edu/~jean/gbooks/manif.html>, Section 1.4),
so it gives you a smooth parametrization of SU(2) in terms of three real parameters.
Recall that su(2) is the set of skew hermitian matrices,
ib c + id -c + id -ib
with a, b, c real.
I dont know if this is what you wanted.
Best,
-- Jean Gallier
On Mar 15, 2018, at 6:51 PM, Henry Baker <hbaker1@pipeline.com> wrote: We know that the circle group is enumerated uniformly by
cos(t)+i*sin(t), for i in [0,2*pi)
using a single (real) parameter t.
Is there a way of uniformly enumerating the unit quaternions * with 1 parameter (probably some sort of random walk) ? * with 2 parameters (probably some sort of random walk) ? * with 1 complex parameter ????? * with 3 parameters ???