[math-fun] complex #'s explore circle group; quaternions explore SU(2) ?
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 ???
Perhaps "space-filling curve" might be better than "random walk" ? At 03:51 PM 3/15/2018, Henry Baker 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 ???
It's a different covering structure, but the products of 2+-i, 2+-j, 2+-k generate a dense covering of all directions. (Divide the generators by sqrt5 to get units.) Except for the cancellations of conjugates, like (2+i)(2-i) = 5, every product is different. You can view the pre-space as a tree with node degree 5, except the root is degree 6. The post-space is points on the unit quaternion sphere. I assume the covering is near uniform, but haven't seen the confirming theorem. Rich PS: Does anyone else miss the XCT instruction? The 7094 & PDP6/10 had it, but it seems to have vanished. I guess it's an architectural nightmare for the hardware folks. --Rich --- Quoting Henry Baker <hbaker1@pipeline.com>:
Perhaps "space-filling curve" might be better than "random walk" ?
At 03:51 PM 3/15/2018, Henry Baker 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 ???
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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 ???
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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 ???
It might be worth saying that that the exponential map for su(2) is particularly nice. If B is a skew Hermitian matrix such that det(B) = 1, i u_1 u_2 + i u_3 B = -u2 + i u_3 -i u_3 with u_1^2 + u_2^2 + u_3^2 = 1, then q = e^{theta B} = (cos theta) I + (sin theta) B, just like the complex numbers! Then we see that exp is injective on the open ball {theta B \in su(2) | det(B) = 1, 0 \leq theta < \pi}. Also, except when q = -I, a principal log of q is found uniquely, unlike the case of SO(3) where the angle \pi causes troubles. A few years ago I gave a long homework about all this, and here is my solution! Best, — Jean
On Mar 16, 2018, at 5:19 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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 ???
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