Tim Wescott wrote:
> On Sun, 21 Jun 2009 02:23:11 +0000, Martin Eisenberg wrote:

>> How have you decided to project the Gaussian from Euclidean
>> space to the 3-sphere?

> You mean from Euclidean 3-space to the surface of the 4-sphere?

I was using the mathematical terminology where the n-sphere is the
shell with intrinsic dimension n, so the set of unit quaternions is a
3-sphere in this lingo.

You can visualize it just like I gather you did your content
integrals: Imagine that a tiny ball appears in midair and grows. It
decelerates, reaches a maximum radius, and collapses back into a
point. What you saw is a 4-ball falling through E^3, with the time
segment isomorphic to a diameter and each (3-ball, instant) pair a
slice through the 4D solid. Likewise, the 3-sphere is the disjoint
union of (2-sphere, instant) pairs, and you can identify the time
segment with the quaternion rotation angle and each sphere with the
set of axis directions.

> I'm still trying to figure that out!

My earlier Riemann sphere idea generalizes to stereographic
projection, but the Von Mises-Fisher distribution might be more
useful in terms of available theory.


Martin

--
Quidquid latine scriptum est, altum videtur.