On Fri, 4 Jul 2003 19:42:27 +0100, Stephen Riley
>I suppose if unit quaternions represent points on a 4-dimensional unit
>hypersphere, it might be hoped (I'd better not say expected!) that a
>slerp on a great circle in that space would also be a minimum distance
>in 2 and 3-dimensions (or rather follow a minimum energy curve/gradient
>or whatever the mathematical term) in lower dimensions, especially since
>the vector part only moves in 2D. Is that right? Are geodesics in higher
>dimensions just 2 dimensional? In any case, I've no doubt coming to the
>correct conclusion rigorously would be a much deeper issue!
On a sphere in of any positive dimension a Slerp gives a uniform speed
great circle arc, which is both planar and a geodesic. Different ways
to characterize the quaternion Slerps are: straightest path, shortest
path, and constant angular velocity. In Lie group terms, that last one
is called "one-parameter-subgroup", multiplying the same infinitesimal
rotation times itself over and over. An ordinary sphere in 3D is not a
group, so we can't use this last idea, but the first two still work.
Now try drawing on a cylinder, which is a rolled up plane; most of the
geodesics look like coiled springs.
Quote:>Is there a way of visualising a 4-dimensional sphere btw?
Yes, but not without distortion. Stereographic projection is a purely
geometric way. Plant a lighthouse at the "south pole", (0,0,-1), and
place a sheet of paper opposite; now trace rays to establish a map
between points on the sphere and points on the paper. This procedure
for the ordinary sphere puts the north pole at the center and circles
of longitude become radiating lines. It works in 2D and 4D as well.
While that has the advantage of mapping circles to circles (counting
those radial lines as really big circles!), it puts the south pole at
infinity. A more compact depiction uses geodesics. Start at the north
pole and walk straight measuring distance. Transfer the measurements
to a flat piece of paper. The ordinary sphere becomes a disk, with all
points on the rim of the disk corresponding to the south pole. When we
apply the same idea in 2D we get a line segment from -pi to pi; in 4D,
we get a solid 3D ball of radius pi.
Since it's boring and unhelpful to draw a picture of a ball, we often
see pictures of the sphere in 4D dissected into circular fibers, the
beautiful Hopf fibration of S3 over S2.