From cbloom@cbloom.com Thu Dec 12 22:38:46 2002
Date: Sun, 10 Nov 2002 12:41:26 -0800
From: Charles Bloom <cbloom@cbloom.com>
To: Casey Muratori <cmu@funkytroll.com>, Jonathan Blow <jon@bolt-action.com>
Cc: cbloom@cbloom.com
Subject: Re: (very) rough draft


At 12:21 PM 11/10/2002 -0800, Charles Bloom wrote:
>Consider two vectors in the exponential map :
>
>r1 = n1 * theta1
>r2 = n2 * theta2
>
>Here the "n" are unit vectors an the theta are angles in [0,pi].
>Each rotation has a "mirror" that must be considered :
>
>r1' = n1 * (theta1 - 2pi)
>r2' = n2 * (theta2 - 2pi)

It's useful to compare this to what you have to do with quats.
With quats, you start with two rotations {q1,q2}.  You need
to consider the mirrors -q1 and -q2.  You have to consider

q1->q2, q1->(-q2), q2->(-q1), and (-q1)->(-q2)

Neatly enough, there are actually only two cases here, since

q1->(-q2) is the same as q2->(-q1)

and (-q1)->(-q2) is the same as q1->q2

So, to get the shortest path, all you have to do is :

z = q1 DOT q2

if z >= 0 , use q1->q2
else , use q1->(-q2)

Note that this is also a better distance metric.  The proper
distance metric for rotations is the length of the geodesic
between them on the surface of S3.  This is just acos(z)
for quats, so using "z" means you are correctly choosing the
shorter geodesic.  In the exponential map, it's not even
clear that you *want* to choose the shorter geodesic, since
interpolation doesn't go across geodesics on S3.


-------------------------------------------------------
Charles Bloom    cb@cbloom.com    http://www.cbloom.com