[Algorithms] Scaled quaternion rotations (was: Representing AnimationKey Frame with Quaternion+Translation Vector)

[Marc B. R. <mar...@or...>]

Cayley in [1,2] demonstrates general 3D rotations via quaternions,
(left-handed and right-handed respectively).
>From [2]:
 
    P' = Q P (1/Q)            (e.1)
 
Notice that if you multiply Q by an non-zero scale factor 's', you get:
 
    P' = (sQ) P (1/(sQ))
        = (s/s) Q P (1/Q)
        = Q P (1/Q)
 
So if we consider some quaternion Q to respresent a rotation using (e.1),
all scalar multiples of Q represent the same rotation. If we additionally
consider Q to be a point in some 4D space, then the point is homogenous and
(since we've dropped one degree of freedom) is a 3D object embedded in the
4D space.  The set of all quaternions which represent the same rotation may
be described as 'sQ', again for all 's' except zero and is a line through
the space.
 
Unit quaterions come into play since the inverse can be replaced by the
conjugate, which has nice algebraic, numeric and computation properties. So
if U is a unit quaternion we can instead use the following:
 
    R' = U R U*                (e.2)
 
The set of all unit quaternions is a sphere and the line of all quaternions
which represent a given rotation intersects the sphere at two points (thus
the double coverage).
 
Continuing to use (e.2),  let's plug in a non unit quaternion: Q=sU 
 
    R''  = Q R Q*
         = (sU) R (sU)*
         = (s^2) U  P U*
         = (s^2) R'
 
So R'' is R' scaled by a factor of (s^2).
 
Composition of rotations is achieved by the product:
 
    C = A B
 
Multiply 'A' and 'B' by non-zero scales 'a' and 'b' respectively:
 
    C = (aA)(bB)
       = (ab)AB
 
Combining all of the above, to effect a scaled rotation, use a quaternion
which has a magnitude of the square-root of the desired scaled factor in
(e.2). 
 
 
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[ 1]    "On certain results concerning quaternions", Arthur Cayley, 1845
[ 2]    "On the Application of Quaternions to the Theory of Rotation",
Arthur Cayley,1848