On 5/3/2010 3:22:27 PM, Tom_Gutman wrote:
>1) The scalar part of a
>quaternion is not the scaling.
>The scaling of a quaternion is
>the magnitude of the
>quaternion -- that involves
>all the components.
>
>2) Using p=qrq-1 the scaling
>(magnitude) of q doesn't
>matter -- it cancels out
>between the direct
>multiplication and the
>multiplicatin by the inverse.
>The p=qrq* applies
>specifically when the
>magnitude (not the scalar
>part) of q is 1. For a unit
>quaternion, q-1=p* (the
>reciprocal is the conjugate).
>
>
>You would not put three angles
>as the three components of a
>vector part of a quaternion.
>For rotation around one of the
>coordinate axes the rotation
>quaternion (using the inverse,
>not conjugate, formulation)
>can be represented with a
>scalar part of 1 (an arbitrary
>reference, not a scale) and
>sin�θ in the
>position of the vector part
>corresponding to the rotation
>axis. For small enouth
>θ one can use the small
>angle approximation to the
>sine. For a general rotation,
>given as three rotations
>around the three coordinate
>axes, you have to create the
>rotation quaternion for each
>axes separately and then
>multiply the quaternions (in
>the proper order, depending on
>the order in which the
>rotations are applies -- note
>that the non-commutativity of
>rotations matches the
>non-commutativity of
>quaternion multiplication).
>The resulting rotation
>quaternion would usually then
>be normalized to unit length
>to allow for the simpler
>conjugate form for the
>rotation.
>__________________
>� � � � Tom Gutman
Tom,
Whilst you are correct in the mathematics, I was referring to how they are used in 'practice' in
engineering/computing.
In particular, it is their use for small angle
stuff (and integral over a path/time to get larger
angles) that I was commenting on.
They do do the 'cheats' by using q* instead of q^-
1, and use 1 instead of cos(theta), and then
theta/2 instead of sin(theta/2).
The big "however" is that even with these rather
gross assumptions (with theta of up to 10 degrees
apparently) they still get remarkable accuracy
overall.
My interest, as an engineer, is to fully
understand just why, and how, that is realized,
and to be able to explain it to others (which is
the hard bit!)
My apologies for being a bit fast and loose with
some of the terminology....
Philip Oakley