Quaternions 1


Rotations
---------

Rotation can be represented by a unit vector and an angle of 
revolution about that vector.

When a sequence of rotations is applied in three dimensions 
and the resultant total rotation is calculated it follows 
laws which may not be intuitive.

The order of rotations is important. If a sequence of rotations
is performed, changing the order will produce different results.
Therefore, rotations form a non-Abelian (non-commutative) group.  
Representations of 3D rotations in this way are very messy so it 
is better to think of rotations in terms of QUATERNIONS or matrices 
where we can combine rotations into a single rotation.

Quaternions
----------

H = a + bi + cj + dk where i² = j² = k² = -ijk) are effectively 
the equivalent of the complex plane in 3D. The following rules 
are used when multiplying 2 or more quaternions together:

    1 |  i |  j |  k
----------------------
1 | 1 |  i |  j |  k
----------------------
i | i | -1 |  k | -j
----------------------
j | j | -k | -1 |  i
----------------------
k | k |  j | -i | -1
----------------------

The formula for 3D rotations is:

P_out = q * P_in * q^*

where,

P_out and P_in are points in 3D space represented by the i, j and 
k parts of a quaternion (real part = 0).  q^* is the complex 
conjugate. q is a quaternion which represents the rotation.  In 
terms of the angle and axis of the rotation, q is:

q = cos(α/2) + i(xsin(α/2)) + j(ysin(α/2)) + k(αsin(a/2))

where,

α = rotation angle
x,y,z = rotation axis

Combining Rotations
------------------

A rotation of q₁ followed by a rotation of q₂ is equivalent 
to a single rotation of q₂*q₁. Note the reversal of order, that 
is, we put the first rotation on the right hand side of the 
multiplication.

The order worked out above assumes that each rotation, q₁ and 
then q₂ is done in the absolute frame of reference so we get 
q₂*q₁. When working in the frame of reference of a moving object 
the order is not reversed so the combined rotation is q₁*q₂.
Summarizing:

absolute frame of reference 	frame of reference of rotating object
          q₂*q₁                                  q₁*q₂

Once we have expressed the rotational quantity as a quaternion,
it is easy to work with it. 

Combining rotations and translations implies that the above 
formulas contain both multiplication and addition.

This means that we cannot necessarily derive a single quaternion 
that can represent a combination of the above transforms. This is 
because a quaternion contains 4 scalar values so there is no way 
it can represent say the 6 degrees of freedom of a solid object.

If we want a single algebraic entity which can represent any 
combination of the above transforms then we would need to 
consider other algebras: 

    - Use matrices
    - Use dual quaternions.
    - Use projective or conformal projections.
    - Use quaternions for the rotation part and handle the 
      translation part separately (see affine translations).

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