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Units, Constants and Useful Formulas
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 i^{2} = j^{2} = k^{2} = -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 q_{1} followed by a rotation of q_{2} is equivalent
to a single rotation of q_{2}*q_{1}. 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, q_{1} and
then q_{2} is done in the absolute frame of reference so we get
q_{2}*q_{1}. When working in the frame of reference of a moving object
the order is not reversed so the combined rotation is q_{1}*q_{2}.
Summarizing:
absolute frame of reference frame of reference of rotating object
q_{2}*q_{1} q_{1}*q_{2}
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).