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Units, Constants and Useful Formulas
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 frame of reference
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).