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Units, Constants and Useful Formulas
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Last modified: July 30, 2018
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:
Pout = q * Pin * q*
where,
Pout and Pin 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).