Conformal Geometric Algebra in Stochastic Optimization Problems ...

3.4. THE OPERATORS OF CGA 125
by an angle θ. Notice that it is refrained from expressing the rotation in terms of
Euler angles.
R =
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n 2 x(1 − cos θ) + cos θ nxny(1 − cos θ) − nz sin θ nxnz(1 − cos θ) + ny sin θ
nxny(1 − cos θ) + nz sin θ n 2 y(1 − cos θ) + cos θ nynz(1 − cos θ) − nx sin θ
nxnz(1 − cos θ) − ny sin θ nynz(1 − cos θ) + nx sin θ n 2 z(1 − cos θ) + cos θ
Both approaches have certain advantages. The matrix representation is probably
more efficient, i.e. less multiplications and additions are required to carry out a
rotation, but using a rotor is much more intuitive and handy. Besides, the representation
by means of a rotor is condensed in that four components hold the three
necessary parameters. Last but not least, the parameters are not so strongly mixed
up.
3.4.4 General Rotor
A so-called general rotor allows for rotations about arbitrary axes in space. It shall
be discussed shortly because it differs only slightly from a rotor or a motor.
As one might expect, a general rotor differs from a rotor only in that the intersecting
planes are not bound anymore to pass through the origin. Like before, the general
rotor rotates by twice the included angle. Similarly, the rotation axis is given by
the line L ≡ −P2 ∧ P1 if the rotor (for brevity, the term ‘general’ may be omitted
from now on) was defined by R = P2P1.
Example
Say a general rotor is given by Rφ = PbPa, where the opening angle between the
planes amounts to 1
2φ. It shall, however, be rotated by the different angle θ. The
rotation axis L can be retrieved by
L ′ = 〈Rφ〉 2 ,
and for a normalization and a correct orientation
L = −L′
�
−L ′2
The new rotor is then obtained with
or equally L = −Pb ∧ Pa
� −(Pb ∧ Pa) 2.
�
Rθ = exp − θ
2 L
�
. (3.61)
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