Conformal Geometric Algebra in Stochastic Optimization Problems ...

126 CHAPTER 3. THE CONFORMAL GEOMETRIC ALGEBRA
Fig. 3.16: Rotation about an arbitrary line in space.
Displacing a Pure Rotation
A general rotor can also be viewed as an offset pure rotation. This can be accomplished
by means of a translator T. Hence if Rp denotes a pure rotor, the shifted
(general) rotor R can be expressed via
Clearly, if Rp = P ′ 2 P ′ 1 then R = (TP′ 2 � T)(TP ′ 1 � T) = P2P1.
R = TRp � T. (3.62)
But transformation (3.62) can also be interpreted stepwise: the leftmost and therefore
first subtransformation � T of R rigidly moves the coordinate frame centered at
x = T eo � T, see figure 3.16, to the origin where the pure rotation of Rp is carried
out. Finally, T undoes the change of the coordinate frame.
Using the pure rotor Rp = ˆm ˆn and the translator T = 1 − 1
2 �te, the general rotor
R becomes
R = � 1 − 1
2 �te � ˆm ˆn � 1 + 1
�
�te 2
= ˆm ˆn − (�te) × −( ˆm ˆn )
in analogy with equation (3.61).
Exponential Representation
= Rp − � �t · ( ˆm ∧ ˆn) � e
� � � � � �
(3.58) θ θ
= cos − sin �L
θ
��t + sin ·
2 2 2
� �
L e
� � � �
θ θ
��L �
= cos − sin − �t ·
2 2
� � �
L e
� � � �
(3.51) θ θ
= cos − sin L, (3.63)
2 2
The exponential representation is now being derived by means of the exponential
representation of Rp as stated in equation (3.60). Let Lp denote the rotation axis