Conformal Geometric Algebra in Stochastic Optimization Problems ...

128 CHAPTER 3. THE CONFORMAL GEOMETRIC ALGEBRA
and setting θ = �r�, Rodrigues’s formula (1840), cf. [56, 37], states that
e A = I3 + sinθ (1 − cos θ)
A +
θ θ2 A 2 =: R, (3.65)
where R denotes a rotation matrix for rotating about the vector r ∈ � 3 by an angle
θ = �r�. Thus with ˆr := r/θ it holds that
Ru = u + sinθ ˆr × u + (1 − cos θ) ˆr × (ˆr × u). (3.66)
Identifying ˆr := ˆr with the direction vector of a line � L ∈ 〈�3〉 2 ⊂ �4,1 passing
through the origin, the line may be expressed in respect of point u := K(�u := u),
that is � L = (ˆnu ∧ ˆno)I −1
E (see page 116), such that
ˆr × �u = ((ˆnu ∧ ˆno)I −1
E
∧ �u)I−1 E = −(�u ∧ (ˆnu ∧ ˆno)I −1
E )I−1
E
= �u · ((ˆnu ∧ ˆno)I −1
E IE) = �u · � L = d [u, � L] Pu.
This shows the connection or equality, respectively, between equation (3.64) and
equation (3.66); only u has to be replaced with x.
The Orthogonal Part
Like in figure 3.16, the translation of T may be separated into a translation Tp
parallel or inside, respectively, the rotation plane Pˆr and an orthogonal translation
To along the rotation axis L. Hence let T = TpTo = ToTp, with To = PvPu where
{Pv,Pu} � Pˆr. Again let Rp = P ′ 2 P ′ 1 . Due to {P ′ 2 ,P′ 1 } ⊥ Pˆr, the planes of To and
Rp, respectively, anti-commute such that
R = (TpTo)Rp( � To � Tp) = TpPvPuP ′ 2
−PuP ′ 1
� �� �
P ′ 1Pu Pv � Tp ...
= TpP ′ 2P ′ 1 � Tp = TpRp � Tp.
Consequently, a general rotor R commutes with an orthogonal translator To
RTo = ToR.
This suggests a new operator, namely the motor, which can be viewed as a screw
motion 19 .
3.4.5 Motor
For an introduction consider at first two congruent objects, or simply two point
clouds, X1 and X2 in 3D-space � 3 ; both have a distinct position and orientation.
19 The alternating execution of arbitrary small portions of general rotor and orthogonal translator
can be thought of as a screw motion.