Conformal Geometric Algebra in Stochastic Optimization Problems ...

88 CHAPTER 3. THE CONFORMAL GEOMETRIC ALGEBRA
A ∗ 〈k〉
xp = (x · A ∗−1
〈k〉 )A∗ 〈k〉
= (x · (A 〈k〉 I −1 ) −1 )A 〈k〉 I −1
=
=
=
=
�
⎡
x ·
⎣x ·
(A 〈k〉 I −1 ) ∼
(A 〈k〉 I −1 ) ∼ (A 〈k〉 I −1 )
⎤
�
A 〈k〉 I −1
−I � A 〈k〉
(−I � ⎦A 〈k〉 I
A 〈k〉 )(−A 〈k〉 I)
−1
�
x · A −1
〈k〉 I
�
A 〈k〉 I −1
�
x ∧ A −1
〈k〉
�
IA 〈k〉 I −1
(3.3)
= (x ∧ A −1
〈k〉 )A 〈k〉 , (3.4)
where it was used that in �4,1: � I = I, I −1 = −I, I 2 = −1 and A 〈k〉 I = IA 〈k〉 .
Hence equation (3.4) (which originally represents a rejection, see equation (2.61))
is used instead of equation (3.3).
Where Points Project to Spheres
Next to conformal points, spheres and planes are equally vectors, that is to say
elements that can make up a null space 5 . Clearly, projection and rejection split
a vector into two orthogonal vector parts. But it will turn out that the manifold
of conformal points is not closed under addition; whence it is not surprising that
projecting a point may end with a plane or a sphere.
On the Sandwich Product
This important product reappears in this chapter being the subject of several sections.
Let A and X be general multivectors. Applying A or its dual to X may at
least change the sign of the result, compare
3.3.2 Useful Notes on CGA
AXA = A ∗ IXA ∗ I = −A ∗ XA ∗ . (3.5)
It is begun by repeating the most important basis blades.
e = e∞ = e+ + e−
5 When mentioning, for instance, the OPNS of a blade A〈k〉, only the underlying set of conformal
points is meant rather than all vectors in �er(A 〈k〉) ⊆ � 4,1 .