Conformal Geometric Algebra in Stochastic Optimization Problems ...

76 CHAPTER 2. GEOMETRIC ALGEBRA
It is now possible to write
I ∝ [A 〈k〉 \n1] ∧ [A 〈k〉 \n1] ∗
∝ [A 〈k〉 \n1] ∧
∝ A 〈k〉 ∧
�
[A 〈k〉 \n1] ∗ \{n1, n †
�
[A 〈k〉 \n1] ∗ \{n1, n †
1 }
�
1 }
�
∧ n †
1
∝ A 〈k〉 ∧ (([A 〈k〉 \n1] ∗ · n †
1 ) · n1) ∧ n †
1
(2.44),5.
∝ A 〈k〉 ∧ ([A 〈k〉 \n1] ∗ · n †
1 )
∝ A 〈k〉 ∧ ([A 〈k〉 \n1] ∧ n †
1 )∗ .
∧ n1 ∧ n †
1
Hence the complement of a null blade A 〈k〉 can be obtained by turning the null
vector into its conjugate and building afterwards the dual. This technique may
also be extended to the case where more than one null vector is involved, i.e. if
A 〈k〉 = N 〈l〉 ∧ A ′ 〈k−l〉 then
I ∝
�
N 〈l〉 ∧ A ′ �
〈k−l〉
∧
�
N †
〈l〉 ∧ A′ �∗ 〈k−l〉 .
This result can immediately be verified by means of equation (2.51). Note that a null
blade has no orthogonal complement w.r.t � p,q because A ∗ 〈k〉 is not a complement
of A 〈k〉 and because (N †
〈l〉 ∧ A′ 〈k−l〉) ∗ is neither orthogonal nor perpendicular to
A 〈k〉 .
The following considerations are in particular significant when dealing with the
factorization of a blade.
Projecting with Geometric Algebra
Let � = {n1...r } ∪ {z1...s } be a sound frame for the null blade B 〈l〉 , specifically
B 〈l〉 = ( �r i=1 ni) ∧ ( �s i=1 zi). An arbitrary vector a ∈ � p,q can then be expressed
as
a = γ i zi + α i n †
i + αini + γ i zi,
such that � = {n † 1...r } ∪ {z1...t } is a sound basis as well. The sets {z1...t } and
{z1...s } can be assumed to be orthogonal, too. Consider the inner product
a · B 〈l〉 = α i (−1) i−1 (n †
i · ni) [B 〈l〉 \ni] + γ j (−1) r+j−1 z 2 j [B 〈l〉 \zj].
Hence the n-parts of a that lie in �er(B 〈l〉 ) have no influence, and the n † -parts
that do not come from �er(B 〈l〉 ) produce components in a ·B 〈l〉 . This explains why
the inner product cannot always be used for projecting in geometric algebra.
This problem cannot be overcome easily. Nevertheless, it is always possible to
invoke the ‘Euclidean alternative’ if A 〈k〉 ·ε B 〈l〉 �= 0
�er(B 〈l〉 ) ⊖ε
�
�
k≤l
�er(B 〈l〉 ) ⊖ε �er(A 〈k〉 ) = �er((A〈k〉 ·ε B 〈l〉 ) ·ε B 〈l〉 )
= �er((A †
〈k〉 · B 〈l〉 )† · B 〈l〉 ).