Conformal Geometric Algebra in Stochastic Optimization Problems ...

6.1. GEOMETRIC ALGEBRA AND ITS TENSOR NOTATION 181
By means of Φf it becomes possible to assign vectors to algebra elements. Hence
regarding the preceding text it is Φf(A) = a, Φf(B) = b and Φf(C) = c.
It is quite scaring that every CGA tensor that corresponds to a binary operation
(a unary operation, by contrast, is e.g. the reverse) consists of 2 53 = 32768 coefficients.
This would cause some CGA expressions to be inoperable. So, in practice,
the complexity of calculations can be reduced considerably by using masked multivectors.
A line L, for example, can internally be stored by six coefficients with
associated mask
where
L = 0.45e12 + 0.89e23 + 0.47e2e
�
↓
�
line mask, [0.45 0.89 0 0 0.47 0] ,
line mask = [ 0
0 0 0 0 0
1 2 3 4 5 6 4 5 6 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
0 ]
Aside: The line mask contains double indices because basis blades like 0.47e2e
correspond to the two components 0.47e2e+ and 0.47e2e−, respectively, due to
e = e+ + e− (internally the e+e−-basis is used), see page 85.
Accordingly, a map Φ can be defined that replaces Φf - the full mapping. The
new Φ only maps the minimum number of coefficients necessary presuming the
corresponding mask is known
Φ : �p,q −→ � k , k ≤ n
X ↦−→ x
To understand this consider an OPNS circle C ∗ from which a line LC = −e · C ∗ is
to be calculated according to equation (3.49). As it is known beforehand that C ∗
is a 3-blade, it follows
Φ(C ∗ ) ↦−→ c ∈ � 10
Φ(e) ↦−→ e ∈ � 1
Φ(L) ↦−→ l ∈ � 6 .
Note that generally the outer product of two vectors, for instance, will not produce
3-vector components such that these can be disregarded. Analogously, with the
help of Φ the dimensions of the product tensor for the inner product −e · C ∗ , in
the following denoted by Q, can be restricted to only those components of the
multivectors that are actually needed. Then Q can be determined such that
l k = −e i Q k ij c j
with Q ∈ � 6×1×10 .
So exploiting the internal structure of CGA elements, the operation can effectively
be carried out by means of a 6-by-10 matrix rather than by a 3-valence tensor with
32768 elements.