Conformal Geometric Algebra in Stochastic Optimization Problems ...

26 CHAPTER 2. GEOMETRIC ALGEBRA
Fig. 2.3: Four identically oriented unit cells. The notion of orientation complies
with the outer product since e1 ∧e2 is identical to e2 ∧−e1, −e1 ∧−e2 and −e2 ∧e1.
All cells exhibit the same orientation.
Figure 2.3 illustrates that the chosen rule to determine the orientation is sound in
that it is consistent with the bilinearity and anti-symmetry of the outer product.
The (thought) extension of −e2 along e1, for example, induces the same orientation
as all remaining unit cells that are equal to e1 ∧ e2. This notion of orientation can
be generalized to higher dimensions. Interestingly, orientations are identical if their
respective ‘constructions’, which arise from successively extending the elements
along each other, are congruent.
Recall now that the geometric product is associative. The respective axiom is
succinct and unobtrusive but it has profound implications: consider the product
ab of two vectors. A right-multiplication with b results in abb = ab 2 = λa, with
λ = b 2 ∈ �. Whenever b is not a null vector, i.e. b 2 = 0, it is possible to define the
inverse of a vector by setting
b −1 = b
b 2 , b2 �= 0.
As a consequence, the multiplication of a with b can be undone and a is reobtained
with
(ab)b −1 = a(bb −1 ) = a.
The above result is now being used. Let, at first, x and n be two linearly independent
vectors of the Euclidean geometric algebra �n. Without loss of generality it
can be assumed that n 2 = 1. From the identity x = xnn the following decompo-
sition of x arises
x = (x · n)n +(x ∧ n)n. (2.17)
� �� �
x� Since, in the case of vectors, the inner product is identical to the scalar product, it is
known from Euclidean geometry that the first term (x ·n)n must be the projection
x � of x onto n. Therefore
(x ∧ n)n = x − x � = x⊥ (2.18)
represents the part of x perpendicular to n. Since x � resides in the plane x ∧ n,
spanned by x and n, x⊥ does so as well. This may also be verified by rightmultiplying
equation (2.18) with n
x ∧ n = x⊥n = x⊥ ∧ n with x⊥ · n = 0.