Conformal Geometric Algebra in Stochastic Optimization Problems ...

3.3. CONFORMAL ANALYTIC GEOMETRY 99
3.3.7 Planes
Planes are the last vector valued elements that are mentioned. Recall that a plane
can be considered a special sphere with infinite radius. This notion has a solid
base; replacing the fourth point d in equation (3.21), the OPNS representation of
a sphere, with ‘the point’ e, should consistently yield a sphere passing through
infinity. The resulting OPNS expression does in effect match the plane equation
(3.20).
The IPNS of planes representation is succinct
P = ˆn + de, d ≥ 0 (3.29)
where ˆn, ||ˆn|| = 1, denotes the normal vector of the plane and d stands for the
distance to the origin eo. If the plane is not normalized right from the beginning,
for example P = �n + te and ||�n|| �= 1, it can be divided by P 2 = �n 2 + t(�ne +e�n) =
�n 2 . A plane, as defined above, has to fulfill two conditions
P 2 = 1 and eo · P ≤ 0.
Notice that planes with d < 0 are simply oppositely oriented and occur equally
frequently.
Distance d and normal vector ˆn can easily be extracted
d = P · e+
ˆn = E · (E ∧ P).
The two parameters ˆn and d appear as well in the OPNS formula (3.20): building
the IPNS representation of the first component e ∧�a ∧�b ∧�c, only, in equation (3.20)
yields
(e ∧�a ∧� �
b ∧�c)I = − �a · ( � �
b × �c) e
For the second component it is
= α (de) α ∈ �.
(( � b −�a) ∧ (�c −�a))E I = (�c −�a) × ( � b −�a) (3.30)
= α ˆn,
for the same α ∈ � as above. Its value is the magnitude of ( � b − �a) ∧ (�c − �a) and
can be computed by means of equation (2.16).
Hence if a plane is created by means of the OPNS representation, its orientation
ˆn can be determined with a hand rule 9 : the fingers of the left hand are curled to
match the direction given by the point sequence a-b-c; then the thumb indicates
the direction of the plane.
9 Right hand sequence: a-c-b.