Conformal Geometric Algebra in Stochastic Optimization Problems ...

98 CHAPTER 3. THE CONFORMAL GEOMETRIC ALGEBRA
Fig. 3.4: Inner product between a sphere and a point inside of it. The right side
depicts a cross-section only.
Point - Sphere
Consider a point p and a sphere S = �s + 1
2 (�s2 − r 2 s)e + eo. Assume that p lies
outside S. Then the inner product p · S =: α r 2 p, with α = −0.5, gives the radius
rp of a sphere Sp centered at p in such a way that Sp is orthogonal 8 to S, i.e.
Sp · S = 0. The sphere Sp can be built by
To see this, compare
Sp = p + (p · S)e. (3.28)
Sp · S = (p + (p · S)e) · S = p · S − p · S = 0.
Note that r 2 p is the power of point p with respect to the sphere S if p lies outside
S, cf. [111, 19, 107]. Thus given a chord passing through the points a, b and p,
where a and b are assumed to lie on S, it holds that
d [p,a] d [p,b]
const
= r 2 p.
If p lies inside S (see below), the inner product becomes positive such that it must
be considered that r 2 p < 0. This implies that the sphere Sp becomes imaginary.
Thus by looking at the sign of the inner product it can be determined whether a
point is inside or outside a sphere
p · S < 0
S 2 >0
⇐⇒ point p lies outside S.
This relationship is exploited by Banarer [9] who utilizes CGA spheres in a neural
architecture to define so-called hypersphere neurones so as to achieve a non-linear,
that is a hyperspherical, separation in classification problems.
Now let p be a point in the inside of the sphere S := s − 1
2r2 se. The scenario is
illustrated in figure 3.4. According to equation (3.14) the inner product is
p · S = 1
2 (r2 s − d 2 1
[p,s] ) =: 2r2 p,
where in this case rp denotes the radius of circle Cp, which is depicted on the left
side of the figure.
8 Recall that this means that the respective surfaces are perpendicular.