Conformal Geometric Algebra in Stochastic Optimization Problems ...

3.3. CONFORMAL ANALYTIC GEOMETRY 113
Fig. 3.9: Circles: projection SP and rejection SR of a point x
SO ≡ x · C LC ≡ e · C ∗ Lx ≡ (e ∧ x ∧ SR)I
SR ≡ (x · C −1 )C SP ≡ (x ∧ C −1 )C (SP imaginary)
through the center C, can thus be determined using
⇐⇒
⇐⇒
L ′ = (e ∧ C)I = e · (C I) = −e · C ∗
L ′2 2
= −(e ∧ C) (3.22) 2 (3.46)
= −(e · C) = − P 2 C 1
= −
r2 r2 L = ±r(e · C ∗ ) ⇐⇒ L 2 = −1. (3.49)
Regarding the choice of the sign it is referred to section 3.3.12, which deals with
lines.
In this way an infinite number of circles, which may vary in their position along the
line or in the radius, correspond to the same line. These degrees of freedom make
the formula attractive for being used to replace lines with circles.
Figure 3.9 may serve as an example. There the perpendiculars LC, w.r.t circle C,
and Lx, respectively are depicted. The latter line belongs to circle K2.
Projection
Figure 3.9 summarizes several products, among others, the projection SP and the
rejection SR of a point x with respect to a circle C. The center sP of the imaginary
sphere SP lies on the plane of the circle, i.e. PC. Note that line Lx does, in general,