Conformal Geometric Algebra in Stochastic Optimization Problems ...

6 CHAPTER 1. INTRODUCTION
Spherical Reflection
This example demonstrates that a spherical reflection is possible, yet its main message
is the simplicity of reasoning; some elementary calculations and assumptions
lead to a profound result.
Consider a circle C in 3D - it can clearly be thought of as an intersection of two
spheres. Given one sphere, which coincides with the circle, the set of suitable
spheres to reconstruct the circle is endlessly big. The set also comprises a plane,
that is a special sphere with infinite radius, which actually is the circle’s plane PC.
Let S denote a different set that contains pairs of spheres such that the intersection
between the respective spheres of each pair yields the circle. Each pair in S for
which the constituting spheres locally intersect at right angles can now be taken to
build the (infinitely big) subset S⊥ ⊂ S. There are at least two special cases in S⊥.
• One sphere is of minimal size the other one is of maximum size: the circle
plane PC intersected with the sphere SC whose center lies on PC yields C,
see figure 1.1. The radius of SC equals the circle’s radius.
• Symmetrical case: both spheres, SI and SJ, have equal radius, see figure 1.2.
Fig. 1.1: Construction of the circle C from sphere and plane.
Because both cases fulfill the requirement of perpendicularity, the results of section
1.1.3 can be used and the circle C can be built in two ways
CSP = SCPC and CSS = SISJ .
Note that, in the symmetrical case, each sphere can be replaced by the reflection of
the opposite sphere in the circle’s plane, for example SJ ≡ PCSIPC. The following
calculation makes use of this fact. It starts from C ≡ [CSP ∝ CSS].
SCPC ∝ SISJ employ SJ ∝ PCSIPC
SCPC ∝ SI(PCSIPC) multiply by PC
SCP 2 C ∝ SIPCSIP 2 C
SC ∝ SIPCSI since P 2 C ∈ � by axiom
(1.2)
Besides, with S 2 I ∈ �, PC ∝ SISCSI is implied. The feasibility of a spherical
reflection has hereby been shown, though solely for the special setup in which the
center of SI is located on SC.