Editing Inversive geometry (section)

=== Reciprocation ===
When a point in the plane is interpreted as a [[complex number]] <math>z=x+iy,</math> with [[complex conjugate]] <math>\bar{z}=x-iy,</math> then the [[multiplicative inverse|reciprocal]] of ''z'' is

:<math>\frac{1}{z} = \frac{\bar{z}}{|z|^2}.</math>

Consequently, the algebraic form of the inversion in a unit circle is given by <math>z \mapsto w</math> where:

: <math>w=\frac{1}{\bar z}=\overline{\left(\frac{1}{z}\right)}</math>.

Reciprocation is key in transformation theory as a [[generating set of a group|generator]] of the [[Möbius group]]. The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space. Introduction of reciprocation (dependent upon circle inversion) is what produces the peculiar nature of Möbius geometry, which is sometimes identified with inversive geometry (of the Euclidean plane). However, inversive geometry is the larger study since it includes the raw inversion in a circle (not yet made, with conjugation, into reciprocation). Inversive geometry also includes the [[complex conjugation|conjugation]] mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements are [[analytic function]]s of the whole plane and so are necessarily [[conformal map|conformal]].