Editing Hyperbolic motion (section)

==Disk model motions==
Consider the disk D = {''z'' ∈ '''C''' : ''z z''* < 1 } in the [[complex plane]] '''C'''. The geometric plane of [[Nikolai Lobachevsky|Lobachevsky]] can be displayed in D with circular arcs perpendicular to the boundary of D signifying ''hyperbolic lines''. Using the arithmetic and geometry of complex numbers, and [[Möbius transformation]]s, there is the [[Poincaré disc model]] of the hyperbolic plane:

Suppose ''a'' and ''b'' are complex numbers with ''a a''*&nbsp;&minus;&nbsp;''b b''* = 1. Note that
:|''bz'' + ''a''*|<sup>2</sup> &minus; |''az'' + ''b''*|<sup>2</sup> = (''aa''* &minus; ''bb''*)(1 &minus; |''z''|<sup>2</sup>),
so that |''z''| < 1 implies |(''a''z + ''b''*)/(''bz'' + ''a''*)| < 1 . Hence the disk D is an [[invariant (mathematics)#Invariant set|invariant set]] of the Möbius transformation
:f(''z'') = (''az'' + ''b''*)/(''bz'' + ''a''*).
Since it also permutes the hyperbolic lines, we see that these transformations are '''motions''' of the D model of [[hyperbolic geometry]]. A complex matrix 
:<math>q = \begin{pmatrix} a & b \\ b^* & a^* \end{pmatrix}</math>
with ''aa''* &minus; ''bb''* = 1, which is an element of the special unitary group [[SU(1,1)]].