Disk Model Motions
Consider the disk D = {z ∈ C : z z* < 1 } in the complex plane C. The geometric plane of 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 Mobius transformations, there is the Poincaré disc model of the hyperbolic plane:
Suppose a and b are complex numbers with a a* − b b* = 1. Note that
- |bz + a*|2 − |az + b*|2 = (aa* − bb*)(1 − |z|2),
so that |z| < 1 implies |(az + b*)/(bz + a*)| < 1 . Hence the disk D is an 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
with aa* − bb* = 1, which represents the Möbius transformation from the projective viewpoint, can be considered to be on the unit quasi-sphere in the ring of coquaternions.
Read more about this topic: Hyperbolic Motion
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