Geometry of Mobius Transformations

  Linear-fractional (Moebius) transformations generate a rich and beautiful
  geometry on the complex plane. They are linked to numerous important
  topics, which are suitable for a detailed separate study:

  1. Non-Euclidean geometry on the Lobachevsky half-plane and Poincare
  disk.

  2. Continuous fractions and best approximations.

  3. Transformations of analytic functions.

  4. Hypercomplex numbers.

  5. Discrete groups and geometries.

References

1. Alan F. Beardon. The Geometry of Discrete Groups. Springer-Verlag, 1983
Front Cover

2. Vladimir V. Kisil, Geometry of Mobius Transformations: Elliptic,
  Parabolic and Hyperbolic Actions of SL2(R). ICP, 2012.