 These Möbius transformations act as the isometries of the upper half-plane model of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the Poincaré disk model.The above formula can be also used to define Möbius transformations of dual and double (aka split-complex) numbers.The corresponding geometries are in non-trivial relations to Lobachevskian geometry.