Relating plane transformations with stereographic projection

Stereographic projection is a type of transformation mapping points on a sphere of dimension n+1 onto a plane of dimension n. It has properties such as continuity and preserving of certain angles which lend to exploring properties of the plane in relation to the sphere. Here stereographic projection is used as a method of drawing relationships between different two-dimensional spaces. Each of the complex plane, the split-complex plane, and the hyperbolic plane are examined in their relation to the Euclidean plane. First, Mobius transformations are considered on the complex plane. It is shown that every Mobius transformation can be represented via a movement of the sphere between two stereographic projections. Second, similar transformations are represented on the split-complex plane. Laguerre Transformations, analogous to Mobius transformations, explored with respect to split-complex numbers. Finally, a model of the hyperbolic plane is constructed using a pair of lateral projection and central projection.