Every Mobius transformation can be constructed by stereographic projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back onto the plane, illustrated in the video Mobius Transformations Revealed. In this article we show that, for a given Mobius transformation and sphere, this representation is unique.
Professor Jonathan Rogness, School of Mathematics, University of Minnesota
"Constructing Mobius Transformations with Spheres,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 13
, Article 8.
Available at: http://scholar.rose-hulman.edu/rhumj/vol13/iss2/8