On a Question of Constructing Mӧbius Transformations via Spheres and Rigid Motions

A Mӧbius Transformation or a Fractional Linear Transformation is a complex-valued function that maps points in the extended complex plane into itself either by translations, dilations, inversions, or rotations or even as a combination of the four mappings. Such a mapping can be constructed by a stereographic projection of the complex plane on to a sphere, followed by a rigid motion of the sphere, and a projection back onto the plane. Both Mӧbius transformations and Stereographic projections are abundantly used in diverse fields such as map making, brain mapping, image processing etc. In 2008, Arnold and Rogness created a short video named as Mӧbius Transformation Revealed and made it available on YouTube which became an instant hit. In answering a question posted in the accompanied paper by the same name, Siliciano in 2012 showed that for any given Mӧbius transformation and an admissible sphere, there is exactly one rigid motion of the sphere with which the transformation can be constructed. The present work is prepared on a suggestion posted by Silciano in characterizing rigid motions in constructing a specific Mӧbius transformation. We show that different admissible spheres under a unique Mӧbius transformation would require different rigid motions.