Some Geometric Properties of the m-Möbius Transformations

Möbius transformations, which are one-to-one mappings of  onto  have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations m f mapping m  onto  . Even for the simplest entity, the pre-image by m f of a unique point, there is no way of visualization. Pre-images by m f of figures from  are like ghost figures in m  . This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most im-portant achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in m  are well known and vector calculus in m  is familiar, yet the pre-image by m f of a vector from  is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in m  . Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points