Some Geometric Properties of the <i>m</i>-Möbius Transformations

Abstract

M &#246; bius 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 -M &#246; bius transformations f m mapping onto . Even for the simplest entity, the pre-image by f m of a unique point, there is no way of visualization. Pre-images by f m of figures from C are like ghost figures in C 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 important achievement is the proof in that context of a theorem similar to the symmetry principle for M &#246; bius transformations. It is like saying that the images by m -M &#246; bius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in C m are well known and vector calculus in C m is familiar, yet the pre-image by f m of a vector from C 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 C m . Finally, we have shown that the uniqueness theorem for M &#246; bius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m -M &#246; bius transformations.