No abstract

In this paper we present a new characterization of Möbius transformations by use of hyperbolic regular polygons.

The complex unit disk D = {z ∈ C : |z| < 1} is endowed with Möbius addition ⊕M defined by w ⊕M z = w + z 1 + wz. We prove that the metric dT defined on D by dT (w, z) = tan −1 | − w ⊕M z| is an invariant of Möbius transformations carrying D onto itself. We also prove that (D, dT) and (D, dP) , where dP denotes the Poincaré metric, have the same isometry group and then classify the isometries of (D, dT) .

In this paper, we define a new geometric concept that we will call "degenerate Saccheri quadrilateral" and use it to give a new characterization of Möbius transformations. Our proofs are based on a geometric approach.

a b s t r a c tIn this paper, we prove that a mapping f : D → D in the unit disc of the complex z-plane preserving the gyrotriangles together with their gyroareas must be Möbius.