Mobius transformation and the Cassinian metric

Abstract

In this thesis, we mainly consider Mobius transformations and the Cassinian met ric in the complex plane. In view of the importance of the hyperbolic metric on the unit disk, we have focused on some results associated with Mobius transfor mation, the Cassinian metric and the hyperbolic metric. The concept of inverse (or symmetric) points in circles plays a crucial role in Mobius transformation setting. Some of the basic results such as Ptolemys identity, characterization of disk automorphism, nding Mobius transformations from a pair of circles onto concentric circles, are proved using the concept of inverse points in circles. The construction of the Cassinian distance between two points in a subdo main of the complex plane is studied in two ways: (i) identifying the maximal Cassinian ovals inside the domain with the two points as its foci, and (ii) com puting the Euclidean distance between the inversions of those two points in the unit circle circle centred at the boundary point at which the maximal Cassinian oval meets. Various classical properties of the Cassinian metric are presented.