M (?) Bius Transformations And Conic

In this paper, we mainly discuss the properties and applications of Mobius transformations inΣand in H 2. T is said a Mobius transformation inΣif The set of Mobius transformations inΣforms a group denoted by _A__u__t (Σ) under composition. The subset consists of transformtions normal subgroup of index 2 in _A__u__t (Σ). We write it as Aut (Σ). It is the set of all automorphisms ofΣ. So we pay more attention to the group theory such as the finite subgroups of Aut (Σ). And the cross-rationλis introduced in order to study the transitivity properties of Mobius transformations. We also consider some of the geometric properties of Mobius transformations (especially their relationship with circles inΣ).Hyperbolic geometry was created in the first half of the 19th century in the midst of attempts to understand Euclid's axiomatic basis for geometry. It is one type of non-Euclidean geometry. In terms of Mobius transformations in H 2, we give an equivalent description of orientation-preserving Mobius transformations in H 2, and discuss the geometric classification of them.It is well known that any circle or line is mapped to a circle or a line under the Mobius transformation. It is natural to ask what does the image look like if we replace circle or line by a non-degenerated conico One can prove that ifεis an ellipse centered at 0∈C , and T ( z ) = 1/z_, then the image T(ε) is not an ellipse. Also one can prove that if two ellipsesε,ε' are M?bius equivalent, i. e. there exists a M?bius transformation T with T(ε) =ε', then T is a similarity. Henceε,ε' have the same eccentricity.