H 2 Of M (?) Bius Transform And Class - Conic

In this paper, we mainly discuss the properties and applications of M(o|¨)bius transformations in H~2. We give an equivalent description of orientation-preserving M(o|¨)bius transformations in H~2, and discuss the convergence of hyperbolic and parabolic orientation-preserving M(o|¨)bius transformations. After defining the sign of parabolic orientation-preserving M(o|¨)bius transformations, we give a characteristic of them. As an application of M(o|¨)bius transformations in H~2, I prove that none of the conic-like curves except the circles in (H~2,ρ) are conics in Euclidean space.