Write down the Riemannian metric in the half-plane model of the hyperbolic plane. Show that MÃ¶bius transformations mapping the upper half-plane to itself are isometries of this model.
Calculate the hyperbolic distance from to , where and are positive real numbers. Assuming that the hyperbolic circle with centre and radius is a Euclidean circle, find its Euclidean centre and radius.
Suppose that and are positive real numbers for which the points and of the upper half-plane are such that the hyperbolic distance between them coincides with the Euclidean distance. Obtain an expression for as a function of . Hence show that, for any with , there is a unique positive value of such that the hyperbolic distance between and coincides with the Euclidean distance.