First of all, the hyperbolic distance between x,y [member of] D is given by 
For both (D, [rho]D) and (H, [rho]H) one can define the hyperbolic distance in terms of the absolute ratio.
Note that [h.sub.1] ([alpha] = 1) is the hyperbolic distance
h on D.
The hyperbolic distance
[rho] in [H.sup.3] (or [H.sup.2] respectively) is determined by cosh [rho](P, P') = 1 + [d[(P, P').sup.2]/2rr'], where d is the Euclidean distance and P = z + rj and P' = z' + r'j are two elements of [H.sup.3] (respectively P = x + ri and P' = x' + r'i are two elements of [H.sup.2]).
These co-founders of non-Euclidean geometry proposed, independently of each other, a 2-body problem in hyperbolic space for which the attraction is proportional to the area of the sphere of radius equal to the hyperbolic distance
between the bodies.
Further, the hyperbolic distance
between two points A and X ([[delta].sub.AX]) is represented with the Euclidean distance AX in the Euclidean map ([[rho].sub.AX]) in Equation 1.
Here [??](z) denotes the hyperbolic distance
of z and 0 in D, namely,
We compute the height of S, that is, the hyperbolic distance
between the horospheres at the levels z([s.sub.1]) and z = 1.