First of all, the hyperbolic distance between x,y [member of] D is given by [7]
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.
