Question

Show that the hyperbolic distance between z1 and z2 equals ln(z2 : z1 : x : y). Given that z1, z2 are points in the upper half-plane lying on a hyperbolic line l which is a circle. Let x, y be the intersection points of this circle with the boundary Imz = 0 so that z1 is located between x and z2 on the circle.

Answer 1

The concept of hyperbolic distance in geometry is related to the function D.

As per the given question, consider two points z1, z2 lies in the upper half plane lying on a hyperbolic line which is also a circle.

A circle in the hyperbolic plane is the locus of all points a fixed distance from the center, just as in the Euclidean plane.

Basic facts about hyperbolic circles

1. any three non collinear points lie on a unique circle,
2. a line and circle meet in at most two points,
3. two distinct circles meet in at most two points. From these facts , it is clear that z1 is located between x and z2.