Apollonian circles and hyperbolic geometry (Q388637)

The Apollonian metric is based on the fact that hyperbolic spheres and Euclidean spheres are essentially the same. In this paper under review, the authors use Apollonian circles to compare the hyperbolic distance with the Euclidean distance and to describe a geometric construction for finding the hyperbolic midpoint. In terms of Euclidean geometry, hyperbolic balls are interpreted. They also discuss lower bounds for the hyperbolic distance by using the Euclidean distance.