Generalized discrete Schwarz-Pick lemma for circle packings

In a paper titled "The Schwarz-Pick Lemma for Circle Packings," Alan Beardon and Kenneth Stephenson proved a discrete analogue of the classical Schwarz-Pick Lemma for circle packings. This paper generalizes this discrete analogue. The circle packings considered by Beardon and Stephenson are restricted in that neighboring circles have overlap angles equal to zero (i.e. neighboring circles are tangent). This restriction is lifted to allow neighboring circles to have overlap angles between 0 and π/2. Also included is a proof, that is independent of Andreev's work, of the existence of circle packings of triangulations of the closed disc in the hyperbolic plane with prescribed overlap angles.