Circle packings into convex domains of the Euclidean and hyperbolic plane and the sphere (Q1085443)

The densest possible packing of equal circles in the euclidean plane has density \(\pi\) /12. Results due to J. Molnár and L. Fejes Tóth show that \(\pi\) /12 is an upper bound - under some restrictions - also for packings on the sphere or on the hyperbolic plane. In the present paper, some of these results are extended (by weakening the restrictions). Also, the following theorem is proved: If in the hyperbolic plane \(r\geq 0.812\) and a circle of radius R(\(\geq 2r)\) is covered by circles of radius r, then the covering density is greater than \(2\pi\) /\(\sqrt{27}\).