Finite Coverings in the Hyperbolic Plane

K�roly B�r�czky
2004 Discrete & Computational Geometry  
We prove that if a (th r )-convex domain in the hyperbolic plane is covered by n ≥ 2 circular discs of radius r , then the density of the covering is larger than 2π/ √ 27. The density bound is optimal, and the condition of (th r )-convexity is essentially optimal. Combining our result with earlier estimates yields that if at least two non-overlapping equal circular discs cover a given circular disc in a surface of constant curvature, then the density of the covering is larger than 2π/ √ 27.
doi:10.1007/s00454-004-1101-y fatcat:a7z2cik6n5fmjabaftv3vqabbm