Finite coverings in the hyperbolic plane
type: Item
instance of: scholarly article
title: Finite coverings in the hyperbolic plane
zbMATH Open document ID: 1081.52018
published in: Discrete \& Computational Geometry
publication date: 2005-02-23T00:00:00Z
review text: In \(H^2\) (the hyperbolic plane) a hypercycle of associated distance \(r\) is the complete curve whose points are at distance \(r\) from a given line. Let \(C\) be a convex domain with the property that any boundary point of \(C\) is contained in some hypercycle of associated distance \(r\) such that \(C\) lies on the convex side of the curve. If such a \(C\) is covered by a finite number of circular discs of radius \(r\) then the density of the covering is larger than \(2\pi/\sqrt{27}\). This is the main theorem. As a corollary the author shows that if at least two non-overlapping equal circular discs cover a given circular disc in Euclidean or spherical or hyperbolic 2-space then the density of the arrangement is larger than \(2\pi/\sqrt{27}\). (Similar results for \(E^2\) and \(S^2\) were established earlier, see [\textit{L. Fejes Tóth}, Comment. Math. Helv. 23, 342--349 (1949; Zbl 0035.10901)].) This paper completes the solving of problems started decades ago and hence it is important, substantial and interesting.
Mathematics Subject Classification ID: 52C15
zbMATH DE Number: 2138094
zbMATH Keywords: hypercycle
zbMATH Keywords: density
author: Károly jun. Böröczky
MaRDI profile type: Publication
full work available at URL: s00454-004-1101-y
OpenAlex ID: W1593302568
DOI: 10.1007/S00454-004-1101-Y
description: scientific article; zbMATH DE number 2138094