Original scientific paper

https://doi.org/10.31896/k.23.1

Lattice Coverings by Congruent Translation Balls Using Translation-like Bisector Surfaces in Nil Geometry

Angéla Vránics orcid.org/0000-0002-1324-8916 ; Institute of Mathematics, Department of Geometry, Budapest University of Technology and Economics, Budapest, Hungary
Jenő Szirmai orcid.org/0000-0001-9610-7993 ; Institute of Mathematics, Department of Geometry, Budapest University of Technology and Economics, Budapest, Hungary

Abstract

In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geometries. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. We develop a method to determine the centre and the radius of the circumscribed translation sphere of a given translation tetrahedron.
A further aim of this paper is to study lattice-like coverings with congruent translation balls in Nil space. We introduce the notion of the density of the considered coverings and give upper estimate to it using the radius and the volume of the circumscribed translation sphere of a given translation tetrahedron. The found minimal upper bound density of the translation ball coverings \(\Delta \approx 1.42783\). In our work we will use for computations and visualizations the projective model of Nil described by E. Molnár in [6].

Keywords

Thurston geometries; Nil geometry; translation-like bisector surface of two points; circumscribed sphere of Nil tetrahedron; Dirichlet-Voronoi cell

Hrčak ID:

230716

URI

https://hrcak.srce.hr/230716

Publication date:

20.12.2019.

Article data in other languages: croatian

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