2017/10/05 by Angéla Vránics, Vránics, A., Jenő Szirmai +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · #52C17 #52C35 #53A20 #53A35 #53B20 #Advanced Materials and Mechanics #Connective tissue disorders research #FOS: Mathematics #Geometric and Algebraic Topology #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1710.02394
openalex publication_date 2017/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. 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 \it 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 amd the volume of the circumscribed translation sphere of a given \it translation tetrahedron. The found minimal upper bound density of the translation ball coverings Δ≈ 1.42783. In our work we will use for computations and visualizations the projective model of Nil described by E. Molnár in \citeM97.