Bisector surfaces and circumscribed spheres of tetrahedra derived by translation curves in \SOL geometry

2017/05/10 by Jenő Szirmai, Szirmai, Jenő · 2 citations
Computer Science · Mathematics · #52C35 #53A20 #53A35 #53B20 #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1705.04207

openalex publication_date 2017/05/10 · openalex created_date 2017/05/19 · openalex updated_date 2026/07/28

Abstract

In the present paper we study the \SOL 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. Moreover, we develop a method to determine the centre and the radius of the circumscribed translation sphere of a given \it translation tetrahedron. In our work we will use for computations and visualizations the projective model of \SOL described by E. Molnár in \citeM97.

Citations

• Curvatures of left invariant metrics on lie groups
• The Geometries of 3-Manifolds

Cited by

• Lattice-like Packings and Coverings with Congruent Translation Balls and Cylinders in Sol geometry
• Menelaus' and Ceva's theorems for translation triangles in Thurston geometries

Related

• Lattice coverings by congruent translation balls using translation-like bisector surfaces in Nil geometry
• Translation-like Apollonius and triangular surfaces in non-constant curvature Thurston geometries
• Translation-like isoptic surfaces and angle sums of translation triangles in \NIL geometry
• Interior angle sums of geodesic triangles and translation-like isoptic surfaces in Sol geometry
• Triangle angle sums related to translation curves in \SOL geometry