Interior angle sums of geodesic triangles and translation-like isoptic surfaces in Sol geometry

2024/04/24 by Géza Csima, Csima, Géza, Jenő Szirmai +1 · 1 citation
Engineering · Mathematics · Social Sciences · #52C35 #53A20 #53A35 #53B20 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Historical Geography and Cartography #Mathematics and Applications #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2405.05266

openalex publication_date 2024/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

After having investigated the geodesic triangles and their angle sums in Nil and Sl×ℝ geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to π. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work we will use the projective model of Sol described by E. Molnár in \citeM97.

Cited by

• Menelaus' and Ceva's theorems for translation triangles in Thurston geometries

Related

• Translation-like isoptic surfaces and angle sums of translation triangles in \NIL geometry
• Triangle angle sums related to translation curves in \SOL geometry
• Bisector surfaces and circumscribed spheres of tetrahedra derived by translation curves in \SOL geometry
• Interior angle sums of geodesic triangles in S² × R and H² × R geometries
• NIL geodesic triangles and their interior angle sums