Moscow Mathematical Journal
Volume 26, Issue 1, January–March 2026 pp. 51–71.
Interior Angle Sums of Geodesic Triangles and Translation-Like Isoptic Surfaces in Sol Geometry
After investigating the geodesic triangles and their angle sums in
Nil and $\widetilde{\mathrm{SL}_2\,\mathbb{R}}$ geometries, we consider the analogous problem in
Sol space, which is one of the eight 3-dimensional Thurston
geometries. We analyze the interior angle sum of geodesic triangles,
and we prove that it can be larger than, less than, or equal to
$\pi$. 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 1997. 2020 Math. Subj. Class. 53A20, 53A35, 52C35, 53B20.
Authors:
Géza Csima (1) and Jenő Szirmai (2)
Author institution:(1) Department of Algebra and Geometry, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary,
Department of Basic Technical Studies, University of Debrecen, Ótemető str. 2–4., H-4028 Debrecen, Hungary
(2) Department of Algebra and Geometry, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
Summary:
Keywords: Thurston geometries, Sol geometry, translation and geodesic triangles, interior angle sum, isoptic curves and surfaces, Thaloid.
Contents
Full-Text PDF