Desargues's and Pappus's hexagon theorems on translation-type surfaces in Thurston geometries

Abstract

In \cite{Sz25} we generalized the famous Menelaus' and Ceva's theorems for translation triangles in each non-constant curvature Thurston geometry. In this paper based on the described method and results, we prove that the classical Desargues's and Pappus's hexagon theorems are true not only in classical geometries with constant curvature, but also in Thurston geometries with non-constant curvature on the translation surfaces. In our work we use the unified projective models of Thurston geometries.