Translation-like Apollonius and triangular surfaces in non-constant curvature Thurston geometries

TL;DR

This paper explores generalized translation-like Apollonius and bisector surfaces, as well as translation-like triangles, within non-constant curvature Thurston geometries using the projective model framework.

Contribution

It introduces definitions and characterizations of translation-like Apollonius and bisector surfaces, and proposes a new concept of translation-like triangles in Thurston geometries.

Findings

Defined generalized translation-like Apollonius surfaces

Characterized bisector surfaces in Thurston geometries

Proposed a new definition for translation-like triangles

Abstract

In the present paper we deal with non-constant curvature Thurston geometries \cite{M97}, \cite{S}, \cite{Sz22-3},\cite{W06}. We define and determine the generalized trans\-lation-like Apollonius surfaces and thus also bisector surfaces as a special case. Moreover, we give a possible definition of the "surface of a translation-like triangle" in each investigated geometry. In our work we will use the projective model of Thurston geometries described by E. Moln\'ar in \cite{M97}.