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

G\'eza Csima; Jen\H{o} Szirmai

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

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arxiv 2410.22955 v1 pith:TEQZXK42 submitted 2024-10-30 math.MG

Géza Csima , Jenő Szirmai This is my paper

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keywords citegeometriessurfacesthurstonapolloniuscurvaturenon-constanttranslation-like

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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}.

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Menelaus' and Ceva's theorems for translation triangles in Thurston geometries
math.GT 2025-06 reject novelty 5.0 of 10

Menelaus' and Ceva's theorems are formulated and proved for translation triangles in Nil, Sol, and ~SL2R spaces using geometry-specific simple ratios.

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