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

Géza Csima, Jenő Szirmai

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. Molnár in \cite{M97}.

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

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. Molnár in \cite{M97}.

Paper Structure

This paper contains 22 sections, 13 theorems, 56 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction, preliminary results
  2. The structures of the investigated geometries
  3. On $\mathbf{S}^2\!\times\!\mathbb{R}$ and $\mathbf{H}^2\!\times\!\mathbb{R}$ geometries
  4. Translation curves
  5. On $\mathbf{Nil}$ geometry
  6. Translation curves
  7. On Sol geometry
  8. Translation curves
  9. On $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ geometry
  10. Translation curves
  11. Translation-like Apollonius surfaces
  12. $\mathbf{Nil}$ geometry
  13. Inverse problem
  14. $\mathbf{Sol}$ geometry
  15. Inverse problem
  16. ...and 7 more sections

Key Result

Lemma 3.2

Figures (10)

  • Figure 1: Translation-like $\mathbf{Nil}$ Apollonius surface of point pairs $(P_1,P_2)$ with coordinates $((1,0,0,0), (1,-1,1,1))$ with parameter $\lambda=2$,
  • Figure 2: Translation-like $\mathbf{Nil}$ bisector (equidistant surface) of point pairs $(P_1,P_2)$ with coordinates $((1,0,0,0), (1,1/2,1,1/2))$.
  • Figure 3: Translation-like$\mathbf{Sol}$ Apollonius surface of point pairs $(P_1,P_2)$ with coordinates $((1,0,0,0), (1,-1,1,1/2))$ with parameter $\sigma=1/2$,
  • Figure 4: Translation-like $\mathbf{Sol}$ bisector (equidistant surface) of point pairs $(P_1,P_2)$ with coordinates $((1,0,0,0), (1,-1,1,1/2))$.
  • Figure 5: Translation-like $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ Apollonius surface of point pairs $(P_1,P_2)$ with coordinates $((1,0,0,0), (1,0,1/6,1/5))$ with parameter $\sigma=2$,
  • ...and 5 more figures

Theorems & Definitions (22)

  • Definition 1.1
  • Definition 1.2
  • Remark 2.1
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • ...and 12 more