Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

Szirmai, Jenő

doi:10.1093/qmath/haab038

Cited by 6 publications

(19 citation statements)

References 27 publications

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“…In this section we recall important notions and results from the papers [30,44,46,47,55,60,61,64,65,81].…”

Section: Geodesic Curves In S 2 ×R Geometrymentioning

confidence: 99%

“…geodesics starting from different vertices and ending at points on the corresponding opposite side usually do not intersect. Therefore, we introduced the definition (see [ [65]) of the surface S A 0 A 1 A 2 of the geodesic triangle using the notion of the generalized Apollonius surfaces: Definition 3.17 The Apollonius surface AS X P 1 P 2 (λ) in the Thurston geometry X is the set of all points of X whose geodesic distances from two fixed points are in a constant ratio λ ∈ R + 0 where…”

Section: Surfaces Of Geodesic Trianglesmentioning

confidence: 99%

“…Figure 7: The Apollonius surface AS S 2 ×R P 1 P 2 (λ) where P 1 = (1, 1, 0, 0), P 2 = (1, 2, 1, 1), λ = 2 (left) and λ = 1 (right, equidistance surface (see [65,44])…”

Section: Surfaces Of Geodesic Trianglesmentioning

confidence: 99%

“…We consider 4 points A 0 , A 1 , A 2 , A 3 in the projective model of the space X (see Section 2, subsections 3.1-2 X ∈ {S 2 ×R, H 2 ×R}). These points are the vertices [65,46] of a geodesic tetrahedron in the space X if any two geodesic segments connecting the points A i and A j (i < j, i, j ∈ {0, 1, 2, 3}) do not have common inner points and any three vertices do not lie on the same geodesic curve. Now, the geodesic segments A i A j are called edges of the geodesic tetrahedron A 0 A 1 A 2 A 3 .…”

Section: Geodesic Tetrahedra and Their Circumscribed Spheresmentioning

confidence: 99%

“…and its circumscibed sphere of radius r ≈ 1.30678 with circumcenter C = (1, ≈ 0.64697, ≈ 0.51402, ≈ 0.15171) [65].…”

Section: Geodesic Tetrahedra and Their Circumscribed Spheresmentioning

confidence: 99%

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Classical Notions and Problems in Thurston Geometries

Szirmai¹

2022

Preprint

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Of the Thurston geometries, those with constant curvature geometries (Euclidean E 3 , hyperbolic H 3 , spherical S 3 ) have been extensively studied, but the other five geometries, H 2 ×R, S 2 ×R, Nil, SL 2 R, Sol have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these -such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.

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“…In this section we recall important notions and results from the papers [30,44,46,47,55,60,61,64,65,81].…”

Section: Geodesic Curves In S 2 ×R Geometrymentioning

confidence: 99%

Section: Surfaces Of Geodesic Trianglesmentioning

confidence: 99%

“…Figure 7: The Apollonius surface AS S 2 ×R P 1 P 2 (λ) where P 1 = (1, 1, 0, 0), P 2 = (1, 2, 1, 1), λ = 2 (left) and λ = 1 (right, equidistance surface (see [65,44])…”

Section: Surfaces Of Geodesic Trianglesmentioning

confidence: 99%

Section: Geodesic Tetrahedra and Their Circumscribed Spheresmentioning

confidence: 99%

“…and its circumscibed sphere of radius r ≈ 1.30678 with circumcenter C = (1, ≈ 0.64697, ≈ 0.51402, ≈ 0.15171) [65].…”

Section: Geodesic Tetrahedra and Their Circumscribed Spheresmentioning

confidence: 99%

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Classical Notions and Problems in Thurston Geometries

Szirmai¹

2022

Preprint

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Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in $$\textbf{Nil}$$ Geometry

Csima,

Szirmai

2023

Results Math

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After having investigated the geodesic and translation triangles and their angle sums in $$\textbf{Sol}$$ Sol and $$\widetilde{{\textbf{S}}{\textbf{L}}_2{\textbf{R}}}$$ S L 2 R ~ geometries we consider the analogous problem in $$\textbf{Nil}$$ Nil space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in $$\textbf{Nil}$$ Nil geometry and we provide a new approach to prove that it can be larger than or equal to $$\pi $$ π . Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of $$\textbf{Nil}$$ Nil isoptic surfaces of translation-like segments and as a special case of this we examine the $$\textbf{Nil}$$ Nil translation-like Thales sphere, which we call Thaloid. In our work we will use the projective model of $$\textbf{Nil}$$ Nil described by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).

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Fibre-Like Cylinders, Their Packings and Coverings in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ Space

Szirmai

2024

Results Math

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In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ space, develop a method to determine its volume and total surface area. We prove that the common part of the above congruent fibre-like cylinders with the base plane are Euclidean circles and determine their radii. Using the former classified infinite or bounded congruent regular prism tilings with generating groups $$\mathbf {pq2_1}$$ pq 2 1 we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in $$\textbf{H}^3$$ H 3 and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R spaces. We prove that in these three previous Thurston geometries, the densities of the optimal fiber-like cylinder packings are equal and the same is true for optimal coverings. In our work we use the projective model of $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).

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Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

Cited by 6 publications

References 27 publications

Classical Notions and Problems in Thurston Geometries

Classical Notions and Problems in Thurston Geometries

Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in $$\textbf{Nil}$$ Geometry

Fibre-Like Cylinders, Their Packings and Coverings in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ Space

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