The sum of the interior angles in geodesic and translation triangles of Sl2(R)~ geometry

Csima, Géza; Szirmai, Jenő

doi:10.2298/fil1814023c

Cited by 7 publications

(11 citation statements)

References 24 publications

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“…In (4) and (5) we can see the 2π periodicity of ϕ. Moreover, we see the (logical) extension to ϕ ∈ R, as real parameter, to have the universal coversH and SL 2 R, respectively, through the projective sphere PS 3 .…”

Section: Introductionmentioning

confidence: 85%

“…x 0 , x 0 ̸ = 0 as well. The π periodicity for the above coordinates in the above maps can be seen from the formula (5).…”

Section: Introductionmentioning

confidence: 98%

“…with unit determinant ad − bc = 1 constitute a Lie transformation group by the usual product operation, taken to act on row matrices as on point coordinates on the right as follows [5] (2)…”

Section: Introductionmentioning

confidence: 99%

“…[5]). The curve C(t), t ≥ 0, is said to be a translation curve iḟC(0).T(t) =Ċ(t), t ≥ 0.The solution, depending on (q : u : v : w) had been determined in[16].…”

mentioning

confidence: 99%

“…Geodesic curves. The second order differential equation system of the SL 2 R geodesic curve is the following:[5] (4r) − sinh(2r))θθ ϑ = − 2ṙ sinh(2r) (θ(3 cosh(2r) − 1) + 2φ) ϕ = 2ṙ(φ + 2θ sinh 2 r) tanh(r).…”

mentioning

confidence: 99%

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Some characterization of curves in $\widetilde{\mathbf{SL}_{2}\mathbb{R}}$

Senoussi¹,

Bekkar²

2019

Sib. Elektron. Mat. Izv.

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Section: Introductionmentioning

confidence: 85%

“…x 0 , x 0 ̸ = 0 as well. The π periodicity for the above coordinates in the above maps can be seen from the formula (5).…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

“…[5]). The curve C(t), t ≥ 0, is said to be a translation curve iḟC(0).T(t) =Ċ(t), t ≥ 0.The solution, depending on (q : u : v : w) had been determined in[16].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Some characterization of curves in $\widetilde{\mathbf{SL}_{2}\mathbb{R}}$

Senoussi¹,

Bekkar²

2019

Sib. Elektron. Mat. Izv.

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Isoptic surfaces of segments in $$\textbf{S}^2\!\times \!\textbf{R}$$ and $$\textbf{H}^2\!\times \!\textbf{R}$$ geometries

Csima

2023

J. Geom.

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In this work, we examine the isoptic surfaces of line segments in the $$\textbf{S}^2\!\times \!\textbf{R}$$ S 2 × R and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R geometries, which are from the 8 Thurston geometries. Based on the procedure first described in Csima and Szirmai (Results Math 78:194-19, 2023), we are able to give the isoptic surface of any segment implicitly. We rely heavily on the calculations published in Szirmai (Bul Acad Ştiinţe Repub Mold Mat 2:44–61, 2020; Q J Math 73:477–494, 2022). As a special case, we examine the Thales sphere in both geometries, which are called Thaloid. In our work we will use the projective model of $$\textbf{S}^2\!\times \!\textbf{R}$$ S 2 × R and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R described by Molnár (Beitr Algebra Geom 38:261–288, 1997).

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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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The sum of the interior angles in geodesic and translation triangles of Sl2(R)~ geometry

Cited by 7 publications

References 24 publications

Some characterization of curves in $\widetilde{\mathbf{SL}_{2}\mathbb{R}}$

Some characterization of curves in $\widetilde{\mathbf{SL}_{2}\mathbb{R}}$

Isoptic surfaces of segments in $$\textbf{S}^2\!\times \!\textbf{R}$$ and $$\textbf{H}^2\!\times \!\textbf{R}$$ geometries

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

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