Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations

Demirel, Oğuzhan

doi:10.36890/iejg.545044

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De Nition 3 [3]1

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2020

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“…In [8,9], O. Demirel presented some characterizations of Möbius transformations by using new classes of geometric hyperbolic objects called "degenerate Lambert quadrilaterals"and "degenerate Saccheri quadrilaterals", respectively.…”

Section: De Nition 3 [3]mentioning

confidence: 99%

The hyperbolic polygons of type (ϵ, n) and Möbius transformations

Demirel

2020

Open Mathematics

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Abstract An n-sided hyperbolic polygon of type (ϵ, n) is a hyperbolic polygon with ordered interior angles $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ, θ1, θ2, …, θn−2, $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ, where 0 < ϵ < $\begin{array}{} \frac{\pi}{2} \end{array} $ and 0 < θi < π satisfying $$\begin{array}{} \displaystyle \sum_{i = 1}^{n-2} \theta_{i}+\Big(\frac{\pi}{2}+\epsilon\Big)+\Big(\frac{\pi}{2}-\epsilon\Big) \lt (n-2)\pi \end{array} $$ and θi + θi+1 ≠ π (1 ≤ i ≤ n − 3), θ1 + ( $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ) ≠ π, θn−2 + ( $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ) ≠ π. In this paper, we present a new characterization of Möbius transformations by using n-sided hyperbolic polygons of type (ϵ, n). Our proofs are based on a geometric approach.