Abstract An n-sided hyperbolic polygon of type (ϵ, n) is a hyperbolic polygon with ordered interior angles π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ, θ1, θ2, …, θn−2, π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ, where 0 < ϵ < π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ and 0 < θi < π satisfying ∑i=1n−2θi+(π2+ϵ)+(π2−ϵ)<(n−2)π $$\begin{array}{} \displaystyle \sum_{i = 1}^{n-2} \theta_{i}+\Big(\frac{\pi}{2}+\epsilon\Big)+\Big(\frac{\pi}{2}-\epsilon\Big) \lt (n-2)\pi \end{array} $$ and θi + θi+1 ≠ π (1 ≤ i ≤ n − 3), θ1 + ( π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ) ≠ π, θn−2 + ( π2 $\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.

中文翻译：

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(ϵ, n) 类型的双曲多边形和莫比乌斯变换

摘要 类型为 (ϵ, n) 的 n 边双曲多边形是具有有序内角 π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ, θ1 的双曲多边形, θ2, …, θn−2, π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ, 其中 0 < ϵ < π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ 和 0 < θi < π 满足 ∑i=1n−2θi+(π2+ϵ)+(π2−ϵ)<(n−2)π $$\begin {array}{} \displaystyle \sum_{i = 1}^{n-2} \theta_{i}+\Big(\frac{\pi}{2}+\epsilon\Big)+\Big(\frac {\pi}{2}-\epsilon\Big) \lt (n-2)\pi \end{array} $$ 和 θi + θi+1 ≠ π (1 ≤ i ≤ n − 3), θ1 + ( π2 $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ) ≠ π, θn−2 + ( π2 $\begin{array}{} \frac{\pi} {2} \end{array} $ − ϵ) ≠ π。在本文中，我们通过使用类型为 (ϵ, n) 的 n 边双曲多边形来展示莫比乌斯变换的新特征。我们的证明基于几何方法。