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

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.


Introduction
A Möbius transformation f : C → C is a map de ned by f (z) = az+b cz+d , where a, b, c, d ∈ C with ad − bc ≠ . They are the automorphisms of extended complex plane C and de ne the Möbius transformation group M(C) with respect to composition. Möbius transformations are also directly conformal homeomorphisms of C onto itself and they have beautiful properties. For example, a map is Möbius if and only if it preserves cross ratios. As for geometric aspect, circle-preserving is another important characterization of Möbius transformations. The following result is one of the most famous theorems for Möbius transformations:

Theorem 1. [1] If f : C → C is a circle preserving map, then f is a Möbius transformation if and only if f is a bijection.
The transformations f (z) = az+b cz+d , where a, b, c, d ∈ C with ad − bc ≠ are known as conjugate Möbius transformations of C. It is easy to see that each conjugate Möbius transformation f is the composition of complex conjugation with a Möbius transformation. Since the complex conjugate transformation and Möbius transformations are homeomorphisms of C onto itself (complex conjugation is given by re ection in the plane through R∪{∞}), conjugate Möbius transformations are homeomorphisms of C onto itself. Notice that the composition of a conjugate Möbius transformation with a Möbius transformation is a conjugate Möbius transformation and composition of two conjugate Möbius transformations is a Möbius transformation. There is topological distinction between Möbius transformations and conjugate Möbius transformations: Möbius transformations preserve the orientation of C, whereas conjugate Möbius transformations reverse it. To see more details about conjugate Möbius transformations, we refer the reader to [2]. The following de nitions are well known and fundamental in hyperbolic geometry.
A Möbius invariant property is naturally related to hyperbolic geometry. To see the characteristics of Möbius transformations involving Lambert quadrilaterals and Saccheri quadrilaterals, we refer the reader to [4]. Moreover, there are many characterizations of Möbius transformations by using various hyperbolic polygons; see, for instance, [5][6][7].

Theorem 7. [8] Let f : B → B be a surjective transformation. Then f is a Möbius transformation or a conjugate Möbius transformation if and only if f preserves all ϵ-Saccheri quadrilaterals.
In the theorems above, B is the open unit disc in the complex plane. Naturally, one may wonder whether the counterpart of Theorem 7 exists for hyperbolic polygons instead of using degenerate Saccheri quadrilaterals. Before giving the a rmative answer of this question let us state the following de nition: De nition 8. Let n be a positive integer satisfying n ≥ . An n-sided hyperbolic polygon of type (ϵ, n) is a convex hyperbolic polygon with ordered interior angles π + ϵ, θ , θ , . . . , θ n− , π − ϵ or π − ϵ, θ , θ , . . . , θ n− , π + ϵ, where < ϵ < π and < θ i < π satisfying Notice that the sides of the n-sided hyperbolic polygon of type (ϵ, n) we mentioned here are hyperbolic line segments.
This paper presents a new characterization of Möbius transformations by use of mappings which preserve n-sided hyperbolic polygons of type (ϵ, n). To do so, we need Carathéodory's theorem which plays a major role in our results. C. Carathéodory [10] proved that every arbitrary one-to-one correspondence between the points of a circular disc C and a bounded point set C such that which circles lying completely in C are transformed into circles lying in C must always be either a Möbius transformation or a conjugate Möbius transformation.
Throughout the paper we denote by X the image of X under f , by [P, Q] the geodesic segment between points P and Q, by PQ the geodesic through points P and Q, by PQR the hyperbolic triangle with vertices P, Q and R, by ∠PQR the angle between [P, Q] and [P, R] and by d H (P, Q) the hyperbolic distance between points P and Q. We consider the hyperbolic plane B = {z ∈ C : |z| < } with length di erential ds = |dz| ( −|z| ) . The Poincaré disc model of hyperbolic geometry is built on B ; more precisely the points of this model are points of B and the hyperbolic lines of this model are Euclidean semicircular arcs that intersect the boundary of B orthogonally including diameters of B . Given two distinct hyperbolic lines that intersect at a point, the measure of the angle between these hyperbolic lines is de ned by the Euclidean tangents at the common point.

A characterization of Möbius transformations by use of hyperbolic polygons of type (ϵ, n)
The assertion f preserves n-sided hyperbolic polygons A A · · · An of type (ϵ, n), n ≥ , with ordered interior angles π −ϵ, θ , θ , . . . , θ n− , π +ϵ means that the image of A A · · · An under f is again an n-sided hyperbolic polygon A A · · · A n with ordered interior angles π − ϵ, θ , θ , . . . , θ n− , π + ϵ and if P is a point on any side of A A · · · An, then P is a point on any side of A A · · · A n . By ψ denote the angle ∠QPA . Since A A A · · · A n− is an ( n − )-sided hyperbolic regular polygon, we immediately get ∠PQA n− = ψ. If ψ > π let's denote ψ = π + α and if ψ < π let's denote ψ = π − α.
Hence we see that PA A · · · A n− Q is an n-sided hyperbolic polygon of type (α, n). By assumption, we obtain that P A A · · · A n− Q is also an n-sided hyperbolic polygon of type (α, n), which implies A ≠ A . Thus f is injective.
Lemma 11. Let f : B → B be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all < ϵ < π . Then f preserves the collinearity and betweenness property of the points.
Proof. Let P and Q be two distinct points in B and assume that S is an interior point of [P, Q]. Let ∆ be the set of all n-sided hyperbolic polygons of type (ϵ, n) such that the points P and Q are two adjacent vertices of these hyperbolic polygons. Then S belongs to all elements of ∆. By the property of f , the images of the elements of ∆ are n-sided hyperbolic polygons of type (ϵ, n) whose vertices contain P and Q . Moreover, the images of the elements of ∆ must contain S . Since f is injective by Lemma 10, we get P ≠ S ≠ Q . Therefore, S must be an interior point of [P , Q ], which implies that f preserves the collinearity and betweenness of the points.
Proof. Let A A · · · An be an n-sided hyperbolic polygon of type (ϵ, n) such that ∠An  (An , S) hold, which implies that the hyperbolic triangles HA M and SAn M are congruent triangles by the hyperbolic side-angle-side theorem. Hence we get ∠A MH = ∠An MS and this yields that the points H, M and S must be collinear and ∠MHA = ∠MSAn. Since ∠MHA < π , we may assume the representation ∠MHA = π − α, where < α < π , which implies ∠MSAn = π + α. Notice that α must be less than ϵ since π − ϵ < π − α. Therefore, one can easily see that HA · · · A n− S is an n-sided hyperbolic polygon of type (α, n). By the property of f , the images of the hyperbolic polygons A A · · · An and HA · · · A n− S are n-sided hyperbolic polygons of type (ϵ, n) and type (α, n), respectively. The hyperbolic polygons A A · · · A n and H A · · · A n− S have n − common angles, which implies ∠A n A A = π ± ϵ and ∠S H A = π ± α. Now assume ∠A n A A = π + ϵ. By ∠S H A = π ± α and α < ϵ hold, we get that the sum of the measures of interior angles of the hyperbolic triangle M A H is ∠A M H + ( π + ϵ) + ( π ± α), which is greater then π. This is not possible in hyperbolic geometry and so we get ∠A n A A = π − ϵ, which yields ∠A n− A n A = π + ϵ.
Lemma 13. Let f : B → B be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all < ϵ < π . Then f preserves the hyperbolic distance.
Proof. Let P and Q be two distinct points in B . Take a point S such that PQS forms a hyperbolic equilateral triangle. By β denote its angles ∠PQS = ∠QSP = ∠SPQ := β. Since β < π let's use the representation β = π − α with < α < π . By Lemma 9, there exists an n-sided hyperbolic polygon of type (α, n), say A A · · · An, such that Then the angle ∠An A A of the hyperbolic polygon A A · · · An can be moved to the point P by using an appropriate hyperbolic isometry g such that g(A ) ∈ [P, S] (or S ∈ [P, g(A )]) and g(An) ∈ [P, Q] (or Q ∈ [P, g(An)]). Since f preserves the angles π + ϵ and π − ϵ of n-sided hyperbolic polygons of type (ϵ, n) for all < ϵ < π by Lemma 12, we get π − α = ∠SPQ = ∠g(An)g(A )g(A ) = ∠g(An) g(A ) g(A ) = ∠S P Q , which implies ∠PQS = ∠P Q S and ∠QSP = ∠Q S P . Because of the fact that the angles at the vertices of a hyperbolic triangle determine its lengths, we get d H (P, Q) = d H (P , Q ); see [11,12].