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

  • Oğuzhan Demirel EMAIL logo
Published/Copyright: March 20, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

An n-sided hyperbolic polygon of type (ϵ, n) is a hyperbolic polygon with ordered interior angles π2 + ϵ, θ1, θ2, …, θn−2, π2 − ϵ, where 0 < ϵ < π2 and 0 < θi < π satisfying

∑i=1n−2θi+(π2+ϵ)+(π2−ϵ)<(n−2)π

and θi + θi+1 ≠ π (1 ≤ i ≤ n − 3), θ1 + ( π2 + ϵ) ≠ π, θn−2 + ( π2 − 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.

Keywords: conformal mapping; Möbius transformations; hyperbolic polygons
MSC 2010: 30C20; 30C35; 30F45; 51M10; 51M15

1 Introduction

A Möbius transformation f : ℂ → ℂ is a map defined by f(z) = az+bcz+d, where a, b, c, d ∈ ℂ with ad − bc ≠ 0. They are the automorphisms of extended complex plane ℂ and define 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¯+bcz¯+d, where a, b, c, d ∈ ℂ with ad − bc ≠ 0 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 reflection 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 definitions are well known and fundamental in hyperbolic geometry.

Definition 2

[3] A Lambert quadrilateral is a hyperbolic quadrilateral with ordered interior angles π2,π2,π2 and θ, where 0 < θ < π2 .

Definition 3

[3] A Saccheri quadrilateral is a hyperbolic quadrilateral with ordered interior angles π2,π2, θ, θ, where 0 < θ < π2 .

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].

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.

Definition 4

[9] A degenerate Lambert quadrilateral is a hyperbolic convex quadrilateral with ordered interior angles π2+ϵ,π2,π2−ϵ, θ, where 0 < θ < π2 and 0 < ϵ < π2−θ2.

Theorem 5

[9] Let f : B2 → B2 be a surjective transformation. Then f is a Möbius transformation or a conjugate Möbius transformation if and only if f preserves all ϵ-Lambert quadrilaterals.

Definition 6

[8] A degenerate Saccheri quadrilateral is a hyperbolic convex quadrilateral with ordered angles π2−ϵ,π2+ϵ, θ, θ, where 0 < θ < π2 and 0 < ϵ < π2−θ2.

Theorem 7

[8] Let f : B2 → B2 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, B2 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 affirmative answer of this question let us state the following definition:

Definition 8

Let n be a positive integer satisfying n ≥ 5. An n-sided hyperbolic polygon of type (ϵ, n) is a convex hyperbolic polygon with ordered interior angles π2 + ϵ, θ1, θ2, …, θn−2, π2 − ϵ or π2 − ϵ, θ1, θ2, …, θn−2, π2 + ϵ, where 0 < ϵ < π2 and 0 < θi < π satisfying

∑i=1n−2θi+(π2+ϵ)+(π2−ϵ)<(n−2)π,

and θi + θi+1 ≠ π (1 ≤ i ≤ n − 3), θ1 + ( π2 + ϵ) ≠ π, θn−2 + ( π2 − ϵ) ≠ π. Notice that the sides of the n-sided hyperbolic polygon of type (ϵ, n) we mentioned here are hyperbolic line segments.

The existence of n-sided hyperbolic polygons of type (ϵ, n) is clear by the following result:

Lemma 9

[3] Let (θ1, θ2, ⋯, θn) be any ordered n-tuple with 0 ≤ θj < π, j = 1, 2, …, n. Then there exists a polygon P with interior angles θ1, θ2, …, θn, occurring in this order around ∂ P, if and only if ∑i=1n θi < (n − 2)π.

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 dH(P, Q) the hyperbolic distance between points P and Q. We consider the hyperbolic plane B2 = {z ∈ ℂ: ∣z∣ < 1} with length differential ds2=4|dz|2(1−|z|2)2. The Poincaré disc model of hyperbolic geometry is built on B2; more precisely the points of this model are points of B2 and the hyperbolic lines of this model are Euclidean semicircular arcs that intersect the boundary of B2 orthogonally including diameters of B2. Given two distinct hyperbolic lines that intersect at a point, the measure of the angle between these hyperbolic lines is defined by the Euclidean tangents at the common point.

2 A characterization of Möbius transformations by use of hyperbolic polygons of type (ϵ, n)

The assertion f preserves n-sided hyperbolic polygons A1A2 ⋯ An of type (ϵ, n), n ≥ 5, with ordered interior angles π2 − ϵ, θ1, θ2, …, θn−2, π2 + ϵ means that the image of A1A2 ⋯ An under f is again an n-sided hyperbolic polygon A1′A2′⋯An′ with ordered interior angles π2−ϵ,θ1′,θ2′,…,θn−2′,π2+ϵ and if P is a point on any side of A1A2 ⋯ An, then P′ is a point on any side of A1′A2′⋯An′.

Lemma 10

Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 . Then f is injective.

Proof

Let A1 and A2 be two distinct points in B2. It is clear that there exists an (2n − 4)-sided hyperbolic regular polygon (n > 4, n ∈ ℕ), say A1A2 ⋯ A2n−4. By β denote the interior angles of A1A2 ⋯ A2n−4. Let M and N be the midpoints of [A2n−4, A1] and [An−2, An−1], respectively. Then the hyperbolic polygons MA1A2 ⋯ An−2N and NAn−1An ⋯ A2n−4M are n-sided hyperbolic polygons satisfying MN ⊥ A1A2n−4 and MN ⊥ An−2An−1. Let P be a point on [M, A1] and let Q be a point on [N, An−1] satisfying dH(P, A1) = dH(Q, An−1). By ψ denote the angle ∠ QPA1. Since A1A2A3 ⋯ A2n−4 is an (2n − 4)-sided hyperbolic regular polygon, we immediately get ∠ PQAn−1 = ψ. If ψ > π2 let’s denote ψ = π2 + α and if ψ < π2 let’s denote ψ = π2 − α. Hence we see that PA1A2 ⋯ An−2Q is an n-sided hyperbolic polygon of type (α, n). By assumption, we obtain that P′A1′A2′⋯An−2′Q′ is also an n-sided hyperbolic polygon of type (α, n), which implies A1′≠A2′. Thus f is injective.□

Lemma 11

Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 . Then f preserves the collinearity and betweenness property of the points.

Proof

Let P and Q be two distinct points in B2 and assume that S is an interior point of [P, Q]. Let Δ 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 Δ. By the property of f, the images of the elements of Δ are n-sided hyperbolic polygons of type (ϵ, n) whose vertices contain P′ and Q′. Moreover, the images of the elements of Δ 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.□

Lemma 12

Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 . Then f preserves the angles π2 + ϵ and π2 − ϵ.

Proof

Let A1A2 ⋯ An be an n-sided hyperbolic polygon of type (ϵ, n) such that ∠AnA1A2 = π2 − ϵ, ∠ Ai−1AiAi+1 = θi (2 ≤ i ≤ n − 1) and ∠ An−1AnA1 = π2 + ϵ. Let us denote the midpoint of A1 and An by M. Clearly, the hyperbolic line A1A2 and the complex unit disc B2 meet at two points, say P and Q. Assume A1 ∈ [P, A2] and ∠ MA2Q = ρ. Let X be a point on [P, A2] moving from P to A2. It is easy to see that if X moves from P to A2, the angle ∠MXQ increases continuously from 0 to ρ. Let H be a point on [A1, A2] such that ∠ MHA2 < π2 . Now take a point on the hyperbolic line AnAn−1, say S, satisfying An ∈ [S, An−1] and dH(S, An) = dH(H, A1). It is easy to see that dH(M, A1) = dH(M, An), ∠ SAnM = ∠ HA1M, and dH(A1, H) = dH(An, S) hold, which implies that the hyperbolic triangles HA1M and SAnM are congruent triangles by the hyperbolic side-angle-side theorem. Hence we get ∠A1MH = ∠AnMS and this yields that the points H, M and S must be collinear and ∠ MHA1 = ∠ MSAn. Since ∠ MHA2 < π2 , we may assume the representation ∠ MHA2 = π2 − α, where 0 < α < π2 , which implies ∠ MSAn = π2 + α. Notice that α must be less than ϵ since π2 − ϵ < π2 − α. Therefore, one can easily see that HA2 ⋯ An−1S is an n-sided hyperbolic polygon of type (α, n). By the property of f, the images of the hyperbolic polygons A1A2 ⋯ An and HA2 ⋯ An−1S are n-sided hyperbolic polygons of type (ϵ, n) and type (α, n), respectively. The hyperbolic polygons A1′A2′⋯An′ and H′A2′⋯An−1′S′ have n − 2 common angles, which implies ∠An′A1′A2′=π2±ϵ and ∠S′H′A2′=π2±α. Now assume ∠An′A1′A2′=π2+ϵ. By Lemma 11, we get H′∈[A1′,A2′],M′∈[A1′,An′] and M′ ∈ [H′, S′]. Since ∠S′H′ A2′=π2 ± α and α < ϵ hold, we get that the sum of the measures of interior angles of the hyperbolic triangle M′A1′H′ is ∠A1′M′H′+(π2+ϵ)+(π2±α), which is greater then π. This is not possible in hyperbolic geometry and so we get ∠An′A1′A2′=π2−ϵ, which yields ∠An−1′An′A1′=π2+ϵ. □

Lemma 13

Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 . Then f preserves the hyperbolic distance.

Proof

Let P and Q be two distinct points in B2. Take a point S such that PQS forms a hyperbolic equilateral triangle. By β denote its angles ∠ PQS = ∠ QSP = ∠ SPQ := β. Since β < π2 let’s use the representation β = π2 − α with 0 < α < π2 . By Lemma 9, there exists an n-sided hyperbolic polygon of type (α, n), say A1A2 ⋯ An, such that ∠A1A2A3 = γ2, ∠A2A3A4 = γ3, …, ∠ An−2An−1An = γn−1, ∠ An−1AnA1 = π2 + α and ∠AnA1A2 = π2 − α. Then the angle ∠AnA1A2 of the hyperbolic polygon A1A2 ⋯ An can be moved to the point P by using an appropriate hyperbolic isometry g such that g(A2) ∈ [P, S] (or S ∈ [P, g(A2)]) and g(An) ∈ [P, Q] (or Q ∈ [P, g(An)]). Since f preserves the angles π2 + ϵ and π2 − ϵ of n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 by Lemma 12, we get

π2−α=∠SPQ=∠g(An)g(A1)g(A2)=∠g(An)′g(A1)′g(A2)′=∠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 dH(P, Q) = dH(P′, Q′); see [11, 12].□

Corollary 14

Let f : B2 → B2 be a mapping which preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 . Then f preserves the measures of all angles.

Theorem 15

Let f : B2 → B2 be a surjective transformation. Then f is a Möbius transformation or a conjugate Möbius transformation if and only if f preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 .

Proof

The “only if” part is clear since f is an isometry. Conversely, we may assume that f preserves all n-sided hyperbolic polygons of type (ϵ, n) in B2 and f (O) = O by composing an hyperbolic isometry if necessary. Let us take two distinct points in B2 and denote them by x, y. By Lemma 13, we immediately get dH(O, x) = dH(O, x′) and dH(O, y) = dH(O, y′), namely ∣x∣ = ∣x′∣ and ∣y∣ = ∣y′∣, where ∣⋅∣ denotes the Euclidean norm. Therefore, we get ∣x − y∣ = ∣x′ − y′∣ since f preserves angular sizes by Corollary 14. As

2〈x,y〉=|x|2+|y|2−|x−y|2=|x′|2+|y′|2−|x′−y′|2=2〈x′,y′〉,

f preserves the Euclidean inner-product this implies that f is a restriction of an orthogonal transformation on B2, that is, f is a Möbius transformation or a conjugate Möbius transformation by Carathéodory’s theorem.□

Corollary 16

Let f : B2 → B2 be a conformal (angle preserving with sign) surjective transformation. Then f is a Möbius transformation if and only if f preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 .

Corollary 17

Let f : B2 → B2 be an angle reversing surjective transformation. Then f is a conjugate Möbius transformation if and only if f preserves n-sided hyperbolic polygons of type (ϵ, n) for all 0 < ϵ < π2 .

Acknowledgement

The author would like to thank the anonymous reviewers for their careful, constructive and insightful comments in relation to this work.

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Received: 2019-10-14
Accepted: 2020-02-05
Published Online: 2020-03-20

© 2020 Oğuzhan Demirel, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
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  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0015
Keywords for this article
conformal mapping; Möbius transformations; hyperbolic polygons
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