CONSTRUCTION OF HYPERBOLIC FUNDAMENTAL REGIONS FOR SOURCE CODINGAPPLICATIONS

Lucila A. Leskow1 and Reginaldo Palazzo Jr.2

1Unicamp, Campinas, Brazil, lucilaleskow@yahoo.com.br2Unicamp, Campinas, Brazil, palazzo@dt.fee.unicamp.br

Abstract: In this paper we propose a method for the con-struction of a new set of tessellations of the hyperbolic planebased on the Farey series. Among the possible applications,these tessellations can be used for source coding.

Keywords: Engineering Applications, Arithmetic Coding,Farey Series.

1. INTRODUCTION

The analysis of hyperbolic tessellations has led to appli-cations in several contexts such as in the construction of ge-ometrically uniform codes [1], topological quantum codes[2], and in signal set designs. Recent results, [3], show thatthe best performance of a digital communication system isachieved when designed on surfaces of constant negative cur-vature. This is the reason why several works have been donein order to show the importance of introducing the conceptsof surfaces with constant negative curvature in the design ofcommunications systems.

One of the reasons to propose the construction of hyper-bolic tessellations is due to the fact that new procedures forcode construction may come at hand such that the problemof coding geodesic may be realized.

By use of the Farey series, in this paper we propose amethod for the construction of new tessellations of the upper-half plane model of the hyperbolic geometry by the union oftriangles.

These tessellations have fundamental regions consistingof polygons with more than 3 edges and their properties arerelated to the characteristics associated with the Farey series,and so are good candidates for use in applications such asin the transmission of information. As an example, in [4] itis made use of tessellations considered in this paper in orderto propose an arithmetic coding method for ternary discretememoryless sources.

This paper is organized as follows. In Section 2 basicconcepts necessary to the development of the paper are pre-sented. In Section 3 new tessellations based on the union of

the triangles derived from the Farey tessellation is shown. InSection 4 an application to a ternary source coding problemis taken into consideration as an example. Finally, in Section5 the concluding remarks are drawn.

2. PRELIMINARIES

Let H2 = {x + iy ∈ C : y > 0} be the upper-half planeequipped with the Poincaré metric ds2 = dx2+dy2

y2 . With thismetric, H2 becomes a model of the hyperbolic geometry. Thegroup PSL(2,R) = SL(2,R)/{±I} acts on H2 by Möbiustransformations given by: z → az+b

cz+d .These transformations are orientation preserving isome-

tries in H2 with the Poincaré metric. Their actions extend upto the boundary of H2, that is, R ∪ {∞}.

A Fuchsian group is a discrete subgroup of PSL(2,R).Every Fuchsian group has associated with it a fundamentalregion or a fundamental domain, [5]. This region is estab-lished by the generators of the group and this region is alwaysconnected and convex.

The action of a Fuchsian group on the fundamental regionimplies in the covering of H2. Geometrically, the tessellationof H2 may be viewed as a partitioning of the hyperbolic planein non-overlapping regions (polygons).

Let Fm be a Farey sequence of order m. Hence, Fm con-sists of a series of irreducible fractions whose denominatordoes not exceed m, that is, P

Q with |P |, |Q| ≤ m arrangedin increasing order. If P

Q belongs to Fm, 0 ≤ P ≤ Q ≤ m,P and Q are coprimes and the value 0 is included under theform 0

1 . Therefore,

F1 ={−∞,−1, 0, 1,∞} (1)

F2 ={−∞,−2,−1,−1

2, 0,

1

2, 1, 2,∞} (2)

and so on. The Farey series has many interesting properties,among them we mention the following results, [6],

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http://dx.doi.org/10.5540/DINCON.2011.001.1.0118

Theorem 2.1 If P/Q, P ′′/Q′′ and P ′/Q′ are three consec-utive terms of Fm, then

P ′′

Q′′=P + P ′

Q+Q′. (3)

Theorem 2.2 If P/Q and P ′/Q′ are two consecutive termsof Fm, then

QP ′ − PQ′ = 1. (4)

If for each value of m, m ≥ 1, the consecutive terms ofFm are joined by a geodesic, then the tessellation of H2 iscalled Farey tessellation. Fig. 1 illustrates part of the tessel-lation.

Figure 1 – Farey tessellation in the interval [0, 1].

3. TESSELLATIONS OF H2 BY ELEMENTARYPOLYGONS

In this section a new set of tessellations will be presentedwhose corresponding fundamental regions consist of poly-gons with 2n edges obtained from the Farey series. Note thatthe fundamental region of the Farey tessellation, shown inFig. 1, is a hyperbolic triangle.

An interesting fact is that by doing the union of the edgesof the corresponding mirrored triangles with respect to themiddle point of the interval [0, 1], leads to a tessellationwhose fundamental regions consist of 2n edges. Fig. 2 il-lustrates the case of a polygon with four edges.

Figure 2 – Tessellation of the interval [0, 1] by polygons with 4edges.

This tessellation whose fundamental region consists of 4edges shows that it is possible to obtain other tessellationsfrom the union of triangles of the Farey tessellation.

In order to analyze the Farey tessellation, we define whatwe mean by an elementary triangle and an elementary poly-gon. Consider the upper-half plane. A geodesic having as itsend points two consecutive terms of the Farey series is calledan elementary edge.

A triangle is called elementary if it consists of elementaryedges. A polygon is called elementary if its vertices belongto Fm for some m > 2. As a consequence, the Farey tessel-lation consists of the union of elementary triangles.

Consider the triangle formed by the geodesicsC(γ(∞, 0)), C(γ(0, 1)) and C(γ(1,∞)). This trian-gle is called fundamental triangle, as shown in Fig. 3.

Figure 3 – Fundamental triangle.

Each triangle in H2 located below the fundamental trian-gle has its vertices in R ∪ {∞}.

With the aim to analyze the existing relationships amongthe fundamental triangles we need to establish a notation forthe identification of each triangle. Let v1, v2 and v3 be thevertices of a triangle such that v1, v2, v3 ∈ Q and v1 < v2 <v3.

If each triangle is identified by (P3)v1,v3 (with v1, v2,

v3 ∈ Fm) where v1 and v3 are the vertices with the great-est Euclidean distance and by making use of the propertiesof the Farey series, it is always possible to obtain v2 fromthe other two vertices. Looking again at the Farey tessella-tion as shown in Fig. 1, we see that below the fundamentalpolygons there are two types of triangles which may be clas-sified according to the distance between their vertices. Thischaracteristic is identified by an index X . Let d(v1v2) be theEuclidean distance between the two vertices v1 and v2. Whend(v1v2) < d(v2v3), X = D and when d(v1v2) > d(v2v3),X = E.

From the index X , each triangle is denoted by (P3)v1,v3X .

Now, several triangles have in common a pair of vertices andan edge, and in some cases, more than one triangle has thesame vertex. In order to refer to each one of these trianglesin a sequence, we consider an index i such that the least itsvalue is, let us say i, the greater the value of d(v1v3). Thevalue of i indicates the position that a triangle occupies in thesequence.

For instance, consider a sequence of triangles that hasa left vertex in common, that is, (P3)

vc,v1E1

, (P3)vc,v2E2

,(P3)

vc,v3E3

,... as can be seen in Fig. 4. For each value of i(i ≥ 1), (P3)

vc,viEi

and (P3)vc,vi+1

Ei+1means that these two tri-

angles have a pair of vertices and an edge in common, thatis, (vc, vi and C(γ(vi+1, vi))).

By having vertices with values in the Farey series, thesetriangles present several interesting properties, many ofwhich will be considered in this paper. The first propertyestablishes how from an elementary triangle, we may know

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Figure 4 – Elementary triangles of the type E.

which triangle type (E or D) is below it in the Farey tessella-tion.

Property 3.1 Let (P3)v1,v3X be an elementary triangle be-

longing to the Farey tessellation and v1, v2 and v3 its ver-tices such that v1 < v2 < v3 and X may assume the valuesE orD. Below the geodesics C(γ(v1, v2)) and C(γ(v2, v3))there are two triangles having two vertices and an edge incommon with that triangle. Independent of the value of X ,the triangle below C(γ(v1, v2)) will always be of the typeE and the triangle below C(γ(v2, v3)) will always be of thetype D.

This property becomes clear when looking at the Fareytessellation. However, it may be proved by using the resultsfrom Theorems 2.1 and 2.2.

It is from the construction of the elementary polygons thatwe obtain new tessellations of H2. In order to identify thesepolygons it will be used a notation similar to the one used inthe case of the triangles, where instead of using (P3) it willbe used (P2n) indicating that a polygon has 2n edges.

Definition 3.1 Consider a set consisting of 2n triangles suchthat its union form a polygon with 2n edges denoted by:

(P2n)pq ,νw

X =n⋃i=1

(P3)v1i,v3iXki

, (5)

where v1i, v3i ∈ Fm for some m > 3, ki ≥ 1 and the valueof n correspond to the number of triangles of the type E or

D. If X = E then (P2n)pq ,νw

Ekdenotes the polygon of the type

FE (Left Bundle). If X = D then (P2n)pq ,νw

Dkdenotes the

polygon of the type FD (Right Bundle).

In this way, every polygon will be classified either as FEor FD. And so, using the previous notation, we may, forinstance, represent the polygons with four edges in the fol-lowing way:

Definition 3.2 A polygon with 4 edges will be denoted by:

(P4)vc,viEk

= (P3)vc,viEi

∪ (P3)vc,vi+1

Ei+1, (6)

when it is of the type E and by,

(P4)vi,vcDk

= (P3)vi,vcDi

∪ (P3)vi+1,vcDi+1

, (7)

when it is of the type D, such that the polygons will alwaysbe disjunct.

Let (P3)0,∞E0

denote the fundamental triangle with verticesin 0, 1 and∞ and by (P3)

0,1E0

the triangle with vertices in 0,12 and 1, the union of these triangles leads to a polygon with4 edges with vertices in 0, 12 , 1 and∞, the elementary square,and denoted by (P4)

0,∞E0

= (P3)0,∞E0∪ (P3)

0,1E0

.The elementary square (P4)

0,∞E0

corresponds to the fun-damental region of the new tessellation. Knowing the ver-tices of this polygon in the axis x, from Theorem 2.1 thevertices of the two other polygons with four edges are de-termined and denoted by (P4)

0, 12E1

= (P3)0, 12E1∪ (P3)

0, 13E2

and

(P4)12 ,1

D1= (P3)

12 ,1

D1∪ (P3)

23 ,1

D2.

By applying Theorem 2.1 to the consecutive vertices ofeach polygon allows us to generalize the previous results. Forinstance, from the case 2n = 4 we arrive at Theorem 3.1,whose proof will be omitted, however it is a consequence ofusing Theorem 2.1 and Property 3.1.

Theorem 3.1 Let (P4)pq ,νw

X be a polygon such that pq and νw

is the pair of vertices with the greatest Euclidean distance.For i ≥ 1, the polygons with 4 edges may be obtained in thefollowing way:

For X = E we have:

• If p 6= 0, then (P4)pq ,νw

Ei= (P3)

pq ,νw

E2i−1∪ (P3)

pq ,p+νq+w

E2i.

• If p = 0, then (P4)0, 1

2i

Ei= (P3)

0, 12i

E2i−1∪ (P3)

0, 12i+1

E2i.

For X = D we have:

• Ifν

w6= 1, then (P4)

pq ,νw

Di= (P3)

pq ,νw

D2i−1∪ (P3)

p+νq+w ,

νw

D2i.

• Ifν

w= 1, then (P4)

2i−12i ,1

Di= (P3)

2i−12i ,1

D2i−1∪(P3)

2i2i+1 ,1

D2i.

In the same way as in the Farey tessellation, we may seethat there also exists an alternate of polygons of the type FEand FD, see Fig. 2.

Using the same procedure for realizing the union of thetriangles employed to generate the polygons with four edges,if the triangles are taking together and joined four-by-four,six-by-six, eight-by-eight, a new set of tessellations of H2

may be obtained with polygons with 2n edges. In this case,one can see by induction that the union of 2n − 2 trianglesresults in a polygon with 2n edges.

Definition 3.3 A fundamental polygon with 2n edges will beformed by the union of the triangles (P3)

0,∞E0

and (P3)0,1E0

,together with the union of n − 2 triangles of the type E andD with consecutive indices. The triangles of the type E willhave a vertex 0 in common and those of the type D will havea vertex 1 in common.

For the polygons located below the elementary edges ofthe fundamental polygon, we have that a polygon with 2nedges will be formed by the union of 2n − 2 triangles withconsecutive indices. Consequently, for the tessellations ofH2 formed by polygons with 2n edges, it is possible to es-tablish the following result.

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Theorem 3.2 The fundamental polygon with 2n edges of atessellation is formed by the union of the following elemen-tary triangles:

(P2n)0,∞E0

= (P3)0,∞E0∪ (P3)

0,1E0

n−2⋃i=1

(P3)0, 1i+1

Ei

n−2⋃j=1

(P3)jj+1 ,1

Dj

(8)

For the case 2n = 4, and by observing the correspond-ing tessellation in Fig. 2, we can see that there exists analternance between the polygons of the type FE and FD.This characteristic will always be present in the tessellationsformed by polygons with 2n edges. As a consequence,

Property 3.2 Let (P2n)0,∞E0

be a polygon with 2n edges. Thepolygons below the fundamental polygon will always be ofthe form: FE, (FD)n−2(FE)n−2FD where If (FD)n−2 =FD, ..., FD︸ ︷︷ ︸n−2 times

. For the subsequent polygons, the type will be

given by FE, (FD)2n−2 whenever the polygon is of the type

FE that is, (P2n)pq ,νw

Ei, and by FD, (FE)2n−2 whenever the

polygon is of the type FD, (P2n)pq ,νw

Di.

A general formula to obtain the vertices for each polygonwith 2n edges can also be obtained.

4. APPLICATION

In this section an application of the new set of tessella-tions for coding of geodesics as tree codes is presented. Werefer the reader to [4] for further details.

From the tessellations formed by polygons with 4 edges,it is possible to obtain a source coding procedure, similar tothe Elias method, for the output of a discrete memorylesssource, [7]. This method, denoted by CPNI, may be seen asa binary representation of segments of a geodesic belongingto PSL(2,Z). By assumption, this geodesic is a hyperbolictransformation, that is, it has two roots. These roots satisfythe property that one of them belongs to the interval [−1, 0]and the other one is greater than 1.

The fractional part of the root greater than 1 may beseen as the probability of occurrence of the correspondinggeodesic. We denote this probability by ρ. Note that there ex-ists only one geodesic associated with these two roots. Fromthe tessellation formed by polygons with 4 edges, we mayinterpret each such a geodesic as a path in a tree with proba-bilities.

To generate a codeword, we have to look up the j-th rep-resentation of ρ in the subinterval in which it is located. Asoccurs with the Elias method, the CPNI method has also aprecision problem, which may be solved by use of a scalestrategy. The results coming out of the application of theElias method and the CPNI method for 600 sequences withlength 5, 7, 9 and 11 from 15 distinct ternary sources may beseen in Table 1.

It may be seen that without the use of the scale factor, theCPNI method presents a performance which is a little bet-ter than the Elias method. In the average, the Elias method

compacts sequences with 1.41 bits with less than the neces-sary for conveying the information in the original messagewhereas the CPNI method compact sequences with 1.71 bitsless than the same previous sequences. Using a scale strategy(third row in Table 1) it is possible to compact messages witha reduction in the mean of 2.60 bits.

Table 1 – Mean number of bits to convey the sequences.

Original Number of bitsMethod5 7 9 11

Elias 3.47 5.81 7.38 9.67CPNI 3.24 5.50 7.10 9.27CPNI - Scale Ft 3.16 4.62 6.34 7.60

5. CONCLUSIONS AND FINAL REMARKS

In this paper we have proposed a systematic procedure inorder to generate new sets of tessellations of H2 based on theunion of triangles whose vertices belong to the Farey series.

A general formula for obtaining the tessellations formedby polygons with n = 4 edges, with n > 1, was establishedin Theorem 3.1, one of the contributions of this paper. FromTheorem 2.2 and Property 3.1 generalized formulas can beobtained for the fundamental polygon as well as for the ver-tices, for the general case.

Therefore, the procedure of constructing this set of tessel-lations provides a new perspective of hyperbolic tessellationsto be used in the design of signal sets, geodesic coding, andso on.

REFERENCES

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[2]PUB C. Albuquerque, R. Palazzo and E. B. Silva, “Ontoric quantum codes", International Journal of Pure andApplied Mathematics, Vol. 50, pp. 221-226, 2009.

[3] R. Cavalcante and H. Lazari and J. Lima and R.Palazzo, “A New Mathematical Approach for theDesign of Digital Communication Systems", AMS-DIMACS Series, Vol. 68, No. 1, pp. 145-177, 2005.

[4] Leskow, L.H.A. and Palazzo Junior, R., “Codificaçãode fontes ternárias no plano hiperbólico", Anais doXXXII Congresso Nacional de Matemática Aplicada eComputacional, Vol. 1, pp. 1-2, Cuiabá, October 2009.

[5] Katok, S.: Fuchsian Groups, The University of ChicagoPress, 1992.

[6] Hardy, G.H., Wright, E.M.:An Introduction to the The-ory of Numbers, Oxford University Press, 1960.

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