Casey Robert Hume- Generalized Quadrilateral Circle Patterns

8/3/2019 Casey Robert Hume- Generalized Quadrilateral Circle Patterns

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GENERALIZED QUADRILATERAL CIRCLE PATTERNSby

CASEY ROBERT HUME, B.A.

A THESISIN

MATHEMATICSSubmitted to the Graduate Faculty

of Texas Tech University inPartial Fulfillment ofthe Requirements forthe Degree of

MASTER OF SCIENCi:

Approved

Co-Chair])erson of the Committee

Co-Chairpjerson of the Committee

Accepted

Dean of the Graduate SchoolAugust, 2003


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ACKNOW LEDGEMENTS

I would like to express gratitude for the generosity of the CSEM ScholarshipProgram, which was funded through a grant from the NSF and proposed by thefaculty of the Colleges of Engine ering an d -Arts and Sciences at Texas Tech University.

.Additionally I would like to than k Dr. Kim berly Drews for her sup po rt andencouragement, and luy family, friends and Eisak and Daphne for their concern andenthusiasm (however difficult it may have been to tell Daphne was enthusiastic).

The graphics in this paper were produced using Maple.


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C O N T E N T S

.ACKNOWLEDGEMENTS i iLIST OF FIGU RE S ivL INTROD UCTION 12. GENER,AL1ZED QUADRIL.ATERAL CIRCL E PATT ERN S 3

2.1 No tation 32.2 Qu adrilateral Circle Patt erns 32.3 Ra nge Co nstructio n 172.4 Possible Benefits 22

3. CON STANT ANGLE CONDITION 233.1 Des cription 233.2 Necessary and Sufficient Cond ition s 233.3 Req uiring a/ / -f a\ = 7r . 253.4 Geometr ic Radii Condit ion . . . 29

4. EXA.MPLES - 34BIBLIOGR,APH^' 38

111


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LIST OF FIGURES

2.1 .A grap hic of the desired prop erties of a circle pa tte rn 42.2 .A circle pa tte rn of the identity m ap 72.3 Exa mp le 2.1 122.4 Exa mp le 2.2 with 9 = 7r/3. . 142.5 Ex am ple 2.3 and zoom of Exa mp le 2.3 152.6 E.xample 2.4 beg inn ing circle and four circles 202.7 O ne possible extension of Exam ple 2.4 213.1 Exa mp le 3.1 284.1 .A circle pa tte rn for f{z) = 2^/^ with cons tant angles 7r/2. 354.2 An other circle pat tern for f{z) = z^^^ with cons tant angles 354.3 A Q C P for f{z) = z^ with constant angles of - / 2 364.4 An other Q C P for f{z) = 2^ with coii.stant angle s 364.5 A Q C P for f{z) = log{z) with co ns tan t rr/2 aiiKl* s 374.6 A Q C P for f{z) = ( with constant - / 2 angles. . 37

IV


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CHAPTER 1INTRODUCTION

.Although much is currently being learned about circle packings and patterns,their usefulness in terms of computation and approximations is restricted by theirappU cability. In specific, one of the gre atest downfalls to th is field of study thusfar has been the difficulty of utilizing current techniques in analyzing quasiconformalmaps. This has caused much difficulty in several applications of these techniques,such as in the field of brain mapping [3, 12]. This paper seeks to examine some ofthe basic aspects of circle patterns in order to expand the situations in which theseapproximations may be utilized, with specific interest in allowing the inclusion of evensuch quasiconformal approximations.

Our method of generalizing circle patterns, herein restricted to generalizing thequadrilateral circle pattern, is achieved by flKll^iIlJ^ entirely un the intersection pointsof the p at te rn and using these to derive aiiv add itiona l information required. By doingthis, we allow a larger category of patterns, including some modelling quasiconformalmapjs.

Following Thurston's Conjet ture [11] and Rodin and Sullivan s subsequent proof[6] that hexagonal c ircle packings can be u.sed to approximate Riemann maps, therehas been a flurry of research activity into the depth of the connection between con-formal m aps and circle packings and pa tte rns . Oded Sc hram [7] first used circlepatterns with combinatorics of a square lattice to approximate entire functions incertain restricted cases (in this paper these conditions are discussed as the result ofrequiring diagonal circles to be tangent). Bobenko, Hoffman, and Suris [4, 1] loosenedSchramm's condition under the hexagonal case to require only that the pattern had aset of global constant angles of intersection. In this thesis, we return to quadrilaterallattic es with much looser restric tions tha n previously considered. In par ticu lar we areable to directly approximate quasiconformal maps for the first time.

In Chapter 2, we give a definition for a new type of circle pattern constructed


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by focusing only on the points of intersection created by the quadrilateral lattice.We also present a sclieme for constructing even the most general examples of such aquadrilateral circle pattern.

In Chapter 3, we discuss Bobenko and Hoffman's constant angle condition as itrelates to our quadrilateral circle patterns. We also prove that Schramm's tangencycondition necessitates the constant emgle condition, giving us another way to forceBobenko and Hoffman's condition for our quadrilateral circle patterns.

Chapter 4 presents several classical examples of mappings which have been previously done by the mentioned authors, using other methods. Here these have beenconstructed utilizing a simple Maple program written based on the work presented inthis paper.


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CHAPTER 2GENERALIZED QUADRIL.ATERAL CIRCLE PATTERNS

2.1 NotationThis paper will utiUze the following symbols with the accompanying definitions:

Z = Th e ring of integersR = Th e field of real num bers 0 and ^ R, arg(z) = 6,(uniquely defined only if we restrict to a specific branch oflogari thm).

Additional notation will be presented as it is discussed.

2.2 Qu adrilateral Circle Pa tter nsDiscretizations of complex functions are sometimes useful as a tool in analysis,

especially when the original function is only theoretically possible, and no clear construction is known.

By discretizing complex functions carefully, the image of a function may be visualized, so as to provide a clue to understanding the nature of the original function.


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Figure 2.1: A graphic of the desired properties of a circle pattern.

.An example of a method of this kind of discretization is the construction of circlepa tte rn s. T he mo st gene ral model of a circle pa tte rn is simply a cluster of overlap)-ping circles which cover the im age of the original function (except p erha ps nea r to theboimdary) by the union of their interiors. This is very similar to the carnival gameinvolving discs which must be dropped to cover a picture on a card.

Instead of looking at these very general patterns, we will focus here on patternshaving the following properties, which are illustrated in Figure 2.1:

1. Every circle (in the in terior of the pa tte rn) has exactly four specified intersectionpoints on it.

2. Every specified point of intersection has at least four circles which pass throughit.

3. Every circle (in the interior of the pattern) intersects exactly four adjacentcircles in two of these specified points, and exactly four adjacent circles in onlyone of these specified points.

4. If two circles Ci and C2 intersect in points pi and p2, then the arc [pi,p2] in eachcircle which lies in the interior of the other circle contains no other specifiedintersection points.


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Note here that we will not be requiring that these specified points of intersectionare the only points of intersection, but merely that our pattern will have at leastthes e po int s of inters ectio n. .All othe r intersection p oints are disregarded as we arenot requiring that our patterns keep track of all arcs of circles in the domain.

-A circle with exa ctly four specified poin ts along it is a special exam ple of a complexqu adr ilate ral. Recall tha t a simple closed curve in C is also called a Jord an curve,and a Jordan curve with four identified distinct points is called a quadrilateral (Inthe following discussion it will sometimes be necessary to distinguish between thisdefinition of a quadrilateral and the typical Euclidean definition, in order to do this,a polygon of four sides will be called a Euclidean quadrilateral).

We now endeavor to establish necessary and sufficient conditions for a scatteringof points in C to m ake up a circle pat tern . This will enable us to add conditions tofind more specific types of patterns. In this interest, observe the following proposition.P r o p o s i t i o n 2 , 1 . Given four distinct complex points, z\. 22, 23, and 21 the followingstatements are pairwise equivalent:

1. zi , 22. 23. and 24 are either collinear or cocircular (in either case they arecontained in the same circle in C^).

2. g ( 2 i . 2 2 , 2 3 , 2 4 ) R .

3. 3m (z,2 j)( |2 2| ' - I24I') +3 m(2 22 T)(|2 , | ' - I23I')= 3 m (2 ,Z i)(|2 3| ' - |24p) + 3m(Z2 Zj)(|Zi| ' - I24I')

-h 3m(232 T)(|2,|2 - I22I') + 3m(2 ,2T )(|22 |' - I23H

21I' 21 27 1I22P 22 2i 1123^ 23 2j 1|Z4|^ 24 27 1

= 0.


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Proof. (1) . (2) This is a well known result of analysis of the cross ratio on Coo-i-) "^^^ (3) Results from simplification and collection of the numerator of the expression 3m(g(2i, 22, ^3. ~4)) = 0. Th e reverse is true since all four poin ts are distin ct,hen ce (21 - 24) (22 - 23) is non zero , so we may d ivide by i t.(3) (4) Is a simple, but lengthy exercise of linear algebra and appears as anexercise in [8]. DD e f i n i t i o n 2 . 2 . Let p he a mapping

p-.ZxZ > Cp{n, m) = p,m-

Then p is a (local) discrete immersion if:1. for every {n,m) Z x Z, the figure Pn,m obtained by the union of the segments

\Pn.m,Pn + l,m]< \Pn+l,m, Pn+l.m+l]- \Pn+l.m+l> Pn.m+l], (^^d [Pn,m+1, Pn.m] (nOUe of

which are single point.'^. i.e. all four points a re distinct) is either:(a) a polygon, in w hich case C \ Pn.m consists of exactly two components,

or(b) a straight line segment between two of the points, in which case each of the

individual line segments are considered to be 'sides" of the line.2. for every (n, m) G Z x Z the set of distinct points

{Pn,m-li P n + l , m - l . P n + 2 , m - l , Pn+2,m, P n + 2 , m + l ) P n + 2 , m + 2 ,P n + l , m + 2 i P n .m + 2 i P n - l , m + 2 ) P n - l , m - f l ) P n - l , m , P n - l , m - l /

IS contained entirely within the unbounded comp onent of C\ Pn,m (in the casethat Pn,m w a '"e this means that none of the points are actually on Pn,m)-


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I W w y V

V 1 \ r \ 7 \ M

1 1 ) I ) )

Figure 2.2: A circle pattern of the identity map.

In other words, each four points of Z x Z which are corners of a unit square aremapped by p to some Euclidean quadrilateral (or line segment). .Moreover, any twoadjacent sets of points are mapped to adjacent Euclidean quadrilaterals (or possiblyline segments) which have exactly one side in common (where for the linear case, sideis explained in the definition).De f in i t i on 2 . 3 . A quadrilateral circle pattern (or QC P) is the image of a (local)discrete immersion

p : Z X Z > Cp{n, m) = Pn,m

under which the equation3m(q{pn,m,Pn+l,m,Pn+l,m + UPn,m+ l)) = 0

holds true for all (n, m) Z x Z .7


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Thus , by Proposition 2.1(1) each such set of four adjacent points may be joinedby a real circle or a straight line.

Furthermore, it is clear one may also require versions (3) and (4) of the givencondition, with version (4) giving the following corollary.Cor o l lar y 2.4. .4 (local) discrete imm ersionp : Z x Z >C such thatp{n,m) = pn,mis a quadrU atend cvxle p attern if and only if

\Pn.ml Pn,m Pn.m2

1| Pn + l , m | P n + l , m P n + l , m 1| P n , m + l | Pn,m+l Pn,m+l 1

iPn+l.r/i + l l p + i , m + l P n + l , m + l 1

= 0 for all {n, m) G Z x

Proof. This is an immediate consequence of 2.1(4), but one must note det{.A) -{l)"det{Ei E2 E3... En A) where E, are elementary row swapping matrices.Thus if det{A) = 0 then det{Ei "2 E 3 . . . .4) = 0, and the converse is true aswell. D

.As a quick remark, it is important to note that the four conditions of Proposition2.1 are independent of the ordering of the four points, which is easily seen from 2.1(4)and the noted fact of linear algebra from the proof of Corollary 2.4. This will be afact used later.

To see how such a (local) discrete immersion relates to a circle pattern, observe thefollowing m eth od of represen tation of circles presented by [8] included here withoutproof

Fac t : Let 2! be a 2 x 2 hermitian matrix, with complex entries A, B, C, and D(where both .4 and D are real, and B = C). Then

'A B"(i = C D,is a matrix representation of the complex circle

{z, l) = Azl+Bz + Cl + D = 0.8


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In thi s rep rese ntatio n, if C represents a real circle then th at circle has center 7 andradius p which can be expressed in terms of .4, B. C, and D as follows,

B = - .4TC = - . 4 ^ = BD = . 4 ( | 7 | - ' - p ^ )

furtherm ore the d eterm inan t A of (!)A = det{) = -A'p'.

(2.1)(2.2)(2.3)

(2.4)In addition, such circles can be fully classified by the real numbers .4 and A by,

.4 5i: 0, A < 0 real circle p > 0A = 0 point circle p = 0A > 0 imaginary circle p < 0

.4 = 0, (A = - 1 5 p < 0 )A < 0 straig ht lineA = 0 0 (D 7i 0) or C (D = 0)

Then, utilizing this notation, we may write each quartet of points based at {n,m) GZ x Z {i.e pn,m, Pn+i,m, Pn.m4i, Pn+i,m+i) in a matrix of formulas suggested byProposition 2.1(3).

' 3 m ( p n + i , m P n , m + l )4 - 3 m ( p + l , m P n + l, m + l )- ^ 3 m ( p + l , m + l P n , m + 1 )

-HPn+l.m+l ( |Pn + l ,m| " |Pn ,m +l | )+P n + l .m ( | P n , m +l l " | P n +l , m - f l | )

+Pn ,r7 i+1 ( |Pn + l ,m +l |^ " |Pn +l ,m | ) j

^ ( p n + l , m . l ( | pn + l , m | ' " K m + l T ) | p + l. m +l P 3 m ( p + i , ^ P , ^ + l )+ P n + l . m ( |P n,r .+ i r - | P n + l , m + l | ' ) + |P n.m +l | ' j ' m ( p + l , ^ p + l , ^ + l )

+ P . m + 1 ( | P n + l . m + l | ' " | P n + l . m r ) j + | p + l , m | ' 3 m ( p + i , ^ + i P n .m + l )


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which is the result of viewing Proposition 2.1(3) as a quadratic form with variableterms of p, , and | v ^ .

Thus we see that ,,,, is indeed hermitian, and represents either a line or a realcircle by Proposition 2.1.

It represents a line when3m(p+l,TO P +l,m +l) -I- 3m (p ,m+ i Pn+l,m) + 3ni (p+i ,m+l Pn,m+l) = 0

and thusPn+l,m Pn+l,m+l + Pn,m-f 1 Pn+l,m + Pn+l,m+l Pn,m+1 ^ R-

I f 3m ( p + i , , P n .m+ l ) + 3m ( p + i , p + , , ^ + l ) + 3m( p + i , m+ l P n , m+ l) " 0.


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when| P n + l . r n + l | 3 m ( p + i _ , n p , ^ i ) - |- | p ^ ^ . i | 3 m ( p + i , i p + i , , j n . i )

+ IP-. + l . m j 3 n i ( p + i , , + i Pn,m+l) " 0 .

2. For 3m(p+,,, p+i,,+,) + 3m(p,,+i Pn+i,m) + 3m (p+ i,^+ i p,m+i) ^ 0 usethe standard representation

(l c\C D^

Notice that each of these representations are unique, depending only on the valuesof certain ratios of the matrix entries and by Proposition 2.1(4) and the proof ofCoroUaiy- 2.4 each is independent of which three points of each quadrilateral areutilized in the formula.

Then if we ensure that no four cocircular points p,,, Pn+i,m, Pn,m+i. and pn+\.,m+\of the quad rilate ral circle patte rn p are collinear. we need use only the unique stan dardrepresentation of each real circle Cn,m as

B,m

where .4. B, C. and D are defined as^.m - I ,. o

.4 = 3 m ( p + i , m P n , m + l ) + 3 m ( p + i . m P n + l . m + l ) + 3 n i ( p + i , m + l P n , m + l ) ( 2 . 5 )

B = - - ( PnH -l.m +l ( |P n + l. m | - | P n , m + l l )

- -p + i, m ( | P n , m + l | ^ " j P n + L m + l D + P n ,m + 1 ( | P n + l . m + l | ^ " | P n + l , m | ) j ( 2 - 6 )

C= - ( p + l , m + l ( | P n + l ,m | - I Pn .m + lj )

+Pn+l,m ( | P n . m + i r " |P n + l , m + i P ) + P r.,m + 1 ( | P n + l , m + l | " | P n + l , m | ) j ( 2 - 7 )

D = | p + l , m + l | ^ 3 ' " ( P n + l , m P n , m + l )+ i P n . m + l l ^ 3 m ( p + i , r P + l , m + l ) + | p + l .m | ^ ^ m ( p + i , m + l P n , m + l ) - ( 2 - 8 )

1 1


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(] (] 7\ A ^

Figure 2.3: Example 2.1.

Furthermore, we have from (2.2) that __Ifn.m Jwhile recalling (2.4) we have

'n,m VJ /

- A

BA

2 -D

(2.9)

(2.10)

E x a m p l e 2 . 1 . Takepn,m = (2n-l-H-t(2m-l-l)) and notice that for any four cocircularpoints p{n, m ), p{n + 1, m ), p{n, m -I-1) and p{n -I-1, m 4-1) we have

(2n 4-1)2-h (2m 4-1)2 (2 n- H i)-I -t(2 m -h 1) (2n-H 1) - i(2 m-h 1) 1(2n-H3)2 4- (2 m 4- l) 2 (2n-H 3)-h i(2m + 1) (2n-H 3) - i(2m-H 1) 1(2n-H 1)2 4-(2m 4-3)2 (2n4-1)-H i(2m + 3) ( 2 n - H ) - z(2m 4-3) 1(2n 4-3)2 4-( 2m 4-3)2 (2n 4-3) 4-i(2 m 4-3) (2n-h 3) - z(2m-h 3) 1

= 0.

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Thus by Prvposition 2.1(4) this foiyns a quadrilateral circle pattern and by (2.5)-(2.8).

.4 = 4B = - 8 ( n - h l ) + 8 (m + l)zC = -8(n 4-1) - 8(m 4- l) iD = 4(4n^ 4- 8n 4- 3 4- 4m2 + 8m 4- 3)

= 8(2n'-' 4- 4n 4- 2m2 -|- 4m 4- 3)and thus by 12.4). (2.9) and (2.10) we have

CIn.m = - - = 2 ( n 4 - l ) 4 - 2 ( m 4 - l ) i.4A = ^ - c

= 2(2n2 + 4u + 2m2 - 4m -h 3) - 4(n + 1)^ - 4(m + 1)= - 2

and

= ^/2.

This example may be generalized to the QCP p{n, m) = (an4-c) 4- {am)i for someQ 6 R \ {0} and c G C Such a quad rilateral circle patt ern would be the discreteanalog of th e ma pping / : C > C with /(z) = Q Z 4- c. It is clear th at the ma pping/ and the QCP p bo th consist of a dilation by the real number Q and a translationby th e complex num ber c. In fact, p is the res triction of / to Z x Z.

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Figure 2.4: Example 2.2 with 9 = 7r/3.

Example 2.2. Let p{n, m) = e'*(n 4- mi) for fixed 9 eR. Then sincen2 -h m2 e'*(n 4- mi) e-'*(n - mi)

(n-H 1)2-I-m2 e*((n 4-1)4-mi) e-*( (n 4-1) - mi) 1n2 4-( m4-l )2 e'(n4 -(m4-l )i) e-**(n - (m4-l)i)

(n 4- l)2 4-(m-Hl)2 e*((n4-l)-K(m4-l)i) e'^Hn + I) - {m + l)i) 1by Proposition 2.1(4), this is a quadrilateral circle pattern, and to locate the centersof the circles, as well as the radii, we apply the same (2.4) and (2.5) through (2.10)to find

= 0

A = \i$B = - ( 2 n 4 - l - ( 2 m 4 - l ) i ),-ieC = - ^ ( 2 n 4 - l 4 - ( 2 m 4 - l ) i )

D = n^ + n-i-m'^-i-m

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r w u / ^v ^

'* >s)sl

Fig ure 2 5 : Ex am ple 2 .3 and zoom of Ex am ple 2 .3 .

an d

an d

hence

oe7 , ^ = ( 2 n + l 4 - ( 2 m + l ) 0

^ n . T n * ~ .-

_ _ i _

T his ex am ple corresp ionds to the ma ppin g / : C C wi th f(z) = ('"z. Thisc + ( m + l ) 2 ( - | - l ) ' + ( m 4 - l ) i ( n 4-1)'^ 4 - ( m + l)z 1

= 0.

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Again we wish to locate the centers and radii of the circles, so we apply the same (2.4)and (2.5) through (2.10) to find

A = 3n2 4- 3n 4- 1^ = ^ [(n + 1) ' -n' + (2m -h 1) ((n -h 1)^ - n') i]^ = ^ [( + 1 ) ' - n - (2m 4-1) ((n -h 1)^ - n') i]D = n^{n + \f ((n -h 1)^ - n^) - 3n m (n m + n + m 4-1)

and

and

hence

ln,m = 2 (( " + 1) ' + (2m 4- l)i)

An.m = ^ (9n '' 4- ISn ' -|- iSn^ + 6n 4- 2)

Pn,m = V ^ - A ,1= - V 9n^ 4- 187i3 -I- lo/i^ + 6n 4- 2.

This example shows one example of a QCP which does not restrict itself to onlyfour intersection points. .As long as only the four specified points of intersection areconsidered we may still utilize the pattern to model some mapping which maps thespecified inters ection po ints app ropria tely. .Notice th at , unlike the othe r pa tter ns,this example corresponds to a quasiconformal map /(z) = {D\e{z))^ 4- i3m(z ).

In general, we wish to utilize quadrilateral circle patterns to discretize a mapping/ : C > C by res trictin g its domain to Z x Z. However, for many possible reasons,this may not be possible. For example, the function may have singularities at somecomplex integer points, or the function may be multivalent.

Exa mp les 2.1 and 2.2 show th at if we are unable to use Z x Z as the dom ain of ourQ C P , we may ins tead consider the image of Z x Z under the m apping / : C > C

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with f{z) = 02 4- 6 for any o G C \ {0} and any 6 G C. By m aking this new setthe domain for our desired QCP, we ma>- be able to avoid the difficulties that thecomplex integer points created. Thus the created QCP would actually correspond tothe composition of the desired function with /(2).

For some other QCP's, it ma>- be necessary to restrict ourselves to subdomains ofZ X Z in order to elim inate th e difficulty associated to the multivalen t functions. Th isis done e.xactly as expected by omitting from the domain all points which lie directlyon the desired branch cut and not considering the mapping of an>- quadrilateralswhich would intereect the branch cut.

2.3 Range Con structionFor most complex functions, there will be no QCP possible. For Mobius functions

everv- set of cocircular points in Z x Z will be mapped to a set of cocircular points inC x , hen ce a Q C P w ill always be possible. Of interest, however are general conformalmapjs and quasiconfo rmal ma ps, for which we would like to find a Q CP . These, howevermay not necessarily map any particular set of cocircular points in the domain to aset of cocircular points in the range. It may be possible, by carefully selecting pointsof C to con stru ct a Q C P based loosely on the desired function. This metho d ofcon struc ting a QC P by selecting its range, point by point, is called range constructionand the general a lgorithm is outlined below.

We will utilize some terminology borrowed from tiling, since the construction of apattern requires the introduction of a beginning circle and requires that all adjacentcircles be completed prior to working on any circles further away (creating a methodver\- similar to tiling the plane with circles). This will allow the entirety of the patternto be constructed, in accordance with the necessary conditions as explained in [2].We will call the circle which we begin with our origin circle, and we shall call eachsuccessive layer around the origin circle a corona.

1. Select two points in the complex plane, call them Zi and 22.

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2. Either :(a) Select any real number p G [1/2 [22 - 2 i| , 00) and co nstru ct one of the two

circles (di stin ct if p # 1/2 I22 - 211) pass ing thr ou gh zi and 22 with radiusp. Select, on the circumference two points 23 and 24 which are pairwisedistinct in the set {21, 22, 23,24}.

O r(b) Select a thir d p oint 23 G C \ {0(23 - 2I) |Q G R } and use this third point in

eq uat ion s (2.5) thro ugh (2.8) along with Zi and 22 to determ ine the circleo,o- Then select k G R \{ 0, 1} and solve the equation 9(24, 21, 22,23) = kfor 24 to determine the last point on the circle. \'arying this k will allow usto alter the shape of the quadrilateral and thus will allow us to constructapproximations to quasiconformal maps.

3. Select any one of 2], 22. 23. 2) and label it poo- Continuing counterclockwisearound the circle label the other points respectively pio, Pi,i and po,i.

It is necessary to create such an origin circle in this process, in order to admit unrestri cte d co ntin ua tion , as in the mono dromy theorem for circle pa tte rns [2, 9, 10].

The construction continues in coronas about the origin circle as follows:4. Either:

(a) Choose six appropriate positive real numbers to serve as the radii of thecircles in the first corona, noting that the four circles adjoining more thanone point of the origin circle must have radii larger than or equal to onehalf the E uclidean distance between the two points of intersection. Thefour co m er circles m ust have radii larger than zero. Affix each of theseap pro pria tely to the poin ts po,o, Po.i, Pi,o, and p i j . Identify th e eightintersection po in ts of these circles which are no t inclu ded in (to,o- Nowchoose, for each comer circle, one point in the arc which lies outside of all

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adjoinin g circles. Now label these four points and the eight intersectionpoints of the circles appropriately, counterclockwise according to Z x Z.

O r

(b) For each edge , for excimple look a t [po,o,Pi,o] of Co,o choose one po int inthe half plane of C divided by {a(po,o - Pi,o) JQ: G M} containing neitherPi,i nor po,i. Utilizing our earlier m eth ods , call this point 23 and select anynumber k G (oc,0) U (0, cx)) and let Z4 be the solu tion to the equation^(-^.Po.o-Pi.OT^a) = k. Beginning with po,o, label the new points po,-i andPi.-i. respectively, moving counterclockwise. Here the num bering systemis clearly mimicking ZxZ. Label these eight poin ts appro priately. Foreach of the following four ordered sots of points:

^l = {Pl,2,Pl,lP2,l}52 = {P-1.1 ,P0,1 ,P0.2}53 = {P-i,o.Po,o.Po.-i}^4 = {Pl , - l .P l .O,P 2,o}

select possibly different kj G (l ,o c) for each ; = 1,2,3,4 (these are arbitrary within this range of R which corresponds to the appropriate arc ofeach circle). Then for each j , if the points in Sj (in appropriate order) arep*, / = 1,2,3 for j = 1.2,3,4, then let Zj be the solution to the equationqizj.p\p^,j^) = kj. (Note that the ordering mentioned is not necessary,bu t in any different ord erin g, the choice of kj must reflect the location ofthe new poin t on the a pp rop ria te arc of the circle.) Finally, label each ofthe four points {ZX,Z2,Z3,ZA} appropriately according to their placements.

Each successive corona is built on the last beginning with any outer section of theprevious corona's quadrilaterals and continuing to choose fc's in appropriate intervals

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/

I t

\ **-* ' " " - -^ . ^ M _ 1 \%

-1 V

4

\ i > - - T Z " > - ^ 4 fbigure 2.6: Example 2.4 beginning circle and four circles.of R until that side has been filled in and continuing onto the next side until the onlyremaining unlabelled points are the four on the corner arcs of each new corona.E x a m p l e 2 .4 . This example is far from a complete example, but it does illustrate theprocess through the first corona. (We shall use method (b) through out the examp lesince it illustrates the point b y point con struction more explicitly.)

Initially, le t po,o = 0 and pi,o = 1 and choose pi.i = 1 4- 2 i. Selecting k = 2 andsolving the equation

9 ( z , 0 , l , l 4 - 2 i ) = 2gives - 3 4 - 4 t2 =which we label po.i- Thus our circle o,o w drawn upon the points {0 ,1,1 4- 2i, ( - 3 4-4 0 / 5 } .

Next, select the point pi._i = -3i and again use k = 2 and solve the equation

9 ( z , 0 , l , - 3 t ) = 2which gives

z = - 9 - 6 z13which we label po.-i- i^ow the circle Co.-i w drawn upon the points {0 ,1 , - 3 i , ( - 9 -6 t ) / 1 3 } .

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/ ( V

T "^ N

- 4 -

/ .-^^y""'^ ^~~"^-. " 5/ \ \

/ \ \

Figure 2.7: One possible extension of Example 2.4.

Moving to the right, select the point p2,i = 2 and set k = 5. Solving the equation7(2,2.1 + 2/. 1) = 5

gives 39 4-3t34

which we label p2,o- H'e then draw the circle Cj.o upon the points {2,1,1 4- 2i, (39 4-3)/34}.

The other sides are done similarly. The circle 1,-1 now must be drawn upon thepoints {-3i. 1, (39 4- 3 0 / 3 4 } , but we require the fourth point of the qu adrilateral, sochoose k = 2 again to solve the equation

9( 2 ,-3 z, 1, (39-^ 3z)/34) = 2to find

which we label p2^-\z = 4 1 l 4 - 5 4 i"313

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2.4 Poss ible Benefits-Although the range construction of quadrilateral circle patterns can be tedious,

it allows the quadrilateral circle pattern to discretize and "graph" the discretizationsof even quasiconformal maps [5, 12]. However, in the quasiconformal QCP the arcsof the circles are not necessarily the images of the arcs of the domain circles, and ingene ral will not b e. .As explained in the range cons tructio n, we may select a ppro pria te%-alues for th e cross ra tio of each qu adr ilate ral in order to co ntrol th e quasiconformalityfactor of the QCP.

If poin ts ar e chosen carefully, it is possible even to con struc t a Q C P of a map /which has singularities. By composing the desired function with a function consistingof some function fo{z) = az -\-c for selected a G C \ {0} and c G C, we may carefullydisplace all domain points away from singularities, and then by restricting our domainalong appr op riat e branch cuts, we may atte m pt to map the new points via / andcreate a QCP.

The most common subclass of the generalized QCP tn be studied are those whichhave certain types of uniformity running throughout. The most important ofthe.se isthe constant angle condition.

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CHAPTER 3CONSTANT ANGLE CONDITION

3.1 DescriptionTh e con stant angle condition requires tha t each intersecting pair of circles intersect

in one of two angles, ay and o// throughout the pattern. Also, it is necessary that theangle an be the intersection angle between all pairs of circles (in,m and +i^m, whilethe angle QV is the intersection angle between all pairs of circles (in,m and Cn^m+i-

.As such , requ iring th e cons tant angle condition reduces some of the para m etersof the choices during range construction.

We now introduce some additional notation which will ease the explanation.For a given quadrilateral circle pattern and, for each {n,m) G Z x Z, the circle

, along with its first corona will be called a patch. The set of circles1 ^ + 1 , m i C n ,m + 1- ^ n - l . m - ' J ^ . r n - l - " f i . m /

in each patch are called a flower with center (tn,m and four petals.It is common to order these, so that the center circle of a flower is numbered 0,

while the p etal s are num bered clockwise 1 through 4.

3.2 Necessary and Sufficient Cond ition sLet us begin by utilizing our previous terminology to describe what the constant

angle condition is and what is necessary to force a QCP to satisfy it. Let1 c[

and ' 1 G^0 2 I>2,

be two circles as described in Ch apte r 2, intersecting in distinct points zy and 22. Thenby 2.5 through 2.10 d has center 71 = -C i and rad ius Pi = ^ ^ ^ 7 = yJ\Cif - Dxand 2 has center n = -C2 and radius p2 = \/-^2 = \]\C2\ - ^2-

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If


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'n,m

and Q = 0, since by the definition of the quadrilateral circle pattern we may not allowintersection angle n as this would leave no outer arc upon which to place the finalthree points on the inner circle. Thus every intersection angle in a quadrilateral circlepattern must come from the interval [0,7r).De f in i t io n 3 .1 . .4 quadrilateral circle pattern satisfies a constant angle conditionfor angles O H and aye (0,7r) if for every (n ,m ) G Z x Z the circles n+i,m and


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This condition produces the extra benefit that any such pattern is made up oftwo packings which are joined at certain common points. The fact that all diagonallyadjac ent circles are required to be tang en t forces each of the two sets of diagona l circles(O ne inclu ding the circle (To.o and the othe r including (Lx,o) to consist ccMiipletely oftangent circles, hence be a packing.L e m m a 3 . 2 . Every QC P w hich rcqums that pairs of diagona l circles be tangentsatKand 3 are tan ge nt, hence the intersection angle between circles (To and 3 must besup plem enta ry to Qv . and thus must be equal to an By assumption (^4 is tangentto both C: and 3. and thus the iiitcscction angle hrtwc.n cin Ics (EQ and Ci must besu pp lem en tar y to Q//. Thu s th e iiitcrsct tion ang lr between ci rd cs (To and (T.i must beequal to ay.

Extending this to the flowers with centers (Ti, 2- 3, and (T, shows that all anglesin one direction must be equal to c.,/. while all other angles must be equal to QV-.

Similarly, this extends to the entire pattern, and therefore p satisii


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Proof Since \Cx - C2I is the distance between circles \ and Cj, if they are taiif^cut,th is d is tance must be equa l to the sum of the rad i i . I f the d is tance be tween the c i rc les1 a nd x is eq ual t o th e sum of their r adii , the n th ere ex ists some p oin t 2 G Ci fl 2such tha t the equa t ion above sa t i sf ies tha t p and the two cente rs of the c i rc les a recol l inear , an d by our equ a t io ns i t is c lear tha t p must fal l in between the two centers.T hi s ha pp en s only when the two c i rcles a re tang ent . CDP r o p o s i t i o n 3 . 4 . .4 QCP unth

C = 1 Bn,rc cSiitisfit\< the con.


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I ( ) ( ) ( ) T '

' 1 \ 1 \ 1 \ /i I ) 1 J 1 r T "

IXJX ''X' '/ \ . y \ y \ / /

H ence,

Figure 3.1: Example 3.1.

^n.m-2 5

ln.m = 3\n+^^ 4 - 2 f 2 m 4 - l j iPn.m.

Thus for the horizontal angle an and the vertical angle ay we haveC O S ( Q W ) = - an d2 5C O S ( Q V ) = 2 ^

which is constant for a// n, m G Z x Z.28


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3.4 Geom etric Radii ConditionAnother subclass of QCP with the constant angle condition is created by imposing

crite ria u pon the radii of the circles of the QC P This creates a pa ttern of circles inwhich rad ii ar e th e geom etric me an of the rad ii of opposing pe tals of the flowercentered on them.

.As a special case of a theorem from [4], we may state the following.C o r o l l a r y 3 . 5 . The quadrilateral circle pattern p satisfies the constant angle condition with ang les an and ay if and only if

arg A Pn.m J \ Pn,m J \ Pn.m J \ Pn.m J = 0

for all (n, m) G Z x Z.This corollary is a consequence of notation only, and is otherwise identical to

Bobenko and Hoffmann's theorem under the special case that one of the angles equalszero.

This condition is identical to the condition

V Pn.m ) \ Pn.m / \ Pn.m / \ Pn.m /-A sufficient co nd itio n for this is sta ted in the following prop osi tion . We first mu stgive one trivial Lemma.L e m m a 3 .6 . For any positive I'cal number 7 ,

-, 4- - > 2 .1Proof Begin with the property of real numbers that for any real number 7

( 7 - l f > 0and expand this and rearrange to find

7^ 4- 1 > 27.29


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Since 7 > 0, dividing through by 7 gives7 + - > 2 .7

P r o p o s i t i o n 3 . 7 . Suppose for all (n, m) G Z x Z and QCP p,_ 2P n + l . m P n - l . m Pn,m

D

and2P n , m + l P n . m - l P n .m '

Then p satisfies the constant angle condition.Proof. Expanding the condition of Corollary 3.5 and using the assumption that

P n + l , m Pn-\.m _ Pn .m+l Pn,m-\ _ .Pn.m Pn.m Pn.m Pn.m

gives

L \ Pn.m Pn.m J \ Pn.m Pn.m J Jand looking at the expression on the right hand side we find that since

e-^-f-e ' = 2iRe(e '")and

| e ^ | < 1,then

e- io ^e'" >-2with equality if and only if Q = TT which, by definition cannot occur in a QCP. Thus,since both an, cky G (0,7r),

29\e{e'") > -2

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and2fHe(c'"* ) > - 2

hence by Lemma 3.6

( e ~ ' 4- e***" -\- ^ " " ' + ' ^ . ^ n , m - l \ I _.^ P n . m + l P n , m - 1 \Pn.m Pn.m J \ Pn.m Pn.m / > - 2 4- 2 = 0,since

P n .m+ l \ _ P n .m - 1Pn,m / Pn.m

Similarly.\ Pn.m Pn,m /

an d therefore the prod uct is also positive. DNote that angles o// and ay are unrelated du ring this entire cons truction . If

we stipulate that an 4- ay = n, then our pattern will satisfy both of these conditions sim ultane ously . It is specifically this type of pa tter n which we present a rangeconstruction for.

1. Select Zx and 22, distin ct com plex num bers, and let d = |2i 22I. Select positivereal numbers ro,r// G [d/2, ->c) with at most one of them equal to d/2.

2 Co nstr uct one of the two possibly dis tiiu t circles pcissing through Zi and 22with radius TQ and label thi s circle the origin circle ir,j,i Con stru ct on e of thetwo possibly dis tinct circles passing through 21 and 2^ with radiu s r//, and labelthis circle ifl. If TQ = r// these circles should be chosen such that tlicy arenot identical. Then the points Zx and 22 should be labelled in an app rop riatemanner as, pi,o and pii, not necessarily respectively.

3. Choose r\ G (0, ocj an d con struc t this circle tang ent (outside) t o (Zx.o at pixwith radius ry . Label this circle Co,\ and label the other intersection point of0,1 and (Io,o as po,i.

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4. By Proposition 3.7, the circles -x.o and (To.-i must have radii rl/rn and /'o/'i ,respcvtively. Th en constru ct circle -X.Q tangent to (To.i at po,i with radius' 'o/'"//, and construct circle (To.-i tangent to (Ti.o it pi,o with radius To/rv.

5. Now select r^ G (0, A. ) and c on struc t the circle tan ge nt to (To.o at pi,i withradius rp - Label this circle \^.6. Lab el th e inte rsec tion points of 1,1 with (Eo.i and 1,0 as p2,i and pi,2, respec

tively.7. To continue this construction, for each {n,Tn) G Z x Z. con struc t the circle ,

tangent to the appropriate circle at the appropriate point with radiusm n - ( m + n ) + I n - m n m - m n mn /n i\Pn.m T^O '^H ' ^V '^D W"^^

and label all points of intersection appropriately.P r o p o s i t i o n 3 . 8 . Thi pn nnusly (/UH 11 construction yu ids a QCP which satisjiis thicon-1ant angle condition.Proof. For each n,m, the circle (t, has radius p., defined in (3.4). Then

/ v 2 2 m n - ( 2 m + 2 n ) + 2 _ 2 n - 2 m n , 2 m - 2 m n _ 2 m n / o c \(Pn.m) = TQ r ry -rp [6.0)and

_ { m ( n - l ) - ( m + n - l ) + I ) + ( m ( r i ^ l ) - ( m + n + l ) + l )P n - l . m P n + l , m '"o| M - l - m ( n - l ) ) + ( n + l - m ( n + l ))' '"//{ m - m ( n - l ) ) + ( m - m ( n - ^ l ) )

' "vm{n- 1 ) + m ( n + l )

2 m n - ( 2 m + 2 n ) + 2 2 n - 2 m n ^ 2 m - 2 m n _ 2m n= To r^ ry rp= {Pn.mf

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and similarly,P n , m - lP n , m + l ~ v P n .m )

Hence, by Proposition 3.7, this construction yields a C^CP satisfying the constantangle cond ition. D

Since the range constructed pattern is uniquely determined by the choicc>s of thereal numbers d = \zx z^]. TQ , r// , ry an d rp . after selecting an orientation for thefirst two circles, and since any pattern constructed as described satisfies the constantangle condition, it should be possible to express the two angles a^ an d ay. whichare constant in such a pattern, in terms of only the numbers d, TQ , r//, ry . an d rp .Furthermore, these five independent numbers along with the beginning orientationfully de ter aii ne th e entire constru ction of the QC P Th us any Q C P constructcHl usingthe same numbers is identical up to some Euclidean transformation of the plane.

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CHAPTER 4EXAMPLES

Certain patterns in complex analysis are typically tried first in order to show thestrength of the circle patterns. .Among these are powers of 2, and log{z) [4, 1].E x a m p l e 4 . 1 . .4 circle pattern for f{z) = z"^!^. In this pattern we will restrictour dom ain away from zero by comp osing this function with the function g{z) =2 -t- 1/2 -t- 1/2. Figure 4-1 has constant angles equal to n/2, while Figure 4-'^ hasalso been composed with a regular quasiconformal mapping similar to the mappingustd in E xample 3.1 in order to give a pattern with constant angles which are notperpendicular.E x a m p l e 4 . 2 . H ere we show two different circle patterns for f{z) = c ' Figure4-3 1- done with constant angles equal to IT/2 whilt Figure 4-4 '' composed with aquasiconformal map m order to make thi constant angles differ from 7r/2.E x a m p l e 4 . 3 . .4 circle pattern for f(z) = log(z). .Again, we take cure to translatethe center of the first circle to the ongm befori beginning.Excmiple 4.4. .4 circle pattern for f {z) = e ' . H ere, since this mapp ing is entire, wedo not need to n -tnct our domain awa y from singularities.

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1 5

1

05

0

- 0 5

/ V

(Ls 1

-v. .7rI /'* \ / ^

^

/ ^^

Figure 4. 1: .A circle pa tte rn for f(z) = z^l^ with constant angles 7r/2.

0 6

0

0 2

0

-0 2

/11

/

1 0>

/ "/ 1

/ 1 ^^^ 1 ;

"* I / '

Figure 4.2: Another circle pattern for /(z) = z2/3 with constant angles.

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Fig ure 4.3 : .A Q C P for f{z) = z^ with constant angles of 7r/2.

6;

- 6

Figure 4.4: Another QCP for /(z) = z2 with constant angles.

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1-

as

0

-0 5

- 1

\0 .4 o .e o i f l^J jBi*^LA*UiAca( 2 i

Figure 4.5: A QCP for f{z) = log(z) with constant 7r/2 angles.

Figure 4.6: A QCP for f{z) = e' with constant 7r/2 angles.

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BIBLIOGRAPHY1] .A.I. Bobenko and Wi.B. Suris, "Hexagonal Circle Patterns and Integrable Syst ems: Patterns with the Multi-Ratio Property and Lax Equations on the Regular Triangular Lattice," preprint.2] .Alan F Be ardon and Ken neth Steph enso n, "Th e Uniformization Th eorem forCircle Packings," Indiana Univ. Ma th. J. 39 (1990), 1383-1425.3] .A.I. Bo benk o and T . Hoffman, "Conformallv Sy mm etric Circle Packings," Experimental Math. 10 (2003), no. 1, 141-150.'4] _. "Hexagonal Circle Patterns and Integrable Systems: Patterns with ConstantAngles," Duke Math. J. 116 (2003, no. 3, 525-566.5] O. Lehto and K.I. Virtanen, Quasiconformal mappings in the plane, second ed.,Springer-\erlag, Berlin, Heidelberg, New \'ork, 1973.6] Burt Rodin and Dennis Sullivan. 'The Convergence of Circle Packings to the

Riemann .Mapping," J. Differential Geometry 26 (1987), 349-360.Oded Schramm, "Packing Two-Dimensional Bodies with Prescribed Combinatorics and .Applications to the Construction of Conformal and QuasiconformalMappings," Ph.D. thesis, Princeton, 1990.Hans Schwerdtfeger, Geometry of Complex .\umbers: Circle Geometry, MobiusTransformations, Non-Euclidean Geometry, Dover, .New 'S'ork, 1979.

[9] Kenneth Stephenson, .Notes for Seminar in .Analysis, Fall 1993 and Spring 1994,University of Tennessee.[10] _. .Notes for Seminar in .Analysis, Fall 1997 and Spring 1998, University of Tennessee.[11] W'ilHam Th urs ton , "The F inite R iemann .Mapping The orem ," 1985, Invitedtalk, an International Symposium at Purdue University on the occasion of theproof of the Bieberbach conjecture, March 1985.[12] G. Brock Williams, ".A Circle Packing Measureable Riemann .Mapping Theorem," preprint.

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Casey Robert Hume- Generalized Quadrilateral Circle Patterns