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GENERALIZED QUADRILATERAL CIRCLE PATTERNS
by
CASEY ROBERT HUME, B.A.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCi:
Approved
Co-Chair])erson of the Committee
Co-Chairpjerson of the Committee
Accepted
Dean of the Graduate School
August, 2003
ACKNOWLEDGEMENTS
I would like to express gratitude for the generosity of the CSEM Scholarship
Program, which was funded through a grant from the NSF and proposed by the
faculty of the Colleges of Engineering and -Arts and Sciences at Texas Tech University.
.Additionally I would like to thank Dr. Kimberly Drews for her support and
encouragement, and luy family, friends and Eisak and Daphne for their concern and
enthusiasm (however difficult it may have been to tell Daphne was enthusiastic).
The graphics in this paper were produced using Maple.
CONTENTS
.ACKNOWLEDGEMENTS ii
LIST OF FIGURES iv
L INTRODUCTION 1
2. GENER,AL1ZED QUADRIL.ATERAL CIRCLE PATTERNS 3
2.1 Notation 3
2.2 Quadrilateral Circle Patterns 3
2.3 Range Construction 17
2.4 Possible Benefits 22
3. CONSTANT ANGLE CONDITION 23
3.1 Description 23
3.2 Necessary and Sufficient Conditions 23
3.3 Requiring a// -f a\ = 7r . 25
3.4 Geometric Radii Condition . . . 29
4. EXA.MPLES - 34
BIBLIOGR,APH^' 38
111
LIST OF FIGURES
2.1 .A graphic of the desired properties of a circle pattern 4
2.2 .A circle pattern of the identity map 7
2.3 Example 2.1 12
2.4 Example 2.2 with 9 = 7r/3. . 14
2.5 Example 2.3 and zoom of Example 2.3 15
2.6 E.xample 2.4 beginning circle and four circles 20
2.7 One possible extension of Example 2.4 21
3.1 Example 3.1 28
4.1 .A circle pattern for f{z) = 2 / with constant angles 7r/2. 35
4.2 Another circle pattern for f{z) = z^^^ with constant angles 35
4.3 A QCP for f{z) = z^ with constant angles of - /2 36
4.4 Another QCP for f{z) = 2 with coii.stant angles 36
4.5 A QCP for f{z) = log{z) with constant rr/2 aiiKl* s 37
4.6 A QCP for f{z) = ( with constant - /2 angles. . 37
IV
CHAPTER 1
INTRODUCTION
.Although much is currently being learned about circle packings and patterns,
their usefulness in terms of computation and approximations is restricted by their
appUcability. In specific, one of the greatest downfalls to this field of study thus
far has been the difficulty of utilizing current techniques in analyzing quasiconformal
maps. 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 of
the basic aspects of circle patterns in order to expand the situations in which these
approximations may be utilized, with specific interest in allowing the inclusion of even
such quasiconformal approximations.
Our method of generalizing circle patterns, herein restricted to generalizing the
quadrilateral circle pattern, is achieved by flK•ll iIlJ entirely un the intersection points
of the pattern and using these to derive aiiv additional information required. By doing
this, we allow a larger category of patterns, including some modelling quasiconformal
mapjs.
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, there
has been a flurry of research activity into the depth of the connection between con-
formal maps and circle packings and patterns. Oded Schram [7] first used circle
patterns with combinatorics of a square lattice to approximate entire functions in
certain restricted cases (in this paper these conditions are discussed as the result of
requiring diagonal circles to be tangent). Bobenko, Hoffman, and Suris [4, 1] loosened
Schramm's condition under the hexagonal case to require only that the pattern had a
set of global constant angles of intersection. In this thesis, we return to quadrilateral
lattices with much looser restrictions than previously considered. In particular we are
able to directly approximate quasiconformal maps for the first time.
In Chapter 2, we give a definition for a new type of circle pattern constructed
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 a
quadrilateral circle pattern.
In Chapter 3, we discuss Bobenko and Hoffman's constant angle condition as it
relates to our quadrilateral circle patterns. We also prove that Schramm's tangency
condition necessitates the constant emgle condition, giving us another way to force
Bobenko and Hoffman's condition for our quadrilateral circle patterns.
Chapter 4 presents several classical examples of mappings which have been pre
viously done by the mentioned authors, using other methods. Here these have been
constructed utilizing a simple Maple program written based on the work presented in
this paper.
CHAPTER 2
GENERALIZED QUADRIL.ATERAL CIRCLE PATTERNS
2.1 Notation
This paper will utiUze the following symbols with the accompanying definitions:
Z = The ring of integers
R = The field of real numbers
<- = The field of complex numbers
^x = The extended complex plane (i.e. C U {co} ), also
called the Riemann Sphere
' = The "positive" root of 2 H- 1 = 0
3 = The conjugate of 2 (i.e. if 2 = x + ly, then 2 = r - iy)
^e(2) = i(2-H2)
M ^ ) = ^ ( z - z ) = - i ( 2 - 2 )
[21. 22] = The line segment (or circular arc according to context)
joining points 2] and 22.
arg(2) = For 2 = re'*, where r > 0 and ^ € R, arg(z) = 6,
(uniquely defined only if we restrict to a specific branch of
logarithm).
Additional notation will be presented as it is discussed.
2.2 Quadrilateral Circle Patterns
Discretizations of complex functions are sometimes useful as a tool in analysis,
especially when the original function is only theoretically possible, and no clear con
struction is known.
By discretizing complex functions carefully, the image of a function may be visu
alized, so as to provide a clue to understanding the nature of the original function.
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 circle
patterns. The most general model of a circle pattern is simply a cluster of overlap)-
ping circles which cover the image of the original function (except perhaps near to the
boimdary) by the union of their interiors. This is very similar to the carnival game
involving 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 patterns
having the following properties, which are illustrated in Figure 2.1:
1. Every circle (in the interior of the pattern) has exactly four specified intersection
points on it.
2. Every specified point of intersection has at least four circles which pass through
it.
3. Every circle (in the interior of the pattern) intersects exactly four adjacent
circles in two of these specified points, and exactly four adjacent circles in only
one of these specified points.
4. If two circles Ci and C2 intersect in points pi and p2, then the arc [pi,p2] in each
circle which lies in the interior of the other circle contains no other specified
intersection points.
Note here that we will not be requiring that these specified points of intersection
are the only points of intersection, but merely that our pattern will have at least
these points of intersection. .All other intersection points are disregarded as we are
not requiring that our patterns keep track of all arcs of circles in the domain.
-A circle with exactly four specified points along it is a special example of a complex
quadrilateral. Recall that a simple closed curve in C is also called a Jordan curve,
and a Jordan curve with four identified distinct points is called a quadrilateral (In
the following discussion it will sometimes be necessary to distinguish between this
definition 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 scattering
of points in C to make up a circle pattern. This will enable us to add conditions to
find more specific types of patterns. In this interest, observe the following proposition.
Proposition 2,1. Given four distinct complex points, z\. 22, 23, and 21 the following
statements are pairwise equivalent:
1. zi, 22. 23. and 24 are either collinear or cocircular (in either case they are
contained in the same circle in C^).
2. g ( 2 i . 2 2 , 2 3 , 2 4 ) € R .
3. 3m(z,2j)(|22|' - I24I') +3m(222T)(|2,|' - I23I')
= 3m(2,Zi)(|23|' - |24p) + 3m(Z2Zj)(|Zi|' - I24I')
-h 3m(232T)(|2,|2 - I22I') + 3m(2,2T)(|22|' - I23H
21I' 21 27 1
I22P 22 2i 1
123^ 23 2j 1
|Z4| 24 27 1
= 0.
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 expres
sion 3m(g(2i, 22, ^3. ~4)) = 0. The reverse is true since all four points are distinct,
hence (21 - 24) (22 - 23) is nonzero, so we may divide by it.
(3) •<=> (4) Is a simple, but lengthy exercise of linear algebra and appears as an
exercise in [8]. D
Definition 2.2. Let p he a mapping
p-.ZxZ —>C
p{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 are distinct) is either:
(a) a polygon, in which 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 . Pn+2,m-l , Pn+2,m, Pn+2,m+l) Pn+2,m+2,
Pn+l,m+2i Pn.m+2i Pn-l ,m+2) Pn-l,m-fl) Pn- l ,m, P n - l , m - l /
IS contained entirely within the unbounded component of C\ Pn,m (in the case
that Pn,m w a '»"e this means that none of the points are actually on Pn,m)-
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 are
mapped by p to some Euclidean quadrilateral (or line segment). .Moreover, any two
adjacent sets of points are mapped to adjacent Euclidean quadrilaterals (or possibly
line segments) which have exactly one side in common (where for the linear case, side
is explained in the definition).
Definition 2.3. A quadrilateral circle pattern (or QCP) is the image of a (local)
discrete immersion
p : Z X Z —>C
p{n, m) = Pn,m
under which the equation
3m(q{pn,m,Pn+l,m,Pn+l,m + UPn,m+l)) = 0
holds true for all (n, m) € Z x Z.
7
Thus, by Proposition 2.1(1) each such set of four adjacent points may be joined
by a real circle or a straight line.
Furthermore, it is clear one may also require versions (3) and (4) of the given
condition, with version (4) giving the following corollary.
Corollary 2.4. .4 (local) discrete immersionp : Z x Z —>C such thatp{n,m) = pn,m
is a quadrUatend cvxle pattern if and only if
\Pn. ml Pn,m Pn.m
2
1
|Pn + l,m| Pn+l,m Pn+l,m 1
|Pn,m+l| Pn,m+l Pn,m+l 1
iPn+l.r/i + ll p„+i ,m+l Pn+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 as
well. D
.As a quick remark, it is important to note that the four conditions of Proposition
2.1 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 a
fact used later.
To see how such a (local) discrete immersion relates to a circle pattern, observe the
following method of representation of circles presented by [8] included here without
proof
Fact: 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
In this representation, if C represents a real circle then that circle has center 7 and
radius p which can be expressed in terms of .4, B. C, and D as follows,
B = -.4T
C = - .4^ = B
D = .4(|7|- '-p^)
furthermore the determinant 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 > 0
A = 0 point circle p = 0
A > 0 imaginary circle p < 0
.4 = 0, (A = - 1 5 p < 0 )
A < 0 straight line
A = 0 0 (D 7i 0) or C (D = 0)
Then, utilizing this notation, we may write each quartet of points based at {n,m) G
Z x Z {i.e pn,m, Pn+i,m, Pn.m4i, Pn+i,m+i) in a matrix of formulas suggested by
Proposition 2.1(3).
' 3 m ( p n + i , m Pn,m+l)
4 -3m(p„+ l ,m Pn+l,m + l )
-^3m(p„+l ,m+l Pn,m+1)
-HPn+l.m+l (|Pn + l,m| " |Pn,m+l| )
+Pn + l.m (|Pn,m+ll " |Pn+l,m-fl| )
+Pn,r7i+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 )
+Pn+l .m (|Pn,r.+ i r - | P n + l , m + l | ' ) + |Pn.m+l | ' j ' m ( p „ + l , ^ p „ + l , ^ + l )
+P„.m+1 ( |Pn+ l .m+l | ' " |Pn+l .mr) j + | p „ + l , m | ' 3 m ( p „ + i , ^ + i Pn.m+l)
which is the result of viewing Proposition 2.1(3) as a quadratic form with variable
terms of p„,„, and | v ^ .
Thus we see that €,,,„, is indeed hermitian, and represents either a line or a real
circle by Proposition 2.1.
It represents a line when
3m(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 thus
Pn+l,m Pn+l,m+l + Pn,m-f 1 Pn+l,m + Pn+l,m+l Pn,m+1 ^ R-
If 3m(p„+i,,„ Pn.m+l) + 3m(p„+i,„ p„+,,^+l) + 3m(p„+i,m+l Pn,m+l) ¥" 0. <2 ,,m
represents a real circle.
We refer to [8] again to see that the matrices (Ti and (^2 represent the same circle
if and only if there exists some A G R\{0} such that (Ti = \€2- Thus we affix a
specific representation of each of these circles or lines to be called the standard form:
1. For 3m(p„+i,^p„+1,^+1) -h 3m(p„,^+ip„+i,,„) -(- 3m(p„+i ,„^ip„,m+i) = 0 either
\C 0
where
iq = i
when
IPn+l.m+ll^^TllPn+l.m Pn,m+l) + |Pn,m+l| 3m(p„ + i,m Pn+l,m+l)
+ |Pn+l,m| 3m(p„+i,m+l Pn,m+l) = 0,
or _ . 0 C
• C 1
10
when
|Pn+l.rn+l| 3m(p„+i_,n p„ „,^i) -|- |p„^^.i | 3m(p„+ i ,„ i p„+i, , jn. i)
+ IP-. + l.mj 3ni (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 use
the standard representation
(l c \C D^
Notice that each of these representations are unique, depending only on the values
of certain ratios of the matrix entries and by Proposition 2.1(4) and the proof of
CoroUaiy- 2.4 each is independent of which three points of each quadrilateral are
utilized in the formula.
Then if we ensure that no four cocircular points p„,,„, Pn+i,m, Pn,m+i. and pn+\.,m+\
of the quadrilateral circle pattern p are collinear. we need use only the unique standard
representation 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 Pn,m+l) + 3m(p„+ i .m Pn + l .m+l) + 3ni (p„+i ,m+l Pn,m+l) (2 .5)
B = - - ( PnH-l.m+l ( |Pn+l.m| - | P n , m + l l )
-»-p„+i,m (|Pn,m+l|^ " j P n + L m + l D + Pn,m+1 (|Pn+l.m+l|^ " |Pn+l,m| ) j (2-6)
C= - ( p „ + l , m + l ( |Pn+l,m| - IPn.m+lj )
+Pn+l,m ( |Pn .m+ir " |Pn + l ,m+iP) + Pr.,m+1 ( |Pn+l ,m+l | " |Pn+l,m| ) j (2-7)
D = | p„+ l ,m+l |^3 '" (Pn+l ,m Pn,m+l)
+ iPn.m+ll^3m(p„+i ,r„ P„+l ,m+l) + |p„+l.m|^ ^m(p„+ i ,m+l Pn,m+l)- (2-8)
11
(] (] 7\ A ^
Figure 2.3: Example 2.1.
Furthermore, we have from (2.2) that
__£ Ifn.m — J
while recalling (2.4) we have
'n,m — V
•J
• /
-A
B A
2 -D
(2.9)
(2.10)
Example 2.1. Takepn,m = (2n-l-H-t(2m-l-l)) and notice that for any four cocircular
points 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 (2n-Hi)-I-t(2m-h 1) (2n-H 1) - i(2m-h 1) 1
(2n-H3)2 4-(2m4-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) (2n-H) - z(2m4-3) 1
(2n 4-3)2 4-(2m 4-3)2 (2n 4-3) 4-i(2m 4-3) (2n-h 3) - z(2m-h 3) 1
= 0.
12
Thus by Prvposition 2.1(4) this foiyns a quadrilateral circle pattern and by (2.5)-
(2.8).
.4 = 4
B = - 8 ( n - h l ) + 8(m + l)z
C = -8(n 4-1) - 8(m 4- l)i
D = 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
C In.m = - - = 2(n4- l )4-2(m4-l ) i
.4
A = ^ - 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 some
Q 6 R \ {0} and c G C Such a quadrilateral circle pattern would be the discrete
analog of the mapping / : C —> C with /(z) = QZ 4- c. It is clear that the mapping
/ and the QCP p both consist of a dilation by the real number Q and a translation
by the complex number c. In fact, p is the restriction of / to Z x Z.
13
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 since
n2 -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) 1
n2 4-(m4-l)2 e'«(n4-(m4-l)i) e-**(n - (m4-l)i)
(n4-l)2 4-(m-Hl)2 e*«((n4-l)-K(m4-l)i) e'^Hn + I) - {m + l)i) 1
by Proposition 2.1(4), this is a quadrilateral circle pattern, and to locate the centers
of 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 )
,-ie C = - ^ ( 2 n 4 - l 4 - ( 2 m 4 - l ) i )
D = n^ + n-i-m'^-i-m
14
rwu/^ v ^ '* >s)sl
Figure 2 5: Example 2.3 and zoom of Example 2.3.
and
and
hence
o«e
7„,^ = —(2n + l 4 - ( 2 m + l ) 0
^ n . T n * ~ .-
_ _i_
This example correspionds to the mapping / : C —» C with f(z) = ('"z. This
c <jrresp.jnrlence is exactIv like the previous example, in that again, p is simply the
restriction of / to Z x Z.
E x a m p l e 2 .3 . Let p{n,m) = n^ 4- mt. Then this forms a QCP by Proposition 2.1,
since
n* -Hm2
( n - t - l j ^ ' - m ^
n-* -I- mi n - rm i
(N-H l)••'4-^^^ (n 4 - 1 ) ' - m i 1
n * ^ 4 - ( m ^ l ) 2 n ' 4 - ( m 4 - l ) i n^ + {711-\-l)i 1
( „ ^ l / > + ( m + l ) 2 ( „ - | - l ) ' + ( m 4 - l ) i (n 4-1)'^ 4-(m + l)z 1
= 0.
15
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^) - 3nm(nm + 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 = - V9n^ 4- 187i3 -I- lo/i^ + 6n 4- 2.
This example shows one example of a QCP which does not restrict itself to only
four intersection points. .As long as only the four specified points of intersection are
considered we may still utilize the pattern to model some mapping which maps the
specified intersection points appropriately. .Notice that, unlike the other patterns,
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 restricting its domain to Z x Z. However, for many possible reasons,
this may not be possible. For example, the function may have singularities at some
complex integer points, or the function may be multivalent.
Examples 2.1 and 2.2 show that if we are unable to use Z x Z as the domain of our
QCP, we may instead consider the image of Z x Z under the mapping / : C —> C
16
with f{z) = 02 4- 6 for any o G C \ {0} and any 6 G C. By making this new set
the domain for our desired QCP, we ma>- be able to avoid the difficulties that the
complex integer points created. Thus the created QCP would actually correspond to
the composition of the desired function with /(2).
For some other QCP's, it ma>- be necessary to restrict ourselves to subdomains of
Z X Z in order to eliminate the difficulty associated to the multivalent functions. This
is done e.xactly as expected by omitting from the domain all points which lie directly
on the desired branch cut and not considering the mapping of an>- quadrilaterals
which would intereect the branch cut.
2.3 Range Construction
For 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 in
Cx , hence a QCP will always be possible. Of interest, however are general conformal
mapjs and quasiconformal maps, for which we would like to find a QCP. These, however
may not necessarily map any particular set of cocircular points in the domain to a
set of cocircular points in the range. It may be possible, by carefully selecting points
of C to construct a QCP based loosely on the desired function. This method of
constructing a QCP by selecting its range, point by point, is called range construction
and the general algorithm is outlined below.
We will utilize some terminology borrowed from tiling, since the construction of a
pattern requires the introduction of a beginning circle and requires that all adjacent
circles be completed prior to working on any circles further away (creating a method
ver\- similar to tiling the plane with circles). This will allow the entirety of the pattern
to 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 each
successive layer around the origin circle a corona.
1. Select two points in the complex plane, call them Zi and 22.
17
2. Either:
(a) Select any real number p G [1/2 [22 - 2i|, 00) and construct one of the two
circles (distinct if p # 1/2 I22 - 211) passing through zi and 22 with radius
p. Select, on the circumference two points 23 and 24 which are pairwise
distinct in the set {21, 22, 23,24}.
Or
(b) Select a third point 23 G C\ {0(23 - 2I) |Q G R } and use this third point in
equations (2.5) through (2.8) along with Zi and 22 to determine the circle
€o,o- Then select k G R\{0,1} and solve the equation 9(24, 21, 22,23) = k
for 24 to determine the last point on the circle. \'arying this k will allow us
to alter the shape of the quadrilateral and thus will allow us to construct
approximations to quasiconformal maps.
3. Select any one of 2], 22. 23. 2) and label it poo- Continuing counterclockwise
around 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 unre
stricted continuation, as in the monodromy theorem for circle patterns [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 the
circles in the first corona, noting that the four circles adjoining more than
one point of the origin circle must have radii larger than or equal to one
half the Euclidean distance between the two points of intersection. The
four comer circles must have radii larger than zero. Affix each of these
appropriately to the points po,o, Po.i, Pi,o, and p i j . Identify the eight
intersection points of these circles which are not included in (to,o- Now
choose, for each comer circle, one point in the arc which lies outside of all
18
adjoining circles. Now label these four points and the eight intersection
points of the circles appropriately, counterclockwise according to Z x Z.
Or
(b) For each edge, for excimple look at [po,o,Pi,o] of Co,o choose one point in
the half plane of C divided by {a(po,o - Pi,o) JQ: G M} containing neither
Pi,i nor po,i. Utilizing our earlier methods, call this point 23 and select any
number k G (—oc,0) U (0, cx)) and let Z4 be the solution to the equation
^(-^.Po.o-Pi.OT^a) = k. Beginning with po,o, label the new points po,-i and
Pi.-i. respectively, moving counterclockwise. Here the numbering system
is clearly mimicking ZxZ. Label these eight points appropriately. For
each 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.Pl.O,P2,o}
select possibly different kj G (l,oc) for each ; = 1,2,3,4 (these are arbi
trary within this range of R which corresponds to the appropriate arc of
each circle). Then for each j , if the points in Sj (in appropriate order) are
p*, / = 1,2,3 for j = 1.2,3,4, then let Zj be the solution to the equation
qizj.p\p^,j^) = kj. (Note that the ordering mentioned is not necessary,
but in any different ordering, the choice of kj must reflect the location of
the new point on the appropriate arc of the circle.) Finally, label each of
the 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 the
previous corona's quadrilaterals and continuing to choose fc's in appropriate intervals
19
/
I t
\ **
-*• ' ""--^.^ M _ 1 \%
-1 V
•4
\ i > - - T Z " > - ^ 4 f b Figure 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 only
remaining unlabelled points are the four on the corner arcs of each new corona.
Example 2.4. This example is far from a complete example, but it does illustrate the
process through the first corona. (We shall use method (b) throughout the example
since it illustrates the point by point construction more explicitly.)
Initially, let po,o = 0 and pi,o = 1 and choose pi.i = 1 4- 2i. Selecting k = 2 and
solving the equation
9(z ,0 , l , l4-2i ) = 2
gives - 3 4-4t
2 =
which we label po.i- Thus our circle €o,o w drawn upon the points {0,1,1 4- 2i, ( -3 4-
40 /5} .
Next, select the point pi._i = -3 i and again use k = 2 and solve the equation
9(z ,0 , l , -3 t ) = 2
which gives
z = - 9 - 6 z
13
which we label po.-i- i^ow the circle Co.-i w drawn upon the points {0,1, -3 i , ( -9 -
6t)/13}.
20
/ ( 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 equation
7(2,2.1 + 2/. 1) = 5
gives 39 4-3t
34
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 the
points {-3i. 1, (39 4- 30/34}, but we require the fourth point of the quadrilateral, so
choose k = 2 again to solve the equation
9(2,-3z, 1, (39-^ 3z)/34) = 2
to find
which we label p2^-\
z = 41l4-54i
"313
21
2.4 Possible Benefits
-Although the range construction of quadrilateral circle patterns can be tedious,
it allows the quadrilateral circle pattern to discretize and "graph" the discretizations
of even quasiconformal maps [5, 12]. However, in the quasiconformal QCP the arcs
of the circles are not necessarily the images of the arcs of the domain circles, and in
general will not be. .As explained in the range construction, we may select appropriate
%-alues for the cross ratio of each quadrilateral in order to control the quasiconformality
factor of the QCP.
If points are chosen carefully, it is possible even to construct a QCP of a map /
which has singularities. By composing the desired function with a function consisting
of some function fo{z) = az -\-c for selected a G C \ {0} and c G C, we may carefully
displace all domain points away from singularities, and then by restricting our domain
along appropriate branch cuts, we may attempt to map the new points via / and
create a QCP.
The most common subclass of the generalized QCP tn be studied are those which
have certain types of uniformity running throughout. The most important ofthe.se is
the constant angle condition.
22
CHAPTER 3
CONSTANT ANGLE CONDITION
3.1 Description
The constant angle condition requires that each intersecting pair of circles intersect
in one of two angles, ay and o// throughout the pattern. Also, it is necessary that the
angle an be the intersection angle between all pairs of circles (in,m and €„+i^m, while
the angle QV is the intersection angle between all pairs of circles (in,m and Cn^m+i-
.As such, requiring the constant angle condition reduces some of the parameters
of 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 circles
1 ^ + 1 , m i Cn,m + 1- ^ n - l . m - 'J^.rn-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 petals are numbered clockwise 1 through 4.
3.2 Necessary and Sufficient Conditions
Let 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. Let
1 c[
and ' 1 G
^02 I>2,
be two circles as described in Chapter 2, intersecting in distinct points zy and 22. Then
by 2.5 through 2.10 d has center 71 = -Ci and radius Pi = ^ ^ ^ 7 = yJ\Cif - Dx
and €2 has center n = -C2 and radius p2 = \/-^2 = \]\C2\ - ^2-
23
If
< = l7i - 72I
= | c , - r 2 |
then, referring to [8] we apply the law of cosines to find that
c ' = p? + /)] - 2pip2 cosu, u.' G (0, n) (3.1)
m which ^ is the supplement of the angle of intersection of these circles.
Let
a = 7T — u;
then
u,' = TT — Q
hence (3.1) may be rewritten as
6^ = p] + p^-2piP2Cos..'
= P\+ p\- 2pxP2Cos(n - <i)
= P? + P2 ^ 2pi/A2(OS(Q).
Thus
<J2 _ „2 „ .2
ri,.ia) = " P^-P' 2plp2 _ _
^ ( C i - c 2 ) ( r : - r : ) ^ A i + A2
2^fK:^2
gives the intersection angle a hftwccn circles Cj and €2.
Note that if we allow Zj =22, then we have
. . (pi -H P2)' - P\-PI rosia) =
2piP2 = 1
24
'n,m
and Q = 0, since by the definition of the quadrilateral circle pattern we may not allow
intersection angle n as this would leave no outer arc upon which to place the final
three points on the inner circle. Thus every intersection angle in a quadrilateral circle
pattern must come from the interval [0,7r).
Definition 3.1. .4 quadrilateral circle pattern satisfies a constant angle condition
for angles OH and aye (0,7r) if for every (n,m) G Z x Z the circles €n+i,m and <in,m
have intersection angle an and the circles €n,m+i and dn.m have intersection angle
ay.
In other words, if
B„
Dn
and we define the horizontal and vertical angle functions
cos(Q//(n,m)) = —• — (3.2) '•Pn,mPn-^\,m
I I \\ r- 'i,m "" C ri,m + l ' •^n,m ' -A,, ,,, + 1 . COS ( Q v ( n , 77))) = : '-^ • : ( 3 . 3 )
- P n , m P n , m + l
then both 3.2 and 3.3 are constant functions.
We adjust the range construction method to include the constant angle function
by simply choosing an and ay (or their cosines) and selecting, at each step, the
radii which wiU allow the appropriate angle. This ver>' general construction is not as
useful immediately, however, as some more specific examples of the constant angle
condition.
3.3 Requiring a// 4- Qv = TT
If, in addition to the constant angle condition, we require that the angles a// and
ay are supplementary angles, we force each pair of diagonal circles to be tangent. So
each circle ^^rn is tangent to all of (t„+i,n+i, €„_i,„+i, €„_i,„_i, and (E„_i,„_i.
25
This condition produces the extra benefit that any such pattern is made up of
two packings which are joined at certain common points. The fact that all diagonally
adjacent circles are required to be tangent forces each of the two sets of diagonal circles
(One including the circle (To.o and the other including (Lx,o) to consist ccMiipletely of
tangent circles, hence be a packing.
Lemma 3.2. Every QCP which rcqums that pairs of diagonal circles be tangent
satK<ji(s the constant angle condition.
Proof Let p be a QCP requiring all pairs of diagonal circles be tangent. Then,
c^xaniine a single flower of the pattern with center Co and petals Ci. iTj. €3, and € |
Let an be the intersection angle of circles €0 and (Ei.
Since Ci and C2 are diagonal, they are tangent. Hence the inttMsection angle ay
between circles (To and €2 must be supplementary to Q//. Furthermore, circles €>
and €3 are tangent, hence the intersection angle between circles (To and €3 must be
supplementary to Qv . and thus must be equal to an By assumption ( 4 is tangent
to both C: and €3. and thus the iiitcscction angle hrtwc.n cin Ics (EQ and Ci must be
supplementary to Q//. Thus the iiitcrsct tion anglr between cirdcs (To and (T.i must be
equal to ay.
Extending this to the flowers with centers (Ti, €2- €3, and (T, shows that all angles
in 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<s the constant
angle condition. D
Lemma 3.3. Two circles
(t, =
are tangent to ejich other if and only if
\Ci - r ' i i = Pi 4-P2-
26
Proof Since \Cx - C2I is the distance between circles €\ and Cj, if they are taiif^cut,
this distance must be equal to the sum of the radii. If the distance between the circles
€1 and €x is equal to the sum of their radii, then there exists some point 2 G Ci fl €2
such that the equation above satisfies that p and the two centers of the circles are
collinear, and by our equations it is clear that p must fall in between the two centers.
This happens only when the two circles are tangent. CD
Propos i t i on 3.4. .4 QCP unth
C = 1 Bn,r
c c
Siitisfit\< the con.<tant angle condition with constant angles Q// and ay. Q// 4- n\-
if and only if both of
\Cn,m ~ C'n + l.m+ll = Pn.m + Pn^l.m + l
\Cn,m - C'„_i,m-n| = Pn.m " Pn-l.m+1
hold for every {n,m) G Z x Z.
Proof This follows immediately from l.immas '5 2 and 3.3.
E x a m p l e 3 .1 . Lit p(n,m) = 3n -h 477it then .since
I 9n^ - 16m2 3u 4- 47nj 3ri - 4771/ 1
'){n ^ 1)^ ^ 16m2 3(n -1- 1) 4- 47777 'Mu ^ 1) - l'7i7 1
971-+ 16(m ^ 1)2 3n-h 4(771 4-1)1 3 7 i - 4 ( m + l ) / 1
9(n - 1 ) ' 4 -16(m + 1)2 3(n + 1) + 4(m + l)i 3(n - 1) - 4(m 4-l) i 1
p is a QCP and thu.s by (2.-5) through (2.10). ur find
A = 12
B = - 1 8 ( 2 7 1 - 1)4-2-1(27/) + 1)7
6' = - 1 8 ( 2 n + l ) - 2 4 ( 2 7 7 7 + 1))
D = 108(n2 + n) + 192(77)' + 77)).
= TT
D
= 0
27
I ( ) ( ) ( ) T'
' 1 \ 1 \ 1 \ /
i I ) 1 J 1 r T "
IXJX ''X' ' / \ . y \ y \ / /
Hence,
Figure 3.1: Example 3.1.
—^n.m -25
ln.m = 3\n+^^ 4 - 2 f 2 m 4 - l j i
Pn.m. —
Thus for the horizontal angle an and the vertical angle ay we have
COS(QW) = — - and 25
COS(QV) = 2^
which is constant for a// n, m G Z x Z.
28
3.4 Geometric Radii Condition
Another subclass of QCP with the constant angle condition is created by imposing
criteria upon the radii of the circles of the QCP This creates a pattern of circles in
which radii are the geometric mean of the radii of opposing petals of the flower
centered on them.
.As a special case of a theorem from [4], we may state the following.
Corollary 3.5. The quadrilateral circle pattern p satisfies the constant angle condi
tion with angles 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 equals
zero.
This condition is identical to the condition
V Pn.m ) \ Pn.m / \ Pn.m / \ Pn.m /
-A sufficient condition for this is stated in the following proposition. We first must
give one trivial Lemma.
Lemma 3.6. For any positive I'cal number 7,
-, 4- - > 2. 1
Proof Begin with the property of real numbers that for any real number 7
( 7 - l f > 0
and expand this and rearrange to find
7^ 4- 1 > 27.
29
Since 7 > 0, dividing through by 7 gives
7 + - > 2 . 7
Proposition 3.7. Suppose for all (n, m) G Z x Z and QCP p,
_ 2 Pn+l.mPn-l.m — Pn,m
D
and
2 Pn,m+lPn.m-l — Pn.m'
Then p satisfies the constant angle condition.
Proof. Expanding the condition of Corollary 3.5 and using the assumption that
Pn+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 J
and 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'" >-2
with 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
30
and
2fHe(c'"* ) > - 2
hence by Lemma 3.6
(e ~ ' ° « 4- e***" -\- ^ " " ' + ' ^. ^ n , m - l \ I _.^ Pn.m+l Pn,m-1 \
Pn.m Pn.m J \ Pn.m Pn.m /
> - 2 4- 2 = 0,
since Pn.m+l \ _ Pn.m-1
Pn,m / Pn.m
Similarly.
\ Pn.m Pn,m /
and therefore the product is also positive. D
Note that angles o// and ay are unrelated during this entire construction. If
we stipulate that an 4- ay = n, then our pattern will satisfy both of these condi
tions simultaneously. It is specifically this type of pattern which we present a range
construction for.
1. Select Zx and 22, distinct complex numbers, and let d = |2i — 22I. Select positive
real numbers ro,r// G [d/2, ->c) with at most one of them equal to d/2.
2 Construct one of the two possibly distiiut circles pcissing through Zi and 22
with radius TQ and label this circle the origin circle ir,j,i Construct one of the
two possibly distinct circles passing through 21 and 2 with radius r//, and label
this circle €ifl. If TQ = r// these circles should be chosen such that tlicy are
not identical. Then the points Zx and 22 should be labelled in an appropriate
manner as, pi,o and p i i , not necessarily respectively.
3. Choose r\ G (0, ocj and construct this circle tangent (outside) to (Zx.o at pix
with radius ry. Label this circle Co,\ and label the other intersection point of
€0,1 and (Io,o as po,i.
31
4. By Proposition 3.7, the circles €-x.o and (To.-i must have radii rl/rn and /'o/'i ,
respcvtively. Then construct circle €-X.Q tangent to (To.i at po,i with radius
''o/'"//, and construct circle (To.-i tangent to (Ti.o i»t pi,o with radius To/rv.
5. Now select r^ G (0, A. ) and construct the circle tangent to (To.o at pi,i with
radius rp- Label this circle €\^.
6. Label the intersection 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. construct the circle €„ „,
tangent to the appropriate circle at the appropriate point with radius
m 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.
Proposition 3.8. Thi pn nnusly (/UH 11 construction yu ids a QCP which satisjiis thi
con-1ant angle condition.
Proof. For each n,m, the circle (t„„, has radius p„.,„ defined in (3.4). Then
/ v2 2mn- (2m+2n)+2 _2n-2mn , 2 m - 2 m n _2mn / 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 ) ) • '"v
m{n- 1 ) + m ( n + l )
2mn- (2m+2n)+2 2n-2mn ^2m-2mn _2mn
= To • r^ • ry • rp = {Pn.mf
32
and similarly,
Pn,m-lPn,m+l ~ vPn.m) •
Hence, by Proposition 3.7, this construction yields a C^CP satisfying the constant
angle condition. D
Since the range constructed pattern is uniquely determined by the choicc>s of the
real numbers d = \zx — z^]. TQ, r//, ry and rp. after selecting an orientation for the
first two circles, and since any pattern constructed as described satisfies the constant
angle condition, it should be possible to express the two angles a^ and ay. which
are constant in such a pattern, in terms of only the numbers d, TQ, r//, ry. and rp.
Furthermore, these five independent numbers along with the beginning orientation
fully deteraiine the entire construction of the QCP Thus any QCP constructcHl using
the same numbers is identical up to some Euclidean transformation of the plane.
33
CHAPTER 4
EXAMPLES
Certain patterns in complex analysis are typically tried first in order to show the
strength of the circle patterns. .Among these are powers of 2, and log{z) [4, 1].
Example 4.1. .4 circle pattern for f{z) = z"^!^. In this pattern we will restrict
our domain away from zero by composing 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-'^ has
also been composed with a regular quasiconformal mapping similar to the mapping
ustd in Example 3.1 in order to give a pattern with constant angles which are not
perpendicular.
Example 4.2. Here we show two different circle patterns for f{z) = c' Figure
4-3 1- done with constant angles equal to IT/2 whilt Figure 4-4 '•'• composed with a
quasiconformal map m order to make thi constant angles differ from 7r/2.
Example 4.3. .4 circle pattern for f(z) = log(z). .Again, we take cure to translate
the center of the first circle to the ongm befori beginning.
Excmiple 4.4. .4 circle pattern for f {z) = e'. Here, since this mapping is entire, we
do not need to n -tnct our domain away from singularities.
34
1 5
1
05
0
- 0 5
/ V
(Ls 1
-v..
7r I /'•* \ / ^
^
/ ^
Figure 4.1: .A circle pattern for f(z) = z^l^ with constant angles 7r/2.
06
0«
02
0
-02
/ 1 1
/
1 0>
/ "
/ 1
/ 1 ^^
^ 1 ;
"•* I / '
Figure 4.2: Another circle pattern for /(z) = z2/3 with constant angles.
35
Figure 4.3: .A QCP for f{z) = z^ with constant angles of 7r/2.
6;
-6
Figure 4.4: Another QCP for /(z) = z2 with constant angles.
36
1-
as
0
-05
-1
\0.4 o .eo i fl^JjBi*^ LA*UiAca( 2i
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.
37
BIBLIOGRAPHY
1] .A.I. Bobenko and Wi.B. Suris, "Hexagonal Circle Patterns and Integrable Systems: Patterns with the Multi-Ratio Property and Lax Equations on the Regular Triangular Lattice," preprint.
2] .Alan F Beardon and Kenneth Stephenson, "The Uniformization Theorem for Circle Packings," Indiana Univ. Math. J. 39 (1990), 1383-1425.
3] .A.I. Bobenko and T. Hoffman, "Conformallv Symmetric Circle Packings," Experimental Math. 10 (2003), no. 1, 141-150.'
4] _. "Hexagonal Circle Patterns and Integrable Systems: Patterns with Constant Angles," 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 Quasiconformal Mappings," Ph.D. thesis, Princeton, 1990.
Hans Schwerdtfeger, Geometry of Complex .\umbers: Circle Geometry, Mobius Transformations, 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 Thurston, "The Finite Riemann .Mapping Theorem," 1985, Invited talk, an International Symposium at Purdue University on the occasion of the proof of the Bieberbach conjecture, March 1985.
[12] G. Brock Williams, ".A Circle Packing Measureable Riemann .Mapping Theorem," preprint.
38
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