Planar Conformal Mappings of Piecewise Flat Surfaces

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1 Plnr Conforml Mppings of Pieewise Flt Surfes Philip L. Bowers n Moni K. Hurl Deprtment of Mthemtis, The Flori Stte University, Tllhssee, FL 32306, USA. owers@mth.fsu.eu mhurl@mth.fsu.eu Introution y There is rih literture in the theory of irle pkings on geometri surfes tht from the eginning hs expose intimte onnetions to the pproximtion of onforml mppings. Inee, one of the first pulitions in the sujet, Roin n Sullivn s 1987 pper [10], provies proof of the onvergene of irle pking sheme propose y Bill Thurston for pproximting the Riemnn mpping of n ritrry proper simply-onnete omin in C to the unit isk. Bowers n Stephenson s work in [4], whih explins how to pply the Thurston sheme on nonplnr surfes, my e viewe s fr rehing generliztion of his sheme to the setting of ritrry equilterl surfes. Further, in [4] Bowers n Stephenson propose metho for uniformizing more generl pieewise flt surfes tht neessittes truly new ingreient, nmely, tht of inversive istne pkings. This inversive istne sheme ws introue in very preliminry wy in [4] with some omments on the iffiulty involve in proving tht it proues onvergene to onforml mp. Even with these iffiulties, the sheme hs een enoe in Stephenson s pking softwre CirlePk n, though ll the theoretil ingreients for proving onvergene re not in ple, it seems to work well in prtie. This pper my e viewe s ommentry on n expnsion of the isussion of [4]. Our purposes re threefol. First, we refully esrie the inversive istne sheme, whih is given only ursory explntion in [4]; seon, we give reful nlysis of the theoretil iffiulties tht require resolution efore onforml onvergene n e prove; thir, we give gllery of exmples illustrting the power of the sheme. We shoul note here tht there re speil ses (e.g., tngeny or overlpping pkings) where the onvergene is verifie, n our isussion will give proof of onvergene in those ses. Eh oriente pieewise flt surfe hs nturl onforml struture efine on its interior y omplex tls with onforml hrts of two types. First, eh interior ege gives rise to n ege hrt tht isometrilly mps the y This work is supporte y NSF grnt DMS , NIH grnt MH n FSU grnt FYAP-2002.

2 4 Philip L. Bowers n Moni K. Hurl interior of the two Eulien tringles meeting long tht ege to the plne, preserving orienttion. Overlp mps etween the intersetions of two suh hrts re Eulien isometries n therefore onforml. Seon, eh interior vertex gives rise to vertex hrt tht uses power mp efine on smll open neighorhoo of the vertex to resle n ngle sum Θ ifferent from 2π to one equl to 2π. The vertex hrts re hosen to hve pirwise isjoint omins n the lol form of the hrt mp is the power mp z z 2π/Θ. The overlp mpping etween ny ege hrt n vertex hrt is onforml s the vertex, where the erivtive is zero, is not in the overlp. In this wy ny orientle pieewise flt surfe eomes Riemnn surfe. Notie tht though there re in generl one points t the verties in pieewise flt surfe, these re singulrities of the pieewise Eulien metri only n not singulrities of the onforml struture. Inee, the totl ngle sum t eh vertex given y the onforml struture is 2π n the Eulien ngle α etween two rs emnting from vertex is mesure s 2π α Θ in the onforml struture. The onforml struture thus mesures the mrket shre of the ngle α with respet to the totl Eulien ngle Θ. Though the inversive istne sheme for onforml mpping my e presente in the full generlity of ritrry pieewise flt surfes, of ritrry genus with n ritrry numer of ounry omponents, we hve hosen to restrit our ttention to the simply onnete se so s to illuminte the essentil fetures of the lgorithm n so tht we my isuss the etils of the proof of onvergene without the e iffiulty of hving to work with mouli spes. In ft, we will onsier pieewise flt qurilterls n sk for metho to onformlly mp them to retngles. γ γ γ δ δ δ δ () () () Fig. 1. Three pieewise flt onforml qurilterls. Perhps n exmple will help illustrte the prolem the lgorithm resses. Consier the simple tringultion K of topologil qurilterl with eight fes with ommon entrl vertex n four istinguishe oun-

3 Plnr Conforml Mppings of Pieewise Flt Surfes 5 ry verties,,, n s in Fig. 1. There re mny wys to efine metri on K mking eh fe flt Eulien tringle. Three exmples re inite in Fig. 1 where eh sie is given unit length, exept for the sies lele with γ n δ, whih re given sie lengths γ = n δ = These exmples re isusse in greter etil in Setion 5, ut for now relize tht these lels enoe pieewise flt metri on K y ientifying the fes with Eulien tringles of sie lengths given y the ege lels. This in turn proues three ifferent onforml strutures on K tht eh relizes K s onforml qurilterl. By stnr theorems on onforml mpping, ny onforml qurilterl mps onformlly to Eulien retngle unique up to sling. Approximtions to this onforml mpping in eh se re inite in Fig. 2, where we see pproximtions to the imge tringultions uner the onforml mpping to retngle. Note tht the first n thir retngles of Fig. 2 re oth squres, ut the onformlly orret shpes of the fes in the two exmples re ifferent, n the onforml moulus of the seon is pproximtely µ = () () () Fig. 2. Uniformiztions of qurilterls from Fig. 1. Complete proofs for onvergene of the sheme in the more speilize setting of equilterl surfes, where eh ege hs unit length n eh fe is ientifie with unit equilterl tringle, re foun in [4]. The originl motivtion for eveloping the sheme of [4] ws to onstrut funmentl omins for the equilterl surfes tht rise in Grotheniek s theory of essins enfnts n to pproximte their ssoite Belyĭ mps. Sine then, Hurl et l [7] hve opte this metho to onstrut flt mppings of surfes in R 3 n hve pplie the metho to otin flt mppings of the humn rin, whih is of urrent interest in the neurosiene ommunity. The esire to otin etter onforml integrity in these rin mppings hs inspire us to investigte further this preliminry suggestion in [4] tht moifition of their sheme using inversive istne pkings oul e use to uil on-

4 6 Philip L. Bowers n Moni K. Hurl forml mppings in this pieewise flt s oppose to pieewise equilterl setting. Stephenson s softwre CirlePk ws use for the rin mppings of [7] s well s for lulting n renering our exmples. We note here tht we o not present the irle pking lgorithm use in CirlePk to lulte the pking for given inversive istne t s this hs een isusse mply in [5]. The ingreients of this onforml mpping sheme re inversive istnes of irles in the Riemnn sphere, irle ptterns in the Riemnn sphere, n hexgonl refinement. The first three setions of the pper re entere roun these three respetive themes. We fin tht mny mthemtiins, even those who speilize in omplex nlysis n onforml geometry, re not fmilir with the inversive istne etween pirs of irles in the Riemnn sphere. In Setion 1, we present n inversive istne primer n prove some results out the onforml plement of irles in the Riemnn sphere. In Setion 2, we review the sis of pieewise flt strutures on surfes n introue irle ptterns with inversive istnes enoe long eges. These ptterns, generliztions of irle pkings where eges enoe tngenies, hve een stuie in the se where neighoring irles overlp with some ngle etween 0 n π/2. Bowers n Stephenson [4] introue the notion of irle ptterns where neighoring irles my not overlp, ut where they o stisfy á priori inversive istne requirements. We emphsize gin tht the theoretil unerpinnings of this topi re not entirely in ple n re mtter of urrent reserh y Bowers, Stephenson, Hurl, n others, ut the goo news is tht the lgorithm seems to work well in prtie. This itertive lgorithm for prouing sequene of ptterns tht re hope to pproximte more n more losely the esire onforml mpping is presente in Setion 3, where hexgonl refinements re introue. The iffiulties in the proof of onvergene to the esire onforml mpping re isusse in Setion 4. We etil three min theoretil prolems tht must e resse for omplete resolution of the question of onvergene to onforml mp, n we prove onvergene uner the ssumption tht these prolems hve een resolve. Setion 5 presents gllery of exmples tht illustrte the lgorithm y pproximting onforml mppings to retngles of onforml qurilterls tht rise from pieewise flt metris on topologil isks, s in the exmples of this introution. This llows us to pproximte the onforml mouli of qurilterls tht rise from pieewise flt metris, n to view the onformlly orret shpes of the fes of the tringultion fter mpping to the plne. We shll point out how well the lgorithm works in prtie, prouing imge tringultions with extly the expete properties. Finlly, Setion 6 isusses prtil implementtion issues in pplitions, n omputtionl n theoretil issues surrouning these.

5 Plnr Conforml Mppings of Pieewise Flt Surfes 7 1 An Inversive Distne Primer The inversive istne etween two oriente irles in the Riemnn sphere Ĉ is onforml invrint of the lotion of the irles in the sphere n their reltive orienttions; see [1]. Inee, given oriente irle pirs C 1,C 2 n C 1,C 2 of Ĉ, thereexistsmöius trnsformtion T of the Riemnn sphere with T (C i )=C i for i =1, 2, respeting their reltive orienttions, if n only if the inversive istne etween C 1 n C 2 equls tht etween C 1 n C 2.AnorienteirleC is the ounry of unique open isk C, lle the interior of C, tht lies to the left of C s C is trverse in the iretion of its orienttion. The preise efinition of inversive istne my e stte elegntly with the i of ross rtios n irle interiors. Definition 1. Let C 1 n C 2 e oriente irles in the Riemnn sphere Ĉ ouning the respetive isks C 1 n C 2,nletD e ny oriente irle mutully orthogonl to C 1 n C 2. Denote the points of intersetion of D with C 1 s z 1,z 2 orere so tht the oriente sur of D from z 1 to z 2 lies in the isk C 1. Similrly enote the orere points of intersetion of D with C 2 s w 1,w 2.Theinversive istne etween C 1 n C 2, enote s InvDist(C 1,C 2 ), is efine in terms of the ross rtio y [z 1,z 2 ; w 1,w 2 ]= (z 1 w 1 )(z 2 w 2 ) (z 1 z 2 )(w 1 w 2 ) InvDist(C 1,C 2 )=2[z 1,z 2 ; w 1,w 2 ] 1. Rell tht ross rtios of orere 4-tuples of points in Ĉ re invrint uner Möius trnsformtions. This implies tht whih irle orthogonl to oth C 1 n C 2 is use in the efinition is irrelevnt s Möius trnsformtion tht setwise fixes C 1 n C 2 n e use to move ny one orthogonl irle to nother. Also, whih one of the two orienttions on the orthogonl irle D is use is irrelevnt s the ross rtio stisfies [z 1,z 2 ; w 1,w 2 ]=[z 2,z 1 ; w 2,w 1 ]. This eqution lso shows tht the inversive istne is preserve when the orienttion of oth irles is reverse so tht it is only the reltive orienttion of the two irles tht is importnt for the efinition. When C 1 n C 2 overlp, the oriente ngle of overlp my e efine unmiguously s the ngle etween the tngents to the irles t point of overlp forme y one tngent pointing long the orienttion of its prent irle n the other pointing ginst the orienttion of its prent irle. We istinguish six ifferent wys tht two irles my overlp n esrie the inversive istne in eh se.

6 8 Philip L. Bowers n Moni K. Hurl 1.1 Six Cses The inversive istne is lwys rel numer sine the ross rtio of four points tht lie on irle is lwys rel. The wy to isset the inversive istne is through the uxiliry funtion T (z) =2[z 1,z 2 ; z,w 2 ] 1=2 (z 1 z)(z 2 w 2 ) (z 1 z 2 )(z w 2 ) 1, whih is Möius trnsformtion tht tkes the triple z 1,z 2,w 2 to the triple 1, 1,. The funtion T tkes the orthogonl irle D to the rel line, the irle C 1 to the unit irle entere t the origin, n the irle C 2 to the vertil line orthogonl to the rel xis t the point T (w 1 ); see Fig. 3. Notie tht InvDist(C 1,C 2 )=T(w 1 ) my tke on ny rel vlue n we istinguish the six ses oring to the two reltive orienttions for eh of the three possiilities for intersetion of C 1 with C 2. Fig. 3 provies snpshot of ll the possiilities lele oring to whether the orienttions re ligne or opposite, n whether the intersetion onsists of none, one, or two points. Figs. 3() n 3() illustrte the possiilities for isjoint irles. If the orienttions re opposite, the inversive istne is in the rnge from to 1 exlusive, n if ligne, in the rnge from +1 to + exlusive. Figs. 3() n 3() illustrte those for tngent irles where the inversive istne is ±1 epening on reltive orienttion. Figs. 3(e) n 3(f) illustrte those for interseting irles where the inversive istne is etween 1 n0for intersetion ngles etween π n π/2 n etween 0 n +1 for ngles etween π/2 n 0. Referring to the ngle lels in Fig. 3, we my re off the inversive istnes s InvDist(C 1,C 2 )=seα for isjoint irles, where α is the inite ngle, n InvDist(C 1,C 2 )=osα for interseting irles, where α is the oriente ngle of intersetion of C 1 with C 2. Notie for interseting irles, sine the overlp ngle α my e etermine without regr to the normlizing trnsformtion T, the inversive istne hs n immeite, esily unerstoo mening. One n look t two overlpping irle pirs n estimte whether they re Möius equivlent, tsk of gret iffiulty for isjoint irle pirs. For those with finely evelope intuition for hyperoli spe n the Poinré extensionsofmöius trnsformtions, there is more geometri unerstning of inversive istne ville.

7 Plnr Conforml Mppings of Pieewise Flt Surfes 9 D w 1 z 1 C 1 z 2 w 2 C 2 T T(w ) T(C ) 2 α T(C ) 1 T(D) () Disjoint irles, opposite orienttions. w 2 D w 1 z T(w ) z 1 C 2 C 1 T α T(C ) 1 T(C ) 1 T(D) 2 () Disjoint irles, ligne orienttions. D w 1 z 1 z = 2 T T(w 1 ) α -1 1 T(D) w 2 C 2 C 1 T(C ) T(C 1) 2 () Tngent irles, opposite orienttions. Fig. 3. Three of six wys tht two irles overlp. Here, T (z 1)= 1,T(z 2)=1.

8 10 Philip L. Bowers n Moni K. Hurl D z 1 w 2 z 2 = w 1 T α T(w 1) -1 1 C 1 C 2 T(C ) 1 T(C ) 2 T(D) () Tngent irles, ligne orienttions. D z 1 w 2 z C 1 2 w 1 C 2 T -1 T(C ) 2 α T(w 1 ) 1 T(C 1) T(D) (e) Interseting irles, opposite orienttions. D z 1 w 1 z 2 C 1 w 2 C 2 T α -1 T(w ) 1 T(C ) 1 1 T(C 2) T(D) (f) Interseting irles, ligne orienttions. Fig. 3. Remining three wys tht two irles overlp. Here, T (z 1)= 1,T(z 2)=1.

9 Plnr Conforml Mppings of Pieewise Flt Surfes An Alternte Desription in Terms of Hyperoli Geometry Notie tht if the orienttion of only one memer of irle pir is reverse, the inversive istne merely hnges sign. This follows from the immeite reltion [z 1,z 2 ; w 2,w 1 ]=1 [z 1,z 2 ; w 1,w 2 ]. We therefore efine Definition 2. The solute inversive istne etween ny pir of unoriente irles is the solute vlue of the inversive istne etween the two irles when given either reltive orienttion. We use the sme nottion, InvDist(C 1,C 2 ), for the solute inversive istne etween unoriente irles C 1 n C 2. It is ler then tht there is Möius trnsformtion tking n unoriente irle pir C 1,C 2 to nother unoriente pir C 1,C 2 if n only if their solute inversive istnes gree. When C 1 n C 2 overlp with ute ngle α the solute inversive istne is os α n when they re tngent it tkes the vlue 1. In this susetion our im is to expose geometri unerstning of the solute inversive istne etween two isjoint irles in terms of hyperoli geometry. This is gret intuitive i for unerstning inversive istnes etween isjoint irles. Towr this en ssume C 1 n C 2 re isjoint n y pproprite hoies of orienttion mp vi T so tht T (C 1 ) is the unit irle n T (C 2 ) is the vertil line through the point = InvDist(C 1,C 2 ) > 1. Consier the extene omplex plne s the sphere t infinity for the hyperoli 3-spe relize s the upper hlf-spe moel with metri s = x /x 3 on H 3 = {x =(x 1,x 2,x 3 ): x 3 > 0}. ThePoinré extension of T, enote T,isnisometryofH 3.TheirleC 1 ouns hyperoli plne P 1 in H 3 tht is relize s the upper hemisphere of the sphere in R 3 with the sme Eulien enter n rius s C 1, n similrly C 2 ouns the hyperoli plne P 2. We lulte the hyperoli istne δ etween the plnes P 1 n P 2. First, sine T is n isometry, we work with T (P 1 ), whih is the upper hemisphere of the unit sphere in R 3,nwith T (P 2 ), whih is the vertil hlf plne {x H 3 : x 1 = }. There is unique geoesi segment Σ in H 3 meeting oth T (P 1 )n T (P 2 ) orthogonlly t the respetive points A n B. This geoesi segment lies on the irle in the vertil x 1 x 3 -plne tht is mutully orthogonl to T (P 1 ), T (P 2 ), n to the x 1 -xis; see Fig. 4. Elementry geometry shows this irle to e entere t the point (, 0, 0) n of Eulien rius 2 1, n the points A n B to e given y A = (os se 1, 0, sin se 1 ) n B =(, 0, 2 1). A lultion of the hyperoli length of Σ y integrting the line element s = x /x 3 long ΣfromA to B gives the vlue of δ s δ =ln =osh 1. This proves tht the solute inversive istne etween C 1 n C 2 is preisely the hyperoli osine of the hyperoli istne etween the plnes P 1

10 12 Philip L. Bowers n Moni K. Hurl n P 2 oune y C 1 n C 2. Experiene with this unerstning of inversive istne for isjoint irles ouple with the ft tht Poinréextensions of Möius trnsformtions re isometries of H 3 hs prove invlule in our reserh, prtiulrly for gining intuition in working with isjoint irle ptterns. x 3 Σ B A -1 1 Fig. 4. The hyperoli length of Σ is osh 1. x A Eulien Formul The simplest formul for the solute inversive istne etween two irles in the omplex plne is the one tht the lgorithm for onforml flttening uses. Though simple, it is not t ll trnsprent tht it shoul yiel Möius invrint for the plement of two irles in the plne. We leve it s n exerise to verify tht if C i is the irle in the omplex plne C entere t i of rius R i,fori =1, 2, then the solute inversive istne is given y R1 2 InvDist(C 1,C 2 )= + R R 1 R 2. (1) 2 Pieewise Flt Surfes n Cirle Pkings A omintoril qurilterl is n strt oriente simpliil 2-omplex K tht tringultes lose topologil isk with four istinguishe ounry verties {,,, } orere respeting the ounry orienttion. The sets of verties, eges, n fes of K re enote respetively s V, E, nf. A pieewise flt struture for K is etermine y n ege length funtion : E (0, ) tht stisfies the tringle inequlity onition, nmely, tht for every three eges e 1,e 2,e 3 tht oun fe of K, the inequlity e 1 e 2 + e 3

11 Plnr Conforml Mppings of Pieewise Flt Surfes 13 hols. An ege length funtion for K etermines pieewise Eulien metri y ssigning the length e to eh ege e of K n ientifying eh fe v 1,v 2,v 3 with flt Eulien tringle of ege lengths e i,wherethe ege e i = v j,v k n {i, j, k} = {1, 2, 3}. The resulting pieewise Eulien metri spe is enote s K n, s expline in the introution, rries the struture of Riemnn surfe. Eh interior vertex of K is one point singulrity for the pieewise Eulien metri ut is not singulrity of the onforml struture. Our im is to esrie sheme for pproximting the onforml mpping of K to retngle tht mps the four istinguishe ounry verties to the four orners of the retngle. Of ourse we o not hve nite for the trget retngle sine we o not know the moulus of the onforml qurilterl K ; however, the lgorithm ielly will proue sequene of trget retngles tht onverges to the orret one s well s urviliner tringultion of the trget retngle with the omintoris of K tht shows the orret onforml shpes of the fes. A plentiful supply of pieewise flt surfes is ville from tringulr gris in R 3 where we re off sie lengths of eges of tul Eulien tringles. This, though, gives ut limite supply of the pieewise flt surfes ville, s mny suh surfes mit no isometri emeing in R 3,n mny mit no emeing in ny Eulien spe tht isometrilly emes eh ege s stright Eulien line segment. The itertive lgorithm for onformlly mpping K to retngle uses s see ertin inversive istne t lulte from the ege length funtion. This gives rise to pieewise flt surfe with inversive istne informtion enoe long eges y funtion Φ: E [0, ). We strt this y not ssuming n á priori pieewise flt struture from whih the ege funtion Φ rises. Definition 3. Let Φ: E [0, ) e funtion on the ege set of the omplex K. A irle pking for (K, Φ) is olletion C = {C v : v V} of irles in the plne C, eh oriente ounterlokwise, suh tht the inversive istne of neighoring irles is given y Φ, i.e., InvDist(C u,c v )= Φ( u, v ) for eh ege u, v in E. Perhps more esriptive term woul e irle pttern, s oppose to pking, whenever the irles re isjoint. Nonetheless, we shll use the term pking to esrie olletion of irles, isjoint or not, tht hs omintoril pttern enoe in omplex K governing the plement of the irles in the plne. Tngeny pkings, whih use only the omintoril informtion enoe in K n not ny vrying inversive istne t, re use in [4] to uniformize pieewise equilterl surfes. When the surfe is pieewise flt where fes re generlly not equilterl, more thn the omintoris of K must e use to uil pproximte onforml mps. We

12 14 Philip L. Bowers n Moni K. Hurl now esrie how the metri informtion of K my e use to emellish the omintoris of K with inversive istne t, whih turns out to e suffiient for generting nites for pproximte onforml mps. Let e n ege length funtion for K n let R: V (0, ) e positive funtion on the verties tht, for eh ege u, v, stisfies the onition R(u) 2 + R(v) 2 u, v 2. (2) This inequlity gurntees tht if the ege e = u, v is rwn in the plne s segmentoflength e, nirlesc u n C v oth oriente ounterlokwise of respetive rii R(u) nr(v) re entere t the verties of e, then the oriente overlp, if the irles interset nontrivilly, is t most π/2, n the interiors, if the irles re isjoint, re lso isjoint. The resulting rius funtion R: V (0, ) etermines n inversive istne funtion Φ R on the ege set y Eqution 1: Φ R (e) =InvDist(C u,c v )= u, v 2 R(u) 2 R(v) 2. (3) 2R(u)R(v) AirlepkingC for (K, Φ R ), if it exists, gives rise to isrete onforml mpping of K to the plne y mpping the verties of K to the enters of their orresponing irles n extening ffinely on the metri fes. The imge of suh isrete onforml mpping is the rrier of the irle pking C, n is the union of the tringles orresponing to the fes of K forme y onneting enters of three mutully neighoring irles y stright line segments. The irle pking C is si to e oriente if the orienttions of ll of these nonegenerte tringles inherite from the orienttion on K re omptile. Equivlently, C is oriente if the isrete onforml mpping f is n orienttion preserving mp from K to the plne. When C is oriente n Φ tkes vlues in the unit intervl, this isrete onforml mpping is qusionforml, ut it my fil to e so for generl Φ vlues. Moreover, when C is oriente, it mps the tringultion of K to tringultion of the imge of this mp, though there my e egeneries. We esrie n lgorithm in the next setion tht proues sequene of these isrete onforml mppings, whih serve s the nites for pproximting the onforml mpping of K to retngle. To fore onvergene we nee to normlize the ounry irles in some wy, n we o so y mking further emn on our irle pkings tht will fore retngulr shpe upon the imge. In generl there re mny ifferent irle pkings for the sme t (K, Φ). For exmple, in the se of tngeny pkings, eh speifition of ounry rii for the irles tht orrespon to ounry verties etermines unique oriente pking with the omintoris of K. Alterntely, eh speifition of ounry ngle sums t ounry verties lso uniquely etermines n oriente pking for K. We shll ll irle pking C for (K, Φ) retngulr pking if it is oriente n the ngle sum of the fes t ounry

13 Plnr Conforml Mppings of Pieewise Flt Surfes 15 vertex in the imge tringultion re ll π, exept t the four istinguishe ounry verties, where the ngle sums re π/2. The rrier of retngulr pking is retngle. Fig. 5 shows two pkings n their rriers for the sme pieewise flt surfe K n inversive istne t Φ; the pking on the right is retngulr. Fig. 5. Two pkings for the sme t (K, Φ). 3 Hexgonl Refinement We now fix omintoril qurilterl K with n ege length funtion tht proues the pieewise flt onforml qurilterl K. LetR e onstnt rius funtion tht stisfies Inequlity 2 t eh ege n C retngulr pking for (K, Φ R ), where Φ R stisfies Eqution 3. It is importnt for proving onvergene tht R e onstnt funtion. Let f e the isrete onforml mpping etermine y C. The see t for our onforml mpping lgorithm is the 4-tuple (K 0, 0,R 0, Φ 0 )=(K,,R,Φ R ), from whih we proue the mpping t (C 0,f 0 )=(C,f). We think of f s the zeroeth pproximtion to the onforml mpping tht mps K to plnr retngle. The onstnt rius funtion my e hosen to hve ny positive vlue etween 0 n λ/ 2, where λ is the minimum of e s e rnges over ll the eges of K.

14 16 Philip L. Bowers n Moni K. Hurl Fig. 6. Hexgonl refinement, K K. For etter pproximtions we employ hexgonl refinement, or hexrefinement for short, whih suivies tringle into four sutringles s in Fig. 6; see [4]. The omplex thus otine from K y suiviing eh fe s in Fig. 6 is enote s K. There is nturl ege length funtion on K otine y reing off the lengths of eges otine y pling vertex t the mipoint of eh metri ege in K to hex-suivie eh metri fe of K into four similr opies of itself, sle y 1/2. Then K n K re isometri n thus inistinguishle s metri spes. If the onstnt rius R = 1 2 R is use for K, the inue inversive istne funtion Φ R replites on the eges of fe of K the three inversive istnes of its prent fe in K. Strting then with the see t (K 0, 0,R 0, Φ 0 ), we generte n infinite sequene reursively y (K n+1, n+1,r n+1, Φ n+1 )=(K n, n,r n, Φ n), for whih K = K n n for ll n. This proues n infinite sequene of mpping t (C n,f n ), where f n : K = K n n C is the isrete onforml mpping of the pieewise flt surfe K to the plne etermine y the retngulr pking C n for (K n, Φ n ). Rell tht there re four istinguishe ounry verties {,,, } of K orere respeting the orienttion of the ounry. We ssume one more normliztion onition, esily omplishe y Eulien similrities, y requiring the first two istinguishe verties n of K to mp to the respetive points 0 n 1 uner eh f n,nthetwo others to mp to the upper hlf plne. The imge of eh isrete onforml mpping f n is then retngle in the upper hlf plne one of whose sies lies long the unit intervl [0, 1]. The min onvergene result of [4] my e use to prove, in the speil se of tngeny pkings where gives unit length to eh ege, R is ientilly 1/2, n Φ R is ientilly 1, tht the sequene of mppings f n : K C exists n onverges uniformly to the unique onforml mpping F of K to retngle in the plne with F () =0,F () = 1, n F () n F () in the upper hlf plne. Moreover, the pointwise qusi-onforml ilttions of the mps f n re oune ove n onverge uniformly to unity on ompt susets of the omplement of the verties of K. Our nlysis of the proof will show in the next setion tht this hols in the pieewise flt se when R n e hosen so tht Φ R hs vlues in the unit intervl, n our gol is to unerstn preisely wht is lking in extening the proof to the generl se.

15 Plnr Conforml Mppings of Pieewise Flt Surfes 17 When the sequene f n oes onverge to the expete onforml mp F, the onforml moulus of the onforml qurilterl K is thus etermine to e µ = F (), the height of the imge retngle F ( K ). One might expet then tht the mximum qusi-onforml ilttions of the sequene f n onverge to unity, ut this is not the se. In ft, the mximum qusi-onforml ilttions of the sequene re in generl oune wy from unity sine, t ny vertex v of K whose ngle sum Θ etermine y is ifferent from 2π, there is lwys high istortion t the verties of K n neighoring v; see [4]. Nonetheless, this high lol istortion is relegte to smller n smller neighorhoos of the originl verties of K s we progress long the sequene f n. The result is tht the limit mpping F hs lol ilttion 1, i.e., is onforml, t every point of K other thn those of the originl vertex set V. Removility of isolte singulrities then omes into ply to gurntee tht the ilttions t the originl verties re 1 n, therefore, the limit mpping F is onforml. 4 Proving Convergene n Conformlity There re three min prolems ssoite with the inversive istne sheme for pproximting the onforml mpping of K to retngle. The first is tht of the existene of retngulr pking C n for (K n, Φ n ), the seon is tht of qusi-onformlity of the mppings f n with glolly oune ilttions, n the thir is tht of the rigiity of infinite hexgonl pkings of the plne with presrie perioi inversive istne t. The first prolem onerns the existene of the pproximting sequene f n, the seon onerns the onvergene of the sequene f n to qusi-onforml mpping F,nthe thir onerns the onformlity of the limit mpping F. We shll isuss eh of these in turn fter some generl omments on inversive istne pkings with Φ-vlues restrite to lie in [0, 1], i.e., in whih two neighoring irles interset nontrivilly. This hs een the sujet of lrge oy of theoretil reserh over the pst ee n hlf n there is n extensive literture on the sujet of existene n uniqueness of pkings, prtiulrly in the tngeny se where Φ is ientilly 1. The unerstning of the existene n uniqueness of tngeny irle pkings with presrie omintoris, s well s rigiity of infinite pkings, is ruil in the work of [10] n [4] where mpping lgorithms re shown to onverge to the orret onforml mppings. Using existene, uniqueness, n rigiity results now in ple llows us to pt the proofs of [4] to the nontngeny ut overlpping se where Φ my tke vlues in the intervl [0, 1]. This setion will give just suh proof tht lso overs the generl se of unrestrite Φ vlues if the three prolems tht we nlyze in this setion re foun to hve pproprite resolutions.

16 18 Philip L. Bowers n Moni K. Hurl The prolem of existene. The existene n uniqueness of tngeny irle pkings for omplex K ws first prove in [2] for ritrrily ssigne ounry rii or ngle sums. This is viewe in [2] s the isrete nlogue of the lssil Perron metho of solving the Dirihlet Prolem on plnr omins. Existene n uniqueness results for overlpping pkings with presrie ngles of overlp, where Φ hs vlues in the unit intervl, re prove in [12] n [6]. It follows from this work tht retngulr pking for the t (K n, Φ n )existsslongsthevluesofφ n lie in the unit intervl n two tehnil onitions first esrie y Thurston in [12] re stisfie. These Thurston onitions re, for n inversive istne ssignment Φ, T1 If simple loop in the omplex K forme y the three eges e 1, e 2, e 3 seprtes the verties of K, then 3 i=1 os 1 Φ(e i ) <π; T2 If v 1,v 2,v 3,v 4 = v 0 re istint verties of K forming eges v i 1,v i n Φ( v i 1,v i )=0fori =1, 2, 3, 4, then either v 0,v 2 or v 1,v 3 is n ege of K. The prolem of existene persists when neighoring irles re llowe to e isjoint, where the Φ vlues my e greter thn unity. In this se the generl ounry vlue prolem is not lwys solvle, i.e., there re exmples of inversive istne ssignments Φ where no irle pking in the plne with the omintoris of K n relize the inversive istne t. Even when suh pkings o exist, they my not exist with preetermine ounry rii or ngle sums. Thus, there re exmples of t (K, Φ) for whih there re no retngulr pkings. These will e etile in forthoming pulitions, ut for now their existene points to the ft tht the mouli spe of t for whih there o exist generl inversive istne pkings is muh more omplite ojet thn those for the speil ses of tngeny n overlpping pkings. For the present work, this lk of omplete unerstning of the existene of irle pking for (K, Φ) mens tht we nnot gurntee tht the see pking C 0 for our lgorithm exists. However, when it oes exist, the lgorithm proues sequene of pproximte onforml mppings. The exmples of inversive istne pking t without retngulr pkings require some gymnstis to onstrut n o not seem to rise nturlly from, for exmple, polyherl surfes emee in R 3. We hve never enountere surfe in prtie where the lk of existene prevente us from uiling see pking for the lgorithm. This prolem oes not seem to e mjor prtil impeiment to the wiespre pplition of the inversive istne sheme for onformlly mpping pieewise flt surfes to the plne. The prolem of qusi-onformlity. Assume the retngulr pkings C n,n therefore the isrete onforml mppings f n,existforlln. The rgument of [4] for proving onvergene of f n to limit mpping F in the tngeny se uses the lssil theory of normlity of fmilies of qusi-onforml mppings foun, for instne, in [9]. The rgument, whih is given for onforml

17 Plnr Conforml Mppings of Pieewise Flt Surfes 19 qurilterls in the proof of the theorem elow, requires tht f n e sequene of qusi-onforml mppings with oune ilttions, mening tht there is glol oun κ on the mximl ilttions κ(f n )ofllthemps in the sequene. In this se, it will e shown tht the sequene f n onverges uniformly to κ-qusi-onforml mpping of K. Qusi-onformlity of eh mp f n s well s glol oun on their ilttions in the tngeny se is gurntee y the ring lemm of [10]. Forthoming pulitions will show tht the ring lemm generlizes to those inversive istne pkings for whih Φ never tkes the vlue 0, i.e., the se of non-orthogonl overlps, n for whih the Thurston onitions hol. However, this generlize ring lemm provies qusi-onformlity only in the overlpping se where the Φ vlues lie in the hlf-lose intervl (0, 1]. The lemm oes not provie qusi-onformlity in the setting of isjoint irle neighors where Φ my tke vlues greter thn unity. In ft, there re exmples of inversive istne ssignments given y Φ where C exists, so tht the isrete onforml mpping f exists, for whih f is not qusionforml. Thus there is no gurntee tht even if the sequene f n exists tht eh mpping is qusi-onforml, n even if eh is, there is no gurntee tht the sequene hs oune ilttions. Agin these exmples require some gymnstis to onstrut n seem not to pper mong, for exmple, polyherl surfes in R 3, n gin this prolem oes not seem to e mjor prtil impeiment to the wiespre pplition of the inversive istne sheme. The prolem of rigiity of infinite hexgonl pkings. Assume now tht the first two prolems hve een resolve for K n we hve sequene of isrete onforml mppings f n, eh qusi-onforml, with ilttions oune y κ. In this se there will exist κ-qusi-onforml limit mpping F. The finl step for uniformizing K is the verifition of the onformlity of F. This step is omplishe for tngeny pkings in oth [10] n [4] y use of the hexgonl pking lemm of [10], whih epens on rigiity result out infinite irle pkings of the omplex plne. The nlogous rigiity result for overlpping pkings is prove in [6], n so the ingreients re in ple to verify onformlity of the limit mpping whenever Φ tkes vlues in the unit intervl. We elieve tht very generl rigiity result hols for ritrry lolly finite inversive istne pkings of the omplex plne, ut for the proof of onformlity, ll we nee is the verifition of the following speilize rigiity onjeture. In the onjeture, H is the onstnt 6-egree tringultion of the plne. The eges of H my e put into three equivlene lsses epening to whih ege of fixe fe τ given ege is prllel. A irle pking of C is lolly finite provie eh point of the plne hs neighorhoo tht meets only finitely mny irles of the pking. Conjeture 1. Let α, β, nγ e the inversive istnes etween respetive pirs of three equi-rii irles in the plne whose enters re the verties of

18 20 Philip L. Bowers n Moni K. Hurl nonegenerte tringle. Let Θ e n inversive istne ege funtion for H tht ssigns the vlues α, β, nγ to the three respetive eges of eh fe so tht Θ is onstnt on eh of the three equivlene lsses. Then lolly finite irle pkings for (H, Θ) re unique up to Eulien similrity, i.e., if C n C re oth lolly finite irle pkings for (H, Θ), then there is similrity S suh tht S(C) ={S(C): C C}= C. α γ β () Inversive istnes > 1. () Hexgonl pttern. Fig. 7. Hexgonl rigiity. If the onjeture is true, then ny irle pking for (H, Θ) hs Z Z- symmetry with funmentl omin the union of ny two tringles forme y onneting neighoring irle enters n tht meet long ommon ege. This follows y pling three irles of equl rii in the plne suh tht the pirwise inversive istnes re given y α, β, nγ. The plne then my e tringulte in the hexgonl pttern with isometri opies of the tringle otine y onneting the enters of these three equi-rii irles. An exmple where α = , β = , n γ = 31 ppers in Fig. 7. When eh of α, β, nγ re no greter thn unity, results of [6] verify the onjeture. The next theorem shows how normlity of qusi-onforml fmily with oune ilttions n the onjeture re use to prove onforml onvergene. We mke the restrition in the theorem n its orollry tht Φ never tkes the vlue 0 so tht the irle pkings hve no orthogonl neighoring irles. Theorem 1. If the sequene of isrete onforml mppings f n exists, is qusi-onforml with oune ilttions, n Conjeture 1 hols, then f n onverges uniformly on ompt susets of the interior of K to onforml mpping F of K to retngle in the omplex plne with F () =0,

19 Plnr Conforml Mppings of Pieewise Flt Surfes 21 F () =1,nF () n F () in the upper hlf plne. Moreover, the mximum ilttion of f n onverges to unity uniformly on ompt susets of the omplement of Vin K. Proof. Suppose the hypotheses hol so tht f n is sequene of qusionforml mppings of K to the plne with glol oun κ on the qusi-onforml ilttions. Let µ e the onforml moulus of the onforml qurilterl K n let F : K Re the unique onforml mpping from K to the retngle in the plne with verties 0, 1, n µi nf () =0, F () =1,F () =1+µi, n F () =µi. As F is 1-qusi-onforml, eh of the mppings f n F 1 is κ-qusi-onforml. Theorem II 5.1 of [9] pplies to show tht the fmily of κ-qusi-onforml mppings F = {f n F 1 : n =1, 2,...} is norml in the interior of R. Letw e ny limit funtion of sequene from F, syw = lim f n(i) F 1 for susequene f n(i), where the limit is uniform on ompt susets of the interior of R. ByTheoremII5.3 of [9], there re extly three possiilities: the limit funtion w on the interior of R is onstnt mpping, mpping onto two istint points, or κ-qusi-onforml mpping. We show next tht the first two possiilities o not our. Let S e the open infinite strip in the omplex plne etween the horizontl lines through ±µi. Sine the four orners n sies of R re mppe y f n F 1 to the four orners n sies of the imge retngle R n = f n ( K ), the refletion priniple for qusi-onforml mppings [9] my e iterte to proue κ-qusi-onforml extension F n of f n F 1 to the omin S, swells κ-qusi-onforml extension w of w. Sinew is limit funtion of sequene from F, w is limit funtion of sequene from F = {F n : n =1, 2,...}. By Theorem II 5.3 of [9], there re extly the sme three possiilities for this funtion w. Notie though tht w is the ientity on the set of integers, whih re ontine in the interior of S, so the first two possiilities re rule out. It follows tht w is κ-qusi-onforml mpping n, s w is the restrition of w to R, sotooisw. The rrier R n of C n is retngle in the upper hlf plne with one sie the unit intervl. Theorem II 5.4 of [9] implies tht the imge of w is the kernel of the interiors of the retngles R n(i), n it is esy to see tht suh kernel must e retngle with one sie the unit intervl. We show elow tht w is onforml, whih immeitely implies tht this imge retngle w(r) muster itself n tht w must e the ientity mpping of R sine it fixes the four orners. It follows tht f n(i) onverges uniformly on ompt susets of the interior of K to F, the unique onforml mpping of K to R with F () =0nF () =1.Asw is n ritrry limit funtion of sequene from F, this rgument shows tht there is only one suh limit funtion, nmely, the ientity funtion on R. As the olletion F is norml fmily of mppings, so tht every infinite suset of F hs limit funtion, it follows tht the sequene f n F 1 itself onverges uniformly on ompt susets of the interior of R to this ientity funtion, or tht the sequene f n onverges uniformly on ompt susets of K to F. This ompletes the

20 22 Philip L. Bowers n Moni K. Hurl proofofonvergeneofthef n to onforml mpping of K moulo the verifition tht w is in ft onforml. This will e omplishe next with the i of Conjeture 1. Let α, β, γ, H, n Θ e s in Conjeture 1 n for eh n, leth n e the suomplex of H forme y n genertions of the hexgonl gri out some fixe vertex v 0.Letσ = v 0,v 1,v 2 e fe of H ontining v 0 n let Θ n e the restrition of Θ to the eges of H n. A proof using the rigiity of Conjeture 1 n the generlize ring lemm, similr to the proof of the hexgonl pking lemm of [10], shows tht there is sequene ε n eresing to zero suh tht, if H n is ny oriente irle pking for (H n, Θ n ), n if τ n is the tringle in C forme y onneting the enters of the irles in H n orresponing to v 0, v 1,nv 2 n τ is the tringle forme y onneting the enters of the irles in the unique pking H orresponing to v 0, v 1, n v 2, then the vertex preserving ffine mp from τ n to τ hs ilttion t most 1 + ε n. This is very strong sttement onerning the shpes of the tringles τ n s the onstnts ε n o not epen on whih pking for (H n, Θ n ) is hosen, ut merely on the ft tht v 0 is n-eep within the omplex H n. Let D e ompt suset ontine in n open fe σ of K. LetN e n ritrry positive integer n hoose n so lrge tht eh point z of D is entere in simply onnete neighorhoo U z forme y N genertions of the hexgonl gri in K n tht results from n hex-refinements of the fe σ. The generliztion of the hexgonl pking lemm of the previous prgrph gurntees tht f n hs mximum ilttion t most 1 + ε N on D, n sine ε N ereses to zero, the mximum ilttion onverges to unity uniformly on D. ThisimpliesyTheoremII5.3 of [9] tht the ilttion of the limit mpping F t ny point in the interior of fe of K is no more thn 1 + ε N, for ll N, n therefore F is onforml on the interiors of the fes of K. By removility of nlyti rs n isolte singulrities, F is onforml on K. We emphsize here tht this rgument with the use of the generlize hexgonl pking lemm requires tht our rius funtion R, fromwhihthe see inversive istne funtion Φ 0 is lulte, e onstnt on the vertex set V. One my run the lgorithm with ritrry vrile rius funtion R, ut the onvergene generlly will not e to onforml mpping. The lst sttement of the theorem requires smll moifition to show uniform onvergene when the ompt set D hits eges, whih we shll not present. Corollry 1. If ll the ege lengths e lie in the hlf-lose intervl (λ, 2λ], for some positive onstnt λ n the Thurston onitions (T1) n (T2) hol, then the funtions f n exist n re qusi-onforml with oune ilttions, n the sequene onverges uniformly on ompt susets of the interior of K to onforml mpping F of K to retngle in the omplex plne with F () =0, F () =1,nF () n F () in the upper hlf plne. Moreover, the mximum ilttion of f n onverges to unity uniformly on ompt susets of the omplement of Vin K.

21 Plnr Conforml Mppings of Pieewise Flt Surfes 23 Proof. If the initil rius funtion R 0 is hosen to hve onstnt vlue λ/ 2, then the initil inversive istne ege funtion Φ 0 tkes vlues in the hlflose intervl (0, 1], sine ll the ege lengths e lie in the intervl (λ, 2λ]. Notie tht the vlues of eh Φ n re the sme s those for the initil inversive istne ege funtion Φ 0, so tht the Φ n vlues re in the unit intervl. ThustheirlepkingsC n re either tngeny or nonorthogonl overlpping pkings. The sequene f n exists y existene-uniqueness results of [2] n [6] tht over the tngeny n overlpping pking ses. Qusi-onformlity of the f n with oune ilttions follows from the ring lemm of [10] for the tngeny se n its generliztion for the overlpping se. The verifition of Conjeture 1 for the tngeny se ppers in [10] n for the overlpping se in [6]; see lso [11]. Theorem 1 pplies. 5 A Gllery of Qurilterls Exmple 1. Our first exmples re those of Figs. 1 n 2. In Fig. 1() ll eges hve unit length n the surfe K is n equilterl surfe forme y gluing eight unit equilterl tringles long eges tht meet t ommon entrl vertex. By onforml symmetry t the entrl vertex, the entrl ngles of ll the tringles hve mesure π/4 in the onforml struture though they ll hve Eulien mesure π/3 in the pieewise flt struture. Notie tht there re nti-onforml refletions ross the igonls from to through the enter n from to through the enter, s well s ross the other two igonls. Thus the iherl group D 4, the symmetry group of the squre, ts s group of onforml symmetries of K. The only retngles on whih D 4 ts onformlly re squres, so we know efore running the inversive istne sheme tht the onforml moulus of K is 1 n K is onformlly equivlent to squre vi mpping tking the equilterl fes to ongruent (2, 4, 4) tringles forme y the igonls n opposite ege isetors of squre. CirlePk onfirms this in Fig. 2(). Sine this is n equilterl surfe, we use tngeny pkings with unit inversive istne funtion. In Fig. 1() ll eges hve unit length exept for the three ounry eges lele y γ, eh of whih hs ege length γ =2sin π This mkes the Eulien ngles opposite γ equl to π/9 so tht the totl Eulien ngle spnne opposite the three lele sies is π/3, the sme s the Eulien ngles of the equilterl tringles t tht vertex. The totl Eulien ngle sum roun the entrl vertex is 2π, so the ngles mesure y the onforml struture t the entrl vertex gree with the Eulien mesures. In prtiulr, onforml mpping to retngle will mp the fes so tht the Eulien ngles t the entrl vertex re preserve. Agin CirlePk onfirms this in Fig. 2(). This time the retngle is not squre n the onforml moulus of the onforml qurilterl K is µ = The fixe vlue we hose for the rius funtion R, fromwhihtheinversive

22 24 Philip L. Bowers n Moni K. Hurl istne ege funtion Φ is lulte y Eqution 3, is γ/2. This mkes the inversive istne vlues unity long the γ eges n otherwise. () One refinement. () Two refinements. () Four refinements. () Close-up. Fig. 8. Converging to onformlity. In Fig. 1() ll eges hve unit length exept for the four ounry eges lele y δ, eh of whih hs ege length δ =2sin π This mkes the Eulien ngles opposite δ equl to π/12 so tht the totl Eulien ngle spnne opposite the four lele sies is π/3. Thus the mrket shre of these four ngles totle equls the mrket shre of eh of the other ngles t the entrl vertex in the unit equilterl tringles. This mens tht the onforml struture on K mesures the totl ngle spnne opposite

23 Plnr Conforml Mppings of Pieewise Flt Surfes 25 the four lele sies s 2π/5 s well s the remining four ngles t the entrl vertex in the four equilterl tringles. In prtiulr, the onforml struture mesures eh ngle opposite δ s π/10 though the Eulien mesure is π/12. Also, K hs n nti-onforml refletion ross the igonl from to n, sine squres re the only retngles with igonl onforml symmetry, we know tht the onforml moulus of K is 1 n K is onformlly equivlent to squre. Agin CirlePk onfirms this in Fig. 2(). The fixe vlue we hose for the rius funtion R is δ/2, whih mkes the inversive istne vlues unity long the δ eges n otherwise. Fig. 8 shows the imge retngulr pkings t stges one, two, n four of the inversive istne itertion, with only the imge of the originl tringultion shown for the fourth stge pking, s well s lose-up of the entrl vertex from the stge four refinement. Fig. 9. Hexgonl gri: lengths of ol eges re 1.1; others re 1.4. Exmple 2. The ege length ssignments use in Fig. 9 for the omplex K hve e equl to 1.1 for the ol eges n 1.4 otherwise. We pproximte the onforml mpping of K to retngle using three ifferent hoies for the initil rius funtion. The rius funtion R(1) tkes the onstnt vlue 1/ 2 where ll neighoring irles overlp nontrivilly. The seon R(2) tkes the onstnt vlue 3/5 where there is mixture of overlpping n isjoint irles in the initil onfigurtion. The thir R(3) tkesthe onstntvlue 1/4 where ll irle pirs re isjoint. The inversive istne lgorithm with ny of the three see rii shoul provie pproximtions tht onverge to the unique onforml mpping of K to retngle of unit horizontl sie length. Fig. 10 shows the fourth iterte of the inversive istne sheme pplie with eh of the three see rii funtions. The irle pkings themselves with the imges of the eges of the initil tringultion K rkene re shown, long with lose-up of neighorhoo of one of the verties. This experimenttion with CirlePk suggests tht the onvergene is inepenent of the initil

24 26 Philip L. Bowers n Moni K. Hurl onstnt rius vlue, s it shoul e. The rnges of the Φ vlues re to for Φ R(1), to for Φ R(2),n to for Φ R(3). Exmple 3. Corollry 1 onfirms tht the inversive istne sheme onverges in se the pieewise flt metri is equilterl, i.e., when the ege length funtion tkes the onstnt vlue 1. Then the metri surfe K is union of equilterl tringles glue sie-to-sie n, when the rius funtion R tkes the onstnt vlue 1/2, the inversive istne funtion Φ R tkes the onstnt vlue 1. The retngulr pkings re then tngeny pkings. Fig. 11 shows three exmples of pieewise equilterl qurilterls n their uniformiztions s retngles. Eh ege in the left-hn figures is given unit length, n four refinements re use to pproximte the retngulr uniformiztions on the right. An interesting feture of equilterl surfes is tht they hve refletive struture in whih eh fe reflets ross ny interior ege to its ompnion fe, see [4]. This refletion is n ntionforml mp n the whole surfe is generte y fixing ny one fe n then refleting ross eges itertively. This is ovious in the pieewise equilterl mnifesttion of the surfe, n this trnsltes into the following property of their retngulr onforml imges. The imge τ = F (τ) ofny equilterl fe τ of K uner the onforml mpping to retngle ontins ll the informtion out the rest of the mp in the sense tht the rest of the mp n the imge urviliner tringultion of the retngle n e reovere y nti-onforml refletions iterte strting with τ. This suggests tht, in priniple, n ritrry finite or even ountly infinite mount of informtion n e represente in the shpe of single urviliner tringle n then reovere y nti-onforml refletions. This is theoretilly interesting n is illustrte in the next exmple. Fig. 12() shows n eight-y-eight squre with eh susqure ivie into two tringles with either right or left slsh. The right slsh enoes zero n the left one, n the rows enoe the iniviul symols of the expression vismth!. The resulting tringultion is given n equilterl metri with ll unit ege lengths n this surfe is mppe onformlly to retngle. The resulting refletive urviliner tringultion is shown in Fig. 12() n the upper left-hn orner tringle is enlrge in Fig. 12(). The whole tringultion in Fig. 12(), n therefore the messge vismth!, my e reovere from the lone tringle in Fig. 12() (or from ny other tringle in the figure) y iterte nti-onforml refletion. Of ourse there is nothing to restrit our ttention to finite tringultions. We might well tringulte the plne, presrie tht eh fe e unit equilterl tringle, then onformlly mp the resulting pieewise equilterl surfe to the plne C or to the unit isk. The imge of ny fe then ontins ll the omintoril informtion of the originl tringultion. The intereste reer might fin the isussion of [3] enlightening.

25 Plnr Conforml Mppings of Pieewise Flt Surfes () R = 1/ 2. () Close-up. () R = 3/5. () Close-up. (e) R = 1/4. (f) Close-up. Fig. 10. Hexgonl gri. 27

26 28 Philip L. Bowers n Moni K. Hurl () A qurilterl n its refletive tringultion. () A ifferent orner point. () A pentgonl pking n its refletive tringultion. Fig. 11. Equilterl surfes n their uniformiztions.

27 Plnr Conforml Mppings of Pieewise Flt Surfes 29 v i s m t h! () Binry enoing. () Refletive enoing. () Top left tringle. Fig. 12. Enoe vismth!. Exmple 4. The left-hn grphi of Fig. 13() (Color Plte 1() on pge 427) shows three-imensionl renering of the surfe of humn ererum otine from the Visile Mn t from the Ntionl Lirry of Meiine. This exmple ontins 52, 360 verties n 103, 845 fes. Hurl et l [8] flttene this mesh qusi-onformlly using tngeny pkings where ll inversive istnes re set to unity. They then ompute texture ump mp using fke iffuse omponent for eh irle using the surfe norml in R 3. The olor for eh irle ws then sle se on the iffuse vlue. In this wy the fissures n suli of the three-imensionl rin t n e represente in the flt mpping, see [8]. The results pper in the left-hn grphis of Figs. 13() n 13() (Color Pltes 1() n 1() on pge 427), where ounry t from the three-imensionl surfe hs een use to normlize the pking. One n see the rmti effet ump mp texturing hs in these flttene imges. In the right hn grphis of Fig. 13 (Color Plte 1 on pge 427), we hve isolte from this rin surfe qurilterl region me up of 2943 fes with 1565 verties, n mppe this susurfe onformlly to retngle. We use the istnes etween neighoring

28 30 Philip L. Bowers n Moni K. Hurl () Right hemisphere n susurfe. () Rii pking of hemisphere n inversive istne pking of susurfe. () Pking with ump mp texture. Fig. 13. Qusi-onformlly mpping of the humn rin to plnr omin.

29 Plnr Conforml Mppings of Pieewise Flt Surfes 31 verties in the three-imensionl grphi of Fig. 13() (Color Plte 1() on pge 427) to ompute n ege-length funtion nthenfltteneusing the inversive istne sheme. The first retngulr mp, Fig. 13() (Color Plte 1() on pge 427), is n inversive istne pking without the ump mp texture, n the seon, Fig. 13() (Color Plte 1() on pge 427), is one with the ump mp texture. This is smple of ongoing work y tem of mthemtiins n neurosientists who re working to uil onforml flttening visuliztion tool for use in neuro-ntomil stuies. Another smple ppers in Fig. 14 (Color Plte 2 on pge 428) where we hve onformlly mppe two ereellum imges otine from MRI sns to isk. The top two imges show the ereellum from two ifferent sujets. The mile two imges show mpping to isk. The ottom two imges orrespon to lose-up view of the isk mpping to highlight some of the etil in the entrl regions of the mppings. The olor oing ientifies regions of interest to neuro-ntomists with the ornge regions initing res of PET tivtion when the sujets perform the sme tsks. 6 Implementtion: Prtil Experimentl, Computtionl, n Theoretil Issues Implementtion of the inversive istne sheme for pproximting onforml mppings requires the evelopment of omputtionl engine tht omputes oriente irle pkings for given inversive istne t (K, Φ). Ken Stephenson hs uilt suh n engine in his progrm CirlePk. Itspking lgorithm for tngeny pkings uses refinement of Thurston s originl ie in [12] s well s moern numeril shemes for fst pproximtion of trnsenentl funtions. The reer my onsult [5] for the ltest etile ount of optiml pking lgorithms. The pking lgorithm generlizes to over ritrry inversive istne pkings, though now there is no gurntee of onvergene s there is in the tngeny se. In ft, s we know of exmples of inversive istne t (K, Φ) tht hve no irle pking reliztion, ny suh see t for CirlePk woul fil to onverge. The pking lgorithm is se on monotoniity results for the hnge in ngle sums out verties s the rius of single irle is hnge while preserving inversive istnes. Agin the intereste reer is irete to [5] for etils. The reer might sk how prtil it is to get relly lose pproximtions to the onforml mpping of K to retngle sine, oviously, the numer of verties grows exponentilly s hex-refinement is iterte. The goo news is tht the experimentl eviene suggests very fst onvergene of the inversive istne sheme. Inee, in ll exmples we hve yet enountere, the ifferene etween the fourth n fifth itertion is so smll s to e unnotiele. This points to theoretil issue whose resolution woul e very vlule for vliting this experimentl oservtion, nmely, tht of eriving nlyti estimtes on the qusi-onforml ilttions of the pproximting mppings

30 32 Philip L. Bowers n Moni K. Hurl Fig. 14. Mpping two ifferent ereellum of the humn rin.

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