Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface

(1)

HAL Id: hal-00804691

https://hal.inria.fr/hal-00804691

Submitted on 22 May 2013

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface

Francis Lazarus, Michel Pocchiola, Gert Vegter, Anne Verroust

To cite this version:

Francis Lazarus, Michel Pocchiola, Gert Vegter, Anne Verroust. Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface. SoCG ’01 - Seventeenth annual symposium on Com- putational geometry, Jun 2001, Medford, United States. pp.80-89, �10.1145/378583.378630�. �hal- 00804691�

(2)

Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface

Francis Lazarus

Michel Pocchiola

y

Gert Vegter

z

Anne Verroust

S

ABSTRACT

Alosed orientable surfaeof genusg anbeobtained

by appropriateidentiation ofpairs of edgesof a4g-

gon(thepolygonalshema). Theidentiededgesform

2gloopsonthesurfae,thataredisjointexeptfortheir

ommonend-point. Theseloopsaregeneratorsofboth

thefundamentalgroupandthehomologygroupofthe

surfae.Theinverseproblemisonernedwithndinga

setof2gloopsonatriangulatedsurfae,suhthatut-

ting the surfaealong these loops yields a(anonial)

polygonal shema. Wepresenttwooptimalalgorithms

forthisinverseproblem. Bothalgorithmshavebeenim-

plementedusingtheCGALpolyhedrondatastruture.

1. INTRODUCTION

LetMg bea regular4g{gon, whosesuessive edges

arelabeleda1;b1;a1;b1;;ag;bg;ag;bg. Edgexisdi-

reted ounterlokwise, edge x lokwise. The spae

obtainedbyidentifyingedges xandx,asindiatedby

theirdiretion,is alosedoriented surfae;Seee.g.[8,

Chapter1.4℄. Thissurfae,alled orientablesurfaeof

genus g, is homeomorphito a 2{sphere with g han-

dles. E.g.,M1 is thetorus;SeeFigure1. Thelabeled

polygonM

g

isalledtheanonialpolygonalshemaof

M

g .

Itiseasytoseethatallvertiesareidentiedtoasingle

pointp0 ofthesurfae. Afteridentiationinpairs,the

CNRSandUniversityofPoitiers,Frane.

E-mail: lazarussi.sp2mi.univ-poit iers. fr

y

Deptd'Informatique,EoleNormaleSuperieure,Paris,

Frane. E-mail: Mihel.Pohiolaens.fr

z

Dept.ofMath. andCS,UniversityofGroningen,The

Netherlands. E-mail: gerts.rug.nl

S

INRIARoquenourt,Frane.

E-mail: Anne.Verroustinria.fr

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.

SCG’01, June 3-5, 2001, Medford, Massachusetts, USA.

a

1 a1

b

1

b

1

Figure 1: Frompolygonal shema to orientable

surfae: the torus.

edgesof thepolygonal shema form2gurvesonMg,

whih are disjoint, exept for their ommon endpoint

p

0

. These 2g loops are generators of the fundamental

groupofMg (andofthersthomologygroup). Inthe

sequelwe dropthe dependeneonthegenus fromour

notation,i.e., Mdenotes alosed orientablesurfaeof

genusg.

Inthispaperweonsidertheinverseproblem: Given

aombinatorial(triangulated)surfae,ndaanonial

set of PL-urves(generators) suh that, after utting

thesurfaealongthesegenerators,weobtainaanoni-

alpolygonalshemaforthesurfae. APL-urveisan

alternatingsequeneofedgesandverties,whereedges

onnettwosuessivevertiesthatlieinthesamefae,

eitherinitsinteriororontheinteriorofoneofitsbound-

aryedges.

In [10℄ an algorithm is skethed that onstruts a

anonialset ofgenerators inoptimal timeand spae.

Inthis paper, wepresent indetailasimpleoptimalal-

gorithm; we allthis theinrementalmethod,sinewe

onstrutthegeneratorswhiletraversingalltrianglesof

thesurfae. Ourmainresultis

(3)

anorientable losed surfae ofgenus g,with atotal of

n verties, edges and faes, an be omputed in O(gn)

time andspae,whihisworst-ase optimal. EahPL-

generatoronsists ofO(n)edges andverties.

. . . . . . ..

.

.. .

Figure 2: A surfae with two groups of dg=2e

and bg=2 handles, separated by a thin tunnel

of size (n). Regardless of the position of the

base-point p

0

, at least half of this tunnel must

be rossed by atleastbg=2 generators.

Optimalityiseasytoestablish;SeeFigure2.

Furthermore,weshowhowtoturnBrahana'smethod[2℄

into a seond algorithm omputinga anonial set of

generators inoptimaltimeand spae. Wehaveimple-

mentedbothmethodsusingtheC++libraryCGAL.For

ommentsontheseimplementations, andtheir perfor-

mane,werefertoSetion6.

There are several reasons for presenting these algo-

rithmshere: (i)ouralgorithmsgreatlysimplifythemethod

of[10℄, (ii) full detailsare presented for thersttime,

(iii)thealgorithmshavebeenimplemented,and(iv)the

algorithmsanbeusedtosolveseveralotherproblems

inomputationaltopology.Amongtheappliationsare

the onstrutionof PL-homeomorphismsbetween sur-

faes, andtheonstrutionof(apart of)theuniversal

overingspaeofthesurfae. Asimilar, non-anonial

polygonalshemahasbeenusedin[6℄todeidewhether

twoPL-urvesonasurfaeare homotopi. Adierent

algorithmforthelatterproblem,basedonmethodsfrom

ombinatorial grouptheory, and abandoninguniversal

overingspaes,is presentedin[4℄. Otherappliations

are oneivable inonnetion with morphing,where a

suitableparametrization of 2-manifolds is providedby

thediskobtainedbyuttingalongtheanonialgener-

ators.

For general bakground material on omputational

topology, also in onnetion with appliations, we re-

fertothesurveys[5℄and[9℄.

2. SURFACES WITH COLLARS

TriangulatedsurfaeswillberepresentedbyDoubly-

ConnetedEdgeList,adatastruturefor representing

subdivisionsofsurfaes. Werefer to[3, Chapter2℄for

details on this data struture. Note that everyundi-

retededgeofthe triangulationorrespondstoexatly

twohalf-edges. The inrementalalgorithm startswith

the open surfae S = Mnft0g, where t0 is an arbi-

trary (losed)triangle, eventuallyontaining the om-

monbasepointoftheonstrutedgenerators. Initially,

thetopologialboundaryBofS is theboundaryoft .

B along at least one edge, and utting these (losed)

triangles fromS. Notethatthenon-visitedpartofM

is anopen subsetof M. The topologial boundary B

is adjustedaordingly. It is represented as a irular

sequeneofhalf-edges,oriented insuhawaythat the

triangletotheleftofahalf-edgebelongstoS. Wesay

thatavertexours inBifitistheoriginofahalf-edge

inB.

Aswewillexplaininmoredetail,theboundaryBmay

beome non-regular during this proess, in the sense

thatavertex oursmultiplyinB,oritontainsboth

a half-edge and its opposite partner (alled its Twin

in [3℄). See Figure 3 (Bottom). Yet, the irregularity

ofB,andhene ofthesurfae S,is restrited. Thisis

mademorepreisebyintroduingthenotionofaollar

ofanopensurfae.

Definition 2. Asurfae withollar inMisa pair

(S;), where S is anopen submanifold of M, and :

S 1

[0;1℄!Misaontinuousmap,suh that

1.(S 1

(0;1℄)S,andtherestritionj

S 1

(0;1℄

:S 1

(0;1℄!S isanembedding;

2.(S 1

f0g)MnS;

3.ThetopologialboundaryofS (vizSnS)istheimage

ofthelosedurve:S 1

f0g!M.

Observethat theurve:S 1

f0g!Misingeneral

not anembedding. Theurve:S 1

f1g!M,whih

isanembedding,maybeonsideredasa`regularization'

ofthe{perhapsnon-regular{boundaryofS.Werefer

to thehalf-openstrip (S 1

(0;1℄) as the ollar of S.

Thisollarhas attahment urve (S 1

f0g),and free

boundary(S 1

f1g).Notethateveryontinuousurve

onnetingapointinSwithapointinMnSintersets

theollarofS.

B

p1

p

2

p3

p

4

Figure 3: Collars. Top: aPL-ollar is obtained

by inserting verties near the tail of half-edges

inident to B,or in a orner ofa triangle. Bot-

tom: aollaron asingular urve B.

AollarS hasastraightforwardrepresentationinthe

(4)

PL-setting. Tothisend,weinsertavertexnearthetail

of eah half-edge in S emanating from a vertex of B.

Notethat inthis way anedge withbothendpointson

B obtainstwoverties. Furthermore,if two suessive

half-edgesofB,sharingaommonvertexv,areinident

to the sametriangle tof S, thereis nohalf-edge of S

emanating from v. In this ase, we insert avertex in

the interiorof t(e.g., onthebisetor ofthe angle oft

at v). Conneting thesequeneof insertedvertiesby

edges we obtain a PL-ollar of S; See Figure3. This

typeofollarwillbeusedinSetion4.

As usual, the Euler harateristi (S) of S is the

alternating sumof the numbers of verties, edges and

faes of S. Cutting the surfae along B we obtain a

boundary of S onsisting of a yli sequene of half-

edges (wheresome pairs ofhalf-edges mayorrespond

tothesameundiretededgeofM). Gluingadiskalong

this yli sequene of half-edges yields a losed ori-

entable surfae. By denition,the genusg ofS is the

genusofthelattersurfae.Itisstraightforwardtohek

that(S)=1 2g.

3. OUTLINE OF THE ALGORITHM

Wenowdesribethealgorithmthatvisitsalltriangles

ofM,startingfromasingletriangle. Thisalgorithmis

thebakbonefortheonstrutionofaanonialsystem

of generators, to be desribed in Setion 4. Globally

speakingthealgorithm proeedsasfollows. Thealgo-

rithmConnetedSum,whihisalled ontheomple-

mentS oftheinitialtriangle,visitsatriangletinident

uponthetopologial boundaryB ofS,updatesS and

B, and alls itself reursively on the updated version

of S. (As we shall explain at the endof this setion,

the algorithm infat deomposes the surfae M as a

onnetedsum oftori, whene itsname.) During this

reursiveproess,Smaybeomedisonneted,inwhih

aseConnetedSumisalledreursivelyoneahon-

neted omponent. It may also happenthat S is not

disonneted,butis notasurfaewithollareither(it

willturn outthatinthe latterasetheollaris split).

BeforewepresentthealgorithminfulldetailinFigure5,

werstspeifytheinputofthealgorithm.

Preondition of ConnetedSum. Algorithm Con-

netedSum takesas inputa pair(S;g), where S is a

surfaewithollar,whihhaspositive genusg.

In partiular, the ondition g > 0 guarantees that

ConnetedSumwillnotbealledondisks,whihwill

be ruialintheanalysis of thetime omplexity. The

proessofvisitingtrianglet,inidentuponthetopolog-

ialboundaryB,isalled anextension. Wedistinguish

twotypesofextensions.

RegularExtension: Triangletshareseithertwover-

tiesandonehalf-edgeh

1

(Figure4,top),orthreever-

tiesandtwohalf-edgesh1,h2(Figure4,bottom),with

B.

WeupdateBintheformerasebyreplaingthehalf-

edge h1 with the two-hain h2;h3, in the latter ase

byreplaingthetwo-hainh1;h2withthehalf-edgeh3.

Notethat thetopologial typesofB and theollardo

not hange upon a regular extension. In partiular,

=)

h

1

h1

h

2 h

2

h

3 3

Figure 4: A regularextension.

S 0

= S nt is a surfaewith ollar (to guarantee that

S 0

isanopensubsetofM,weonsiderthetriangletto

belosed). Therefore,ConnetedSumisalled reur-

sivelyonS 0

. ItisobviousthattheEulerharateristi,

andhenethegenus,doesnothangeunderregularex-

tension.

Splitting Extension: Triangletsharesthreeverties

andonehalf-edgewithB(Figure6,upperpart).

Thevertexoft,notadjaenttotheommonhalf-edge

ofBand t,is alledthesplit vertex,and isdenotedby

v

s

. Let the verties of t be v

1 , v

2 and v

3

, suh that

v1v2 is ahalf-edge ofB, and henev3 =vs. Let L be

thepartofB betweenv

3 andv

1

,andletRbethepart

betweenv

2 andv

3

. ThenBissplitintoB

l

=v

1 v

3 Land

Br=v3v2R . Wedistinguishtwosub-ases:

Sntisnotonneted. InthisaseSntonsistsoftwo

onnetedomponents,S

l andS

r

say,withtopologial

boundaryB

l

and Br,respetively. BothS

l

and Sr are

surfaeswithollars,withattahmenturvesBlandBr,

respetively.

Sntisonneted. Inthisasethetopologialboundary

ofSntisBl[Br,so Sntis notasurfaewithollar.

Inpartiular,ConnetedSumdoesnotaeptSntas

input. Toremedythissituation,letbeasimpleedge-

pathinS nt onneting Bl and Br, alled a join-path

(ofB

l andB

r

). SeeFigure6,wherev

l 2B

l andv

r 2B

r

aretheextremalvertiesof.

Thefollowingresult,whose(straightforward)proofis

omittedfromthisversionofthepaper,guaranteesthat

inaseofasplittingextensiontheConnetedSuman

bealledreursively:

Lemma 3. Supposeproessingtausesasplittingex-

tension.

1.IfSntisonneted,andisajoin-path,thenSn(t[)

isasurfae withollar,havinggenus g 1.

2. If S nt is not onneted, its onneted omponents

S

l and S

r

are surfaes with ollar. Moreover, if their

genusesare g

l

andgr, respetively, theng=g

l +gr.

Thealgorithmthatonstrutsaanonialsetofgen-

eratorsispresentedinFigure5. Chekingwhether the

urrentextensionisregular(line2)anbedoneinO(1)

time,bysettingamarkbitfor eahvisitedvertex. To

determine whether S 0

= Sntis onneted,we try to

onstrut a join-path by performing a breadth-rst

searh on the 1-skeleton of S 0

(we have to say more

(5)

ConnetedSum(S)

1 LettbeatriangleofS inidenttoollar (S)

2 if tausesregularextension

3 thenConnetedSum(Snt)

4 else ollar (Snt)isdisonneted

5 if Sntisonneted

6 thenonstrutPL-pathjoining

omponentsofollar(Snt)

7 onstrutpairofgeneratorsalong

andpartofollar(Snt)

8 if genus(S)>1

9 thenConnetedSum(Sn(t[))

10 elseletS

l andS

r

betheomponentsofSnt

11 if genus(S

l

)>0thenConnetedSum(S

l )

12 if genus(Sr)>0thenConnetedSum(Sr)

Figure 5: Algorithm ConnetedSum

about this presently). Depending onwhether we su-

eed in onneting the two omponents Bl and Br of

thetopologialboundaryofS 0

,wedeidewhetherS 0

is

onnetedornot. IfS 0

isonneted,apairofgenerators

isonstruted(details arepresentedinSetion4),and

ConnetedSumis reursively alledon Sn(t[), if

g 0

=g 1>0. IfS 0

isnotonneted,ConnetedSum

isreursivelyalledontheonnetedomponentsSl,if

gl>0,andSr,ifgr>0.

Lemma 4. 1. IfSntis onneted, establishing on-

netednessandomputingajoin-pathanbeperformed

intimeproportionaltothesizeof S.

2. If S nt has two onneted omponents, establish-

ingnon-onnetednessandomputingthegenusesofthe

onnetedomponentsanbeperformedintimepropor-

tionaltothesizeof thesmalleronnetedomponent.

We only give a skethof the proof. Whena split o-

urs,wetrytoonstrutthejoin-pathbymeansofa

tandemsearhtraversingtheedgesofthesurfaeinpar-

allel,startingfromthesouresBl andBr. Theneither

thetandemsearhsueedsinonnetingB

l andB

r by

the join-path , or it detets that S nt has two on-

netedomponentsSlandSrbyexhaustingthesmaller

ofthesetwoomponents. Inthelatteraseweompute

thegenusofthesmalleromponentbydeterminingthe

numberofverties,edges andfaes. Lemma4,part2,

givesthegenusoftheother onnetedomponent.

Lemma's3and4allowustoanalyzethetimeomplex-

ityofthetraversaloftheinitialsurfaeM. Tothisend,

lett

0

beanarbitrarytriangleofM.

Corollary 5. The all of ConnetedSum on the

surfae with ollar S

0

= Mnt

0

is exeuted in time

O(gn), plus the time needed to onstrut the g pairs

of (line 7) generators upon a non-disonneting split.

Hereg isthegenus ofMandn isthetotalnumber of

verties,edgesandtrianglesinM.

Againthe proof isstraightforward,exept fora minor

subtlety. If all reursive alls eitherresult inthe on-

v

l

v

l

vl

vr

v

r

vr v1

v1

v

1

v2

v

2 3

v3 =vs

v

3

=v

s

Figure6: A splittingextension.

strution of a pair of generators (line 5{9), or in re-

ursiveallsofConnetedSumonollaredsurfaesof

lower genus (i.e., g

l

> 0 and g

r

> 0), the total time

omplexity obviouslyis O(gn)(sinethe totalnumber

ofallsisginthislukyase).

However,ConnetedSummaybealled reursively

on a ollared surfae of the same genus as S in ase

the genus of Sl or Sr is zero. Note that in this ase

theomponentwithgenuszero isdisarded. Sinethe

algorithmspendstimeproportionaltothesmallerofthe

sizesofSlorSr,wehargetheostofthereursiveall

to thedisarded omponent. Therefore, thetotal ost

ofallsofthistypeisO(n).

4. CONSTRUCTING GENERATORS

It remainsto llinthedetailsoftheonstrutionof

apairofgenerators,f Figure5,line7. Thesegenera-

torswill berouted alonganapproah path AP,whih

onnetsthebasepointwiththe boundaryofthenon-

visitedpartofthesurfae. Asthealgorithmproeeds,