Geometric Fairing of Irregular Meshesfor Free-Form Surface Design

RobertSchneider�, Leif Kobbelt1

Max-Planck Institutefor ComputerSciences,Stuhlsatzenhausweg 85,D-66123Saarbrucken,Germany

Abstract

In this paperwe presenta new algorithmfor smoothingarbitrary trianglemesheswhilesatisfyingG1 boundaryconditions.Thealgorithmis basedon solvinga non-linearfourthorderpartial differentialequation(PDE) thatonly dependson intrinsic surfacepropertiesinsteadof beingderived from a particularsurfaceparameterization.This continuousPDEhasa (representation-independent) well-definedsolutionwhichweapproximateby our tri-anglemesh.Hence,changingthemeshcomplexity (refinement)or themeshconnectivity(remeshing)leadsto just anotherdiscretizationof the samesmoothsurfaceand doesn’taffect the resultinggeometricshapebeyond this. This is typically not true for filter-basedmeshsmoothingalgorithms.To simplify the computationwe factorizethe fourth orderPDEinto asetof two nestedsecondorderproblemsthusavoiding theestimationof higherorderderivatives.Furtheraccelerationis achieved by applyingmultigrid techniqueson afine-to-coarsehierarchicalmeshrepresentation.

Key words: Discretefairing,Free-Form modeling,PDEmethod,Parameterizationindependence

1 Introduction

Althoughpiecewisepolynomialpatchesarestill thedominatingfree-formsurfacerepresentationin engineeringapplications,theuseof trianglemesheshasbecomeincreasinglyimportant– especiallywith the rising complexity of 3D models.Ageneralapproachto generatefree-formsurfacesthatsatisfyaestheticrequirementsis surfacefairing where auxiliary degreesof freedomare usedto improve the�

Correspondingauthor. E-mail: schneider@mpi-sb.mpg.de1 E-mail: kobbelt@mpi-sb.mpg.de

Preprintsubmittedto Elsevier Science 8 May 2001

global distribution of curvature(or optimizeany otherquality criterion). For tri-anglemeshesthis typeof optimizationhastwo differentaspects.First, thetrianglemeshshouldhaveouterfairness, i.e. theimaginarysurfacethatis approximatedbythemeshshouldbeoptimalwith respectto curvaturedistribution.Additionally, themeshshouldhave inner fairnesswhichmeansthatthedistributionof meshverticeswithin thesurfaceandtheshapeof the individual facesshouldbegoodaccordingto applicationdependentrequirements.

The result of recentmeshfairing algorithmsusually dependson the underlyingmeshconnectivity, thus the inner fairnessinfluencesthe resultingouter fairnessin someway andeven for the most sophisticatedschemesthereis no guaranteethat the surfacesarefree of parameterizationartifacts(Fig. 1). A solutionto thisshortcomingis to usea fairing algorithmthat is ableto separatethe two fairnesstypes(Fig. 2). This is achievedby usinganouterfairnessmeasurethatis basedonintrinsic surfacepropertiesonly, i.e. propertiesthatdependon thegeometryalone.Unfortunately, intrinsic fairing is a nonlinearproblem,andwhile thelinearfairingoperatorsarehighly efficient andin generalmathematicallywell understood,theanalysisin the non-linearcaseis muchmoredifficult. To the authorsknowledge,evenfor therathersimplefairing functional�

A

κ21 � κ2

2 dA � (1)

leadingto aminimalenergy surface(MES), it is still unknown if asolutionalwaysexistswithin specificsmoothnessclasses.

In this paperthe separationof outerandinner meshfairnessandparameterinde-pendenceis achievedby usinganouterfairnessconceptthatis basedon a discretesolutionof anintrinsicPDE.ThePDEwechooseis of fourthorderandleadsto sur-facesof high quality. To speedup theconstructionscheme,we factorizethefourthorderPDEinto twosecondorderproblemsanduseahierarchicmeshrepresentationto enablemultigrid techniques.

Thepaperis organizedasfollows:Section2 reviewsrelatedwork onmeshfairing.In section3 we presentthe conceptof our intrinsic fairing approach.Section4definesthenotationwe usethroughoutthepaper. In section5 and6 we show howto discretizethenecessaryintrinsics.Finally, in section7 we presentthedetailsofour algorithm.

2 Mesh fairing - previous work

Mostmeshfairingschemesarebasedonlinearoperators.While thisleadsto simpleandfastalgorithms,thereis aseriousconsequence:Theouterfairnessandtherefore

2

(a) (b) (c)

(d) (e) (f)

Fig. 1. Mesh fairing basedon discretizingthe equation∆2 f � 0 using two different lo-cal parameterizationstrategies.Theboundaryconditionis determinedby 3 cylindersthatarearrangedsymmetrically. a) shows the original mesh(920 vertices)andb) a reducedversionwith 118 vertices.In c), d) ande) we seethe resultsif local uniform parameter-ization is assumed.In f) we useda discretizationof a laplacianthat wasderived from adiscreteharmonicmapto fair theoriginalmesh.As wecansee,themeshsizeandthelocalparameterizationstrategy heavily influencetheresultingsurfaces.Therectangularpatchin-troducedin theoriginalmeshleadsto localaswell asglobalshapedistortionsandpreventsasymmetricsolution.

the shapeof the resultingmeshhighly dependson the chosenparameterizationstrategy. As aconsequenceit is notpossibleto separatetheouterandinnerfairnessconceptin thiscase.If wechangetheinnerfairnessstrategy wehave to changetheparameterizationandwill thereforealsochangethe outer fairness(seeFig. 1 (d)and(f)).

Therearetwo typesof fairingalgorithms,dependingonwhetherthefairingis basedon thecalculationof a well definedsurfaceor whetherit is basedon filtering oper-ations.

Thestandardapproachfor fair surfaceconstructionis basedontheideato minimizea fairnessmetric,punishingfeaturesthatareinconsistentwith the fairnessprinci-ple of the simplestshape(Burchardet al., 1994).Applied to meshesthis leadstothediscretefairingapproachproposedby Kobbelt(1997).Besidesof energy mini-mization,duringthelastyearsvariousotherlinearmeshfairingschemeshavebeen

3

(a) (b) (c)

(d) (e) (f)

Fig. 2. Our new fairing approachappliedon theexampleshown in Fig 1. In a), b) andc)we optimizedthe inner fairnesswith respectto a local uniform parameterization.In d), e)andf) themeshis discreteconformalto theoriginal meshin Fig. 1 a).We canseethat thechoseninner fairnessstrategy andthemeshstructurehave only marginal influenceon theshape.The influenceof themeshsizeis alsomuchsmallerthanin the linearsetting.Thecontinuityof therefectionlinesin c) indicatesG1 continuityat theboundary.

developed.

A veryeffectivemethodfor smoothingpolyhedralsurfacesis thediscretediffusionflow (Taubin,1995a).Heretheideais to iteratively updateeachvertex

q�i � qi � λ∆qi (2)

by addingadisplacementvectorthatis ascaleddiscretelaplacian∆qi. For stabilityreasonsthescalefactorλ hasto satisfy0 � λ � 1. A diffusionflow thatis uncondi-tionally stableandhenceenableslargerscalefactorswaspresentedby Desbrunetal. (1999).

Themainpurposeof thediffusionflow is to remove thehigh frequenciesin noisymeshes.Sincetheequilibriumsurfaceof theflow only enablesC0 boundarycon-ditions, (2) is of only limited usein surfacedesign.To enablesmoothboundaryconditionsonehasto considerdiffusionequationsof higherorder. Taubin(1995a)proposedto combinetwo suchsmoothingstepswith positive andnegative scalefactorsanddevelopedan algorithmthatenablesvariousinterpolationconstraints.

4

Anotherideathatenablessmoothboundaryconditionsis to usehigherpowersofthelaplacianin thediffusionflow. As agoodtrade-off betweenefficiency andqual-ity onecanchosethebilaplacianflow, enablingC1 boundaryconditions.

Anothermeshfairingapproachis basedontheideato discretizethePDEapproachof Bloor andWilson(1990).In (Kobbeltetal.,1998b)it wasproposedto discretize

∆2 f � 0 � (3)

where∆ is thelaplacianoperator, to createsurfacessatisfyingprescribedC1 bound-ary conditions.This equationresultsif we apply variationalcalculusto the thinplateenergy��

f 2xx � 2 f 2

xy � f 2yy dxdy

to describetheminimumof thefunctional.

Fairingbasedonsomekind of diffusionflow andfairingbasedonthediscretizationof thePDE(3) thatdescribesthesmoothsolutionaretightly connectedproblems,since(3) is the equilibrium of the bilaplacianflow. In both casesthe discretiza-tion of the laplacianplays the central role. During the last yearsvarious lineardiscretizationsof thelaplacianhavebeendeveloped(Taubin,1995a;Kobbeltetal.,1998b;Desbrunetal.,1999;Guskov etal.,1999),differing in how thegeometryofthemeshaffectsthediscretization.Thechosendiscretizationdeterminestheinnermeshfairness,but it alsogreatlyinfluencestheshapeof theresultingmesh(Fig.1).

To make the outer fairnessindependentof the meshparameterization,other ap-proachesarebasedon intrinsic surfaceproperties.Theseapproachesleadto non-linearfairingschemes.

In (WelchandWitkin, 1994)a meshfairing algorithmwaspresentedthatenablesG1 boundaryconditionsbasedon theideato minimizethetotal curvature(1) thuspunishinglargecurvaturevalues.For an isometricparameterizationthis approachcoincideswith minimizing thethin plateenergy (3). Thenecessaryintrinsic curva-turevalueswereestimatedusinglocal quadraticapproximationsover local planarparameterizations.However, in constellationswherethe local quadraticapproxi-mationsarenotuniquelydefined,theirapproachmayleadto stabilityproblems.

An intrinsic diffusionoperatorusinga discretemeancurvatureflow waspresentedby Desbrunet al. (1999),leadingto anexcellentnoisereductionalgorithm.Whilethis fairingalgorithmis mainlydesignedfor outerfairnesswithout tangentialshift,in (Ohtakeetal.,2000)thisalgorithmwascombinedwith aninnerfairnesscriterionthat leadsto moreregular meshes.Sincethesealgorithmsconverge to a discreteminimalsurfacesatisfyingH � 0, they only enableC0 boundaryconditionsandare

5

therefore,astheir linearcounterparts,of only limited usein surfacedesign.Again,the solution is to usecurvatureflows of higherorder (Brakke, 1992;Hsu et al.,1992;ChoppandSethian,1999).

In (SchneiderandKobbelt,2000a)it wasproposedto approximatea PDE basedon intrinsicsto createfair mesheswith G1 boundaryconditionsin thespecialcasewherethemesheshavesubdivisionconnectivity. As wewill seein thenext section,our methodsharestheprincipal ideato solve an intrinsic PDE,but enablesmuchgreaterflexibility .

3 Our fairing concept

Wepresentafairingalgorithmfor arbitrarytrianglemeshesthatenablesG1 bound-ary conditions(prescribedverticesandunit normals)andallows us to completelyseparateouterandinnerfairness.To achievethis,theouterfairnessstrategy is basedon the ideato discretizean intrinsic PDE.GivenG1 informationat theboundary,thePDEdefinesasmoothsurfacesatisfyingtheconstraints,alteringthemeshsize,theconnectivity or theinnerfairnessconditiononly producesanotherdiscretizationof thesamesmoothsurfaceandhenceleadsto geometriesthatwill becloseto eachother(Fig. 2).

In practicethis allowsusto chooseour innerfairnesscriterionfreely. In this paperwe restrictedourselvesto two especiallyimportantcases.Oneleadsto a regularmeshparameterization,theotherproducesmeshesthatareconformallyparameter-ized to a giveninitial polyhedron.The latter inner fairnessmethodplaysa funda-mentalrole for fairing of texturedmeshesandin meshediting,sinceit minimizeslocaldistortions.An otherconsequenceis thatacoarsemeshalreadyapproximatestheshapeof thesmoothsurfacethat is implicitly definedby thePDE,so increas-ing themeshsizemainly improvesthequalityof theapproximationnot thequalityof theunderlyingshape.We exploit this propertyto improve theefficiency of ourconstructionalgorithmby usingmultigrid methodsfor arbitrarymeshes.Thenec-essarymeshhierarchiesarecreatedusingthe progressive meshrepresentationasintroducedby Hoppe(1996).

Following the set-uppresentedin (SchneiderandKobbelt,2000a)to definefairsurfacessatisfyingG1 boundaryconstraints,the PDE that determinesour outerfairnessconceptin this paperis definedas

∆BH � 0 � (4)

which canbe interpretedassurfaceanalogonto the planarequationκ � � 0 (thederivative of the curvature κ is with respectto arc length) leading to clothoidsplines.Here∆B is theLaplace-BeltramioperatorandH themeancurvature.The

6

PDEonly dependson geometricintrinsicsandis comparatively simplefor a forthorderequation.Becauseof themeanvaluepropertyof the laplacian,it is guaran-teedthattheextremalmeancurvaturevaluesof a solutionof (4) will bereachedattheborderandthat thereareno local extremain the interior. Sinceconstantmeancurvaturesurfacessatisfy this equation,importantbasicshapesasspheres,cylin-dersand minimal surfaceswith H � 0 canbe reconstructed.Although the PDEwe proposecanbe derivedasa simplificationof the Euler-Lagrangeequationre-sultingfrom MESs(1), this is a fairing techniquein its own right, which thereforedoesn’t have to be inferior to the MES approach.In fact, this approacheven hassomeadvantagesoverMESs,e.g.it reproducescylinders,while it followsfrom theEuler-Lagrangeequationof (1) thatMESsdonot reproducethatsurfaceclass.Thisagainis completelyanalogousto the planarcase.It is well known, that minimalenergy curvesdonot reproducecircles,while aclothoidsplineobviouslydoes.

In (SchneiderandKobbelt,2000a)solutionsof (4) wereapproximatedby meshesinthespecialcasewherethemesheshave subdivisionconnectivity andtheboundaryverticescould be regularly sampledon a smoothcurve. The constructionschemewasbasedontheideato designaninnerfairnesscriterionthatpartitionsthesurfacein regularregions.Exploiting this regularityknowledgein advance,it waspossibleto assigna local planarparameterizationto eachvertex. Using thesedomains,theintrinsic valuescouldbeapproximatedby local quadraticapproximations,thusthequality of the discretizationdependson the quality of the estimatedlocal planarparameterizations.

Thealgorithmpresentedin this paperdoesn’t havesuchlimitations.Therearealsono restrictionsconcerningthemeshstructureandtheboundaryverticesandwearefree to choosean inner fairnesscriteria. Nevertheless,the resultingconstructionalgorithmis fastandcanbe implementedcompactly. Insteadof trying to simulatethecontinuouscaseusinglocal quadraticapproximations,we usethediscretedataof our minimizationprocessdirectly.

4 Notation

Wepartitiontheverticesof ameshM into two classes,denotingthesetof all borderverticeswith VB � M andthesetof all verticesin theinterior of M with VI � M . Foreachvertex qi of M let N � qi bethesetof verticesq j thatareadjacentto qi andletD � qi � N � qi ���� qi � be the according1-disk.Let Hi � H � qi denotethediscretemeancurvatureand �ni � �n � qi thediscreteunit normalvectorat thevertex qi . Whenit is clearwhichmeancurvaturevalueor normalvectoris meant,in somecasesweomit theindex to increasethereadability.

7

qj

q

qj-1

j+1

iq

j-1

j+1

j

β j

α

i

qq

q

j

q

Fig. 3. TheLaplace-Beltramioperatorat thevertex qi canbediscretizedusingD � qi � .5 Discretization of the Laplace-Beltrami operator

The discretizationrelies on the fact that thereis a tight connectionbetweentheLaplace-Beltramioperator∆B andthemeancurvaturenormalof asurface.In (Des-brun et al., 2000)alsoan improved discretizationcanbe found that usesa moredetailedareacalculationbasedon voronoiregions,but thefollowing discretizationis sufficient for ourneeds.Let f : � 2 � � 3 beaparameterizationof asurface,thenit is well known thatthefollowing equationholds

∆B f � 2H �n �Exploiting this relation,a discretizationof ∆B follows directly from themeancur-vatureflow approachfor arbitrary meshesthat was presentedby Desbrunet al.(1999).They showedthat themeancurvaturenormalat a vertex qi of a triangularmeshM canbediscretizedusingits 1-neighborhoodby

H �n � 34A ∑

q j � N � qi � � cotα j � cotβ j � q j � qi �� (5)

whereA is the sumof the triangleareasof the 1-disk at qi andα j andβ j arethetriangleanglesasshown in figure3. Sincetheverticescanbeinterpretedassamplepointsof asmoothsurfaceparameterizedby f andexploiting thefactthatascalingfactordoesnot influencethe result,we candiscretizethe equation∆BH � 0 at avertex qi as

∑q j � N � qi � � cotα j � cotβ j � Hi � H j � 0 �

If this equationis satisfiedat all innerverticesqi � VI � M andif we furtherknowall meancurvaturevaluesfor theboundaryverticesVB � M , this leadsusto asparse

8

linearsystemin theunknownsHi � VI � M , whosematrixShasthecoefficients

Sii � ∑q j � N � qi � � cotα j � cotβ j �� (6)

Si j � � "! �#� cotα j � cotβ j : q j � N � qi %$ VI � M 0 : otherwise.

(7)

Thematrix S is symmetricand– aslongasno triangleareasof themeshM vanish– positive definite.To seethis, we notethat S alsoappearsin a paperby Pinkalland Polthier (1993),wherea stableconstructionalgorithm for discreteminimalsurfacesbasedon theideato minimizethediscreteDirichlet energy of ameshwaspresented.Hence,for anelegantproof of themathematicalstructureof this matrixwe canreferto this paper. It shouldbementionedthatS furtherappearsin a paperaboutpiecewiselinearharmonicfunctions(Duchampetal., 1997).

6 Discretization of the mean curvature

In this sectionwe presenta discretizationof the meancurvatureHi at a vertex qi

thatdependson theverticesin a local neighborhood.Therearevarioustechniquesto discretizesurfacecurvatures,but to beapplicablefor ourconstructionalgorithm,it is importantthat – for a givenmeshconnectivity – the discretizationof Hi is acontinuousfunctionof thosevertices.

Not all curvaturediscretizationschemessatisfythisproperty. In (WelchandWitkin,1994) it was proposedto discretizecurvature information using local quadraticleastsquareapproximationover localplanarparameterizations.However, it is wellknown that leastsquareapproximationfails, if the points in the parameterizationplanelie on a curve of degree2. To beableto detectsuchcases,theauthorsesti-matedtheconditionnumberof theleastsquareproblemin theFrobeniusNormandreducedthenumberof thebasisfunctionsif theproblemwasill-conditioned.Thisapproachis notonly costly, but makesthediscretizationprocessdiscontinuousandthusleadsto apotentialinstability in themeshfairingalgorithm.

To avoid analogousstability problems,in thefollowing we proposea meancurva-turediscretizationtechniquethatsatisfiesthecontinuitycriterion.Moreover, aswewill show in section7.2, thepresentedschemehasotherfavorablepropertiesthatsimplify theconstructionprocess.

At first glance,it seemstemptingto useequation(5) to discretizethe meancur-vature,but this would only beapplicablefor innervertices.However, in our con-structionalgorithmpresentedlaterin section7, wehave to beableto discretizethe

9

meancurvaturenot only for inner vertices,but alsofor boundaryvertices,wherewe haveacompletelydifferentsituation.Dueto theboundaryconstraints,herewealreadyknow the tangentplaneof thefinal surface,but we don’t have a complete1-disk. To avoid having meancurvaturediscretizationsof differentaccuracy, wechooseonetechniquethatis ableto handlebothcases,soourmethodexpectsthatatangentplaneis known ateveryvertex. At interiorverticeswherenonormalvectoris known in advance,wedefineit to bethenormalizedsumof thevectorcrossprod-uctsof theincidenttrianglefaces,in orderto minimizesquarerootoperations.

qq

q

q

t

n

t

j

j-1

i

.....

j+1

t j-1

jj+1

qi

qj

jt

r

n

Fig. 4. Left: Projectingtheneighborhoodof qi ontotheplanedefinedby &n andnormalizingtheresultswe getthenormalcurvaturedirections&t j . Right:Thenormalcurvatureκ j alongthedirection&t j is discretizedby interpolatingqi andq j with a circle andusingtheinverseof thecircle radiusr asnormalcurvature.Thecenterof thecircle lies on the line definedby qi and &n.

6.1 MoretonandSequin’scurvaturediscretizationalgorithm

A curvaturediscretizationalgorithmthatseemsideal for our needswaspresentedby MoretonandSequin(1992).The ideaof their approachis to usethe fact thatthenormalcurvaturedistribution cannotbearbitrary, but is determinedby Euler’stheorem(doCarmo,1993).

Let �bx and �by beanarbitraryorthonormalbasisof theplanedefinedby thenormal�n. To eachvertex q j � N � qi we canassigna unit directionvector �t j by projectingq j into theplaneandscalingthis projectionto unit length.For eachq j wecannowestimatea normalcurvatureκ j asthe inverseof thecircle radiusdefinedby qi � q j

andt j (Fig. 4)

κ j � 2 ' q j � qi �n (' q j � qi q j � qi ( � (8)

Using Euler’s theorem,we canexpressthe normalcurvatureκn for a direction �tby theprincipalcurvaturesκ1 andκ2 andtheprincipalcurvaturedirections�e1 and

10

�e2. Let tx andty be thecoordinatesof �t in thebasis�bx � �by andlet ex andey be thecoordinatesof �e1, thenthenormalcurvaturecanbeexpressedas

κn �*) txty + t ,

K, ) tx

ty + � (9)

with

K � -. ex ey� ey ex

/0 , -. κ1 0

0 κ2

/0 , -. ex ey� ey ex

/021 1 �The idea of Moreton and Sequin is to usethe normal curvaturesκ j to createalinear systemand find estimatesfor the unknown principal curvaturevaluesbydeterminingthe leastsquaresolution.Let t j 3 x andt j 3 y denotethecoordinatesof �t j

andlet m bethevalenceof qi , thenwegetby evaluating(9)

A�x � �bwhere

A � -444444.t21 3 x t1 3 xt1 3 y t2

1 3 yt22 3 x t2 3 xt2 3 y t2

2 3 y, , ,t2m3 x tm3 xtm3 y t2

m3 y/65555550 � �b � -444444.

κ1

κ2,κm

/65555550and

�x � 78889 x0

x1

x2

:<;;;= � 78889 e2xκ1 � e2

yκ2

2exey � κ1 � κ2 e2

xκ2 � e2yκ1

:<;;;= �Sincex0 � x2 � κ1 � κ2 this meansthemeancurvatureis determinedby

H � 12 � x0 � x2 �� (10)

In ourcasethemostefficientmethodto solvetheleastsquareproblemis to usethenormalequationsapproach(GolubandVanLoan,1989),sincethismainly involves

11

to calculatetheinverseof a symmetric3 > 3 matrix.Usingthis approach,theleastsquaresolution �x canbeexpressedas�x � � AtA 1 1At �b � (11)

Thecaseswhenthematrix AtA becomessingularcanbedetectedby a simplecri-terion:

Lemma 1 TheMatrix AtA is singularif andonly if all points � ti 3 x � ti 3 y areintersec-tion pointsof theunit circle with two straight linesthroughtheorigin.

Proof: Sincesingularityof AtA is equivalentwith rankA � 3, singularityoccursif f any 3 row vectorsof A arelineardependent.Therefore,theproof is completeifweshow thatamatrixof type

-444. t21 3 x t1 3 xt1 3 y t2

1 3 yt22 3 x t2 3 xt2 3 y t2

2 3 yt23 3 x t3 3 xt3 3 y t2

3 3 y/ 5550

with t2i 3 x � t2

i 3 y � 1 is singularif andonly if the treepointslie on two lines throughtheorigin. This matrix is singular, if therearecoefficientsai which arenotall zerosuchthat

a1

78889 t21 3 x

t22 3 x

t23 3 x

:<;;;= � a2

78889 t1 3 xt1 3 yt2 3 xt2 3 yt3 3 xt3 3 y

:<;;;= � a3

78889 t21 3 y

t22 3 y

t23 3 y

:<;;;= � 78889 0

0

0

:<;;;= �but thismeanseachpoint � ti 3 x � ti 3 y liesonacurvethatsatisfiesa1x2 � a2xy � a3y2 �0. Dependingon thecoefficientsai this equationcharactererizesasinglepoint (theorigin) or two linesthroughtheorigin. ?6.2 Singularityhandling

Perhapsthemostobviousstrategy to avoid singularitiesof AtA is to simply checkwhetherthe criterion presentedin Lemma1 is satisfiedand to apply a specialmethodonly to suchcasesnearsingularity. Suchan approachis straightforward,but it can lead to instabilities.A small perturbationof one vertex can trigger aspecialcasehandlingand thus it could switch the meancurvaturediscretizationmethod,makingtheprocessdiscontinuous.

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q

q

j

q

p

iq

j+1

j

j−1j2

nj1

Ej

n

t

2

1pq

q

2

1t

s

1c

2cc

2

1q

q

2

2t

1t

s

1c

p

I

(a) (b) (c)

Fig. 5. a) At verticesqi with valence3 or 4, betweenall q j @ q j A 1 B N � qi � thatsharea com-monedge,new verticesp j areintroduced.The p j lie on a planeE j determinedby q j @ q j A 1

andthevector &n j1 C &n j2, where &n j1 and &n j2 arethetrianglenormalsof thefacesadjacenttotheedgeq jq j A 1. b) Approximationof a spiral (planarcurve with monotonenonzerocur-vature)by theareaenclosedby two arcs.c) This approximationalsoproducesreasonableresultsfor planarcurveswith monotonecurvatureandaninflectionpoint (denotedasI ).

An elegantsolutionto this discontinuityproblemis to exploit the connectionbe-tweenpossiblesingularityof AtA andthevertex valence.Assumingthatall points� ti 3 x � ti 3 y aredistinct,a simpleconsequenceof Lemma1 is that thematrix AtA canonly becomesingular, if thevalenceof qi is 3 or 4.

For all verticesof valence3 or 4 we increasethedataquantitythatservesasinputfor ouralgorithm.Insteadof enlargingthevertex neighborhood– which lackssym-metry if thevalenceof theneighborverticesvarieslargely – we increasethelocalinput databy estimatingnew verticesp j betweenadjacentq j � N � qi . The p j andq j thenserve asinput for themeancurvaturediscretizationasdescribedin section6.1,makingtheproblemwell posed.

Simply settingp j � � q j � q j D 1 FE 2 would be fastandconvenient,but suchanap-proachdistortstheresultingmeancurvaturediscretizationconsiderablyandshouldnot beusedif high quality resultshave to begenerated.A schemethathasprovento beadequateduringour numericalexperimentsis basedon theideato determinep j by samplinga planarcurve with monotonecurvature,that interpolatesthever-ticesandnormalsat q j andq j D 1 (Fig. 5 a). Sinceit is not obvious what type ofspiral(or pair of spirals,if thecurvehasaninflectionpoint) is mostpromisingandthecomputationof suchaninterpolatingcurvecanbeexpensive,weexploit a niceapproximationpropertyof spirals(MarciniakandPutz,1984):Giventwo planarpointsq1 andq2 with tangentvectors�t1 and �t2, let s beanarbi-trary spiral thatsatisfiesthis G1 interpolationproblem.Let c1 bethecircle definedby q1 � q2 and �t1 andc2 bethecircle definedby q1 � q2 and �t2 (suchthat �t1 and �t2 arecircle tangents).Thenthespirals canbeapproximatedby theareaenclosedby thecirculararcsof c1 andc2 betweenq1 andq2, if thetangentanglebetween�t1 and�t2doesnotchangeexceedingly(Fig. 5 b). Thisargumentis alsoreasonablefor planarcurveswith monotonecurvatureandaninflectionpoint (Fig. 5 c).

Thetangentvectors�t1 and�t2 arecomputedby intersectingthetangentplanesat theverticesq j andq j D 1 (definedby their normalvectors)with theplaneE j (Fig. 5 a),

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wherewe choosein eachcasethedirectionthathasthesmallerangleto thevectorq j D 1 � q j . After intersectingthe enclosedareawith the perpendicularbisectorofq1 andq2, we set p j � p to be the centerpoint of the intersectioninterval. Eachcircle hastwo intersectionpointswith theperpendicularbisector, herewe chooseonly thatintersectionpoint thatis closerto theline definedby q1 andq2.

7 Construction algorithm

In thissectionwefinally presenttheconstructionalgorithmfor ameshMS thatis adiscretesolutionof equation(4), i.e. it satisfies

∆BH � qi � 0 G qi � VI � MS (12)

plusanadditionalinnerfairnesscriterion.Theinputdatafor ouralgorithmconsistsof verticesand unit normalsthat form the G1 boundarycondition and an initialmeshM0 that interpolatesthe boundaryvertices.The ideaof the constructional-gorithmis to createa meshsequenceMk � k � 0 � 1 � 2 �F�H�F� by iteratively updatingthevertices,until theouterandinner fairnessconditionsaresufficiently satisfied.Al-thoughin our implementationthemeshconnectivity variesduringtheconstructionalgorithm(seesection7.4), let usfirst assumethatonly thepositionof theverticeschanges,while theconnectivity remainsconstant.

Onepossiblestrategy to transformtheinitial meshM0 into thesolutionMS wouldbe to usemotion by intrinsic laplacianof curvature(ChoppandSethian,1999),exploiting the fact that (4) describesthe equilibrium of that flow. Here, for finemeshesverysmalltimestepsareneededto achievestabilityandit is hardto decidewhatconcretetimestepvaluesshouldbeusedin anapplication.

To avoid suchproblems,in this paperwe do not usecurvatureflow by the lapla-cian of curvaturedirectly, insteadwe adaptthe ideaspresentedin (SchneiderandKobbelt,1999)to our case.In thatpaperanalgorithmfor the fastconstructionofdiscreteplanarclothoid splines– the planaranalogonto our problem– waspre-sented.The key ideaof that algorithmwasto factorizethe fourth orderprobleminto two problemsof secondorder. In combinationwith a multigrid schemeandan iterationstepthat alternatesbetweenlocal andglobal updatestrategies,an ef-ficient andreliablealgorithmfor the discretizationof planarclothoid splineswaspresented.

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7.1 Factorization

Insteadof solvinga fourth orderproblemdirectly, we factorizeit into two secondorder problemswhich are solved sequentially. The factorizationidea is inspiredby the following observation: Given a fixed Laplace-Beltramioperatorandfixedmeancurvaturevaluesat the boundaryverticesVB � Mk of a meshMk, (12) canbe interpretedasa Dirichlet problemfor the Hi , wherethe unknown scalarmeancurvaturevaluesat theinnerverticesaredeterminedby anonsingularlinearsystemwith a symmetricandpositive definitematrix S whosecoefficientsaredefinedin(6) and(7). SolvingtheresultingnonsingularlinearsystemyieldsscalarvaluesHi

at all inner verticesqi � VI � Mk , that representa discreteharmonicfunction.Theideais now to usethis calculatedscalarvaluesHi to updateeachinner vertex qi

suchthatH � qk D 1i � Hi, which is againa secondorderproblem.Expressedin two

formulas,this factorizationof Mk � MkD 1 becomes

I � ∆BHi � 0

I I � H � qkD 1i � Hi

I"JK G qki � VI � Mk ��

WedeterminetheLaplace-Beltramioperatorandtheboundarymeancurvatureval-uesby calculatingthe accordingvaluesof the currentmeshMk. In practice,it isnot necessaryto solve the Dirichlet problemexactly. Whenwe have determinedthelinearsystem,we applysomeiterationstepsof aniterative linearsolver, usingthecurrentmeancurvaturevaluesasstartingvalues.Soonestepto updatea meshMk � MkD 1 becomes:L All meancurvaturevaluesH � qk

i atall verticesof thecurrentmeshMk arecalcu-lated,themeancurvaturediscretizationtechniquedependson thevertex valenceasdescribedin section6. Also the discreteLaplace-Beltramioperatoris deter-mined,thatmeanswecalculatethecotangentweights(6) and(7) for every innervertex.L Themeancurvaturevaluesat theboundaryandthediscretizedLaplace-BeltramioperatordetermineaDirichlet problemthatcanbeformulatedasa linearsystemfor the interior meancurvaturevalues.Use the meancurvaturevaluesat theinterior verticesasinitial valuesanditeratethe linear systemn timesusinganiterative linearsystemsolver.L Thesecondstepresultsin improvedscalarvaluesHi for theinterior verticesandthis scalarvaluesareusedto updateall qk

i � VI � Mk . The updateis donein aGauss-Seidellike manner, that meansadjacentverticesthat have alreadybeenupdatedareusedwith their new position.

While discreteharmonicfunctionsdonotalwayshaveto shareall themathematicalpropertiesof their continuouscounterparts,e.g.the convex hull property(PinkallandPolthier, 1993),they will neverthelessapproximatecontinuousharmonicfunc-

15

tions. This meansour scalarvaluesHi will approximatea function that doesnothave local extremaandwhosemaximal valuesoccur at the boundary, so the Hi

will behavewell andcanbeapproximatelyboundedby thecurrentmeancurvaturevaluesat theboundaryvertices.

As in (SchneiderandKobbelt,1999),we noticedthatwe canimprove theconver-gencerate,if wemix highandlow frequency smoothingsteps.For thatpurpose,inour implementationwe usedGauss-Seidelandconjugategradient(GolubandVanLoan,1989)iterations.Both schemesenableespeciallyefficient codingif appliedon meshesandbothwould convergefor n � ∞, sincethematrix S is positivedefi-nite.WealternatebetweenameshupdateMk � MkD 1 basedonGauss-Seidelwithn � 1 andaconjugategradientupdatewheren is chosenlarger. As anoninteractivecriterionto terminatethefairingalgorithm,wecaniterateuntil MNM∆BH MNMO� ε ateveryvertex for aprescribedε.

7.2 Vertex updates

The innerverticesareupdatedqki� qkD 1

i , usingthescalarmeancurvaturevaluesHi resultingfrom theiterativesolverdescribedabove.Theaimof theupdatestepisto produceanew meshMk D 1 whosemeancurvaturevaluesH � qkD 1

i at theverticesqi � VI � Mk D 1 arecloserto thecalculatedHi valuesthanthoseof thepreviousmeshMk. In orderto beableto separatebetweeninnerandouterfairness,weonly allowthevertex to movealongthesurfacenormalvector. This meanswe searcha scalarvaluet suchthat

H � qk D 1i � Hi with qkD 1

i � qki � t �n � (13)

If wetakealook atthemeancurvaturediscretizationalgorithmpresentedin section6.1, we find that the matrix A doesnot changeif we move qi along the normalvector. Soonly theright sideof equation(11) is influencedby sucha motionandfor thenormalcurvatures(8) weobtain

κ j � 2 ' q j � qi � t �n �n (' q j � qi � t �n q j � qi � t �n ( �This normalcurvaturediscretizationis nonlinearin t, but sincethe positionof qi

will not changemuchduring theupdatestep,we assumethe distancebetweenqi

andq j to remainconstant.With this linearizationtechniqueweget

κ j P 2 ' q j � qi �n (' q j � qi q j � qi ( � 2t1' q j � qi q j � qi ( �

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Usingthis assumption,thediscretizationof H � qkD 1i determinedby (10) and(11),

becomesa linearfunctionin t. Solvingthis linearequationfor t, we finally updateqkD 1

i � qki � t �n whichapproximatelysolves(13).

7.3 Inner fairnesscondition

To controlhow theverticesaredistributedon thesurfaceof thesolution,for everyinner vertex qi � VI we assigna generalizeddiscretelaplacian∆ that determinesthe local parameterizationof the surface.Assigninga scalarweight λi j to everyq j � N � qi with theconstraint∑ j λi j � 1, thediscretelaplacian∆ is definedas

∆ � qi � � qi � ∑q j � N � qi � λi jq j �

A meshM is saidto satisfytheinnerfairnesscondition,if all ∆ � qi havevanishingtangentialcomponents,sothattherearescalarvaluesti suchthat∆ � qi � ti �ni for allqi � VI � M .We integratetheinnerfairnessconditioninto theconstructionalgorithmby includ-ing it into theupdatestepdescribedin section7.2. Insteadof updatingqk

i� qk D 1

ialongtheline qk

i � t �ni, weupdatealongtheline qki � t �ni, whereqk

i is theprojectionof ∑λi jq j ontotheplanedefinedby qk

i and �ni

qki � ∑

q j � N � qi � λi jq j � ' ∆ � qi �ni ( �ni �In ourexamplesweusedtwo differentgeneralizedlaplacians∆. If weassumelocaluniform parameterization,we arrive at a laplacianwith weightsλi j � 1

m, wheremis the valenceof the vertex qi . This laplacianleadsto meshesthat have a regularvertex distributiononthesurface.However, in someimportantcasessuchakind ofmeshparameterizationis not wanted.For examplein multiresolutionmeshmod-eling (Kobbeltet al., 1998b;Guskov et al., 1999),onewould like to have a fairedmeshthatminimizeslocaldistortionswhencomparedto theoriginalmeshM0.

For meshesit is usuallynotpossibleto getamapthatis conformalin thesensethatanglesarepreserved,but wecanusethefactthatthediscretizationof ∆B f � 0 canbeinterpretedasadiscreteharmonicmap(Pinkall andPolthier,1993;Duchampetal.,1997)andthusapproximatesacontinuousconformalmap.Soto minimizelocaldistortions,we usethe weightsresultingfrom the discretizationof the Laplace-Beltramioperatorof theoriginalmeshM0. Theweightsλi j at thevertex qi arethen

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L Createa progressive meshrepresentationof thegivenmeshandsub-divide thevertex splitsinto hierarchiesL Solve thefairingproblemon thebasemesh,

Createa smoothinitial meshusing an algorithm that can handlenoisymeshes,Solve intrinsic fairingproblemon thebasemeshL For eachhierarchylevel,Add verticesof the progressive meshrepresentationuntil the nexthierarchylevel is reachedasdescribedin section7.4,Smooththecompletemeshonthecurrenthierarchylevel. Whende-terminingthe estimatednew meancurvaturedistribution, alternatebetweenlocalandglobaliterativeschemes

- local:DeterminetheHi valuesusingoneGauss-Seideliterationandupdatethevertices

- global:Determinethe Hi valuesusingn conjugategradientit-erations(with diagonalpreconditioner)andupdatethevertices

Fig. 6. Themultigrid fairing algorithm

givenby

λi j � cotα j � cotβ j

∑q j � N � qi � � cotα j � cotβ j �whereα j andβ j arethetriangleanglesasshown in figure3.

Other promising parameterizationsare e. g. the weightedleast square and theshape-preservingparameterizationspresentedby Floater(1997).Comparedto thediscreteconformalparameterizationthey havetheadvantageto bebasedonconvexcombinationsonly.

7.4 Multigrid approach

It is well known, that the convergenceof meshfairing algorithmscanbe dramat-ically accelerated,if multigrid techniquesareintegratedinto theconstructionpro-cess(Kobbelt,1997;Guskov et al., 1999).In our implementationwe followedthisidea.An overview of thealgorithmis shown in figure6. Thenecessaryhierarchylevels are constructedusing the progressive meshapproach(Hoppe,1996) withhalf-edgecollapses(Kobbeltet al., 1998a).However, insteadof reducinga meshwhile trying to keepthedetails,we aremoreinterestedin creatinga meshwhosesmallestedgelengthis maximalwhile avoiding distortedtriangles(long triangleswith small inner circle). The numberof hierarchylevels canbe specifiedby the

18

user. A reasonablestrategy to fix thenumberof verticesperhierarchyis exponen-tial growth.

Our multigrid algorithmexploits thefact thata coarsemeshalreadyapproximatesthe shapeof the smoothsurfacethat is implicitly definedby the PDE, increasingthemeshsizemainly improvesthesmoothnessof theapproximation(Fig.2 and8).Therefore,westartwith theconstructionof adiscretesolutiononthecoarsestlevelof the progressive meshrepresentationand then eachsolution on a coarselevelservesasstartingpoint for theiterationalgorithmon thenext finer hierarchylevel.Betweentwo hierarchylevelsweneedaprolongationoperatorthatintroducesnewverticesusingthevertex split informationof theprogressivemesh.Whenaddinganew vertex qi , we have to take carethat theouterfairnessis not destroyedat thatposition.This is achievedin threesteps,wherethefirst two stepsaresimilar to theprolongationoperatorusedin (Guskov et al., 1999):L First we updatethemeshtopologyandintroduceqi at thepositiongivenby its

innerfairnesscriterion

qi � ∑q j � N � qi � λi jq jL In somecasesthefirst stepis not enoughto avoid triangledistortions,therefore

in the secondstepwe further updatethe complete1-ring of qi . This meanswesolvethelocal linearproblem∆ql � 0 for all ql � D � qi .L Sincethesecondstepdisturbstheouterfairness,we finally solve ∆BHl � 0 forall ql � D � qi by applyingtheconstructionalgorithmlocally onthe1-diskD � qi .

Our meancurvaturediscretization(section6) aswell as the updatestep(section7.2) assumethat the meshis not a noisy surface.Therefore,beforestartingthemultigrid algorithmwefirst constructat thecoarsestlevel thelinearsolutionof theproblem∆2 f � 0 usingthelaplaciandefinedby thechoseninnerfairnesscriterion(seeFig. 1 c). We further developedan improved bilaplacianmeshfairing algo-rithm (Schneideret al., 2000b)thatproducesaninitial meshwith superiorqualitythan thoseresultingfrom linear schemes.For sparsemesheswith very irregularstructure,this schemeis muchbettersuitedto createaninitial mesh(Fig. 7). Laterat eachhierarchylevel our meshis alreadypresmoothedby the multigrid fairingconcept.

7.5 Examplesandremarks

Becauseof the importanceof the discretemeancurvatureat the boundaryver-ticesfor the constructionalgorithmandthe fact that at suchpointswe only haveone-sidedvertex information,in our exampleswe calculatedthe meancurvatureat every boundaryvertex usingthespecialcasediscretizationasfor innervertices

19

Fig. 7. Comparisonof theresultsfrom a linearbilaplacian(left), our improvedbilaplacian(middle) andour intrinsic method.In the upperrow the meshhasa regular subdivisionconnectivity structureandin thelowerrow it is sparseandhasavery irregularconnectivity

of valence3 or 4. For coarsemeshes,we alsousedthe specialcasehandlingforinterior verticesof highervalence.

Insteadof updatingqk D 1i � qk

i � st �n with s � 1 � 0 asdescribedin section7.2, inpractices hasto lie in therange0 � s � 1 � 0. While s � 1 � 0 usuallywill work well,we noticedthat in somerareconstellationsit is possiblethatverticescanalternatebetweentwopositions.All of ourexamplespresentedin thispaperwereconstructedusings � 0 � 9.

Theconformalinnerfairnessis only hardlymoretimeconsumingto constructthantheuniform parameterization,but requiresmuchmorememory. Heretheoriginalmeshhasto bereconstructedfrom theprogressive meshrepresentationparalleltothe fairedmeshto be ableto updatethe necessarycoefficientsof the laplacian–whichalsohave to bestoredin memory– at everyhierarchylevel.

Thelinesof reflectionin ourexamplepicturesindicateG1 continuityat thebound-ary (continuouslines)andat leastG2 continuity in the interior (smoothlines).Ascanbeseenin examples9 and10theboundarydoesnothaveto besmooth.In suchcasesthefixedunit normalinformationat theboundaryis derivedby averagingtheadjacenttrianglenormalvectorsbeforetheconstruction.

Theconceptbehindourconstructionalgorithmworksbestfor ∆BH � 0,but it is notrestrictedto that.WealsoconstructedMESsby solvingtheirEuler-Lagrangeequa-tion ∆BH � � 2H � H2 � K andsurfacessatisfying∆BH � const. However, theseinhomogenproblemsaremorecostly to solve anda stableconstructionalgorithmis moreinvolved.

In this paperwe assumedthat thefinal meshshouldhave a prescribedvertex con-nectivity. If this is not thecase,flipping of edgesshouldbeenabledto avoid longtriangles,new verticesmaybe introducedwherenecessaryor removedwherethevertex densitybecomestoohigh (seeHsuet al., 1992;WelchandWitkin, 1994).

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Table1Runningtimesof theintrinsic fairing algorithm

Dataset Vertices Triangles Vertices Time

completemesh completemesh basemesh sec.

cylinderblend 920 1746 140 1.9

bustface 2926 5720 256 12

humanear 4665 9220 248 21

tetrathing 13308 26624 308 57

(a) (b) (c)

Fig. 8. 6 circlesareusedto definea tetrathing.a) andb) show thesolutionof a meshwith500resp.13308vertices.c) showsthereflectionlinesof themeshb). Dueto thesymmetricconstellationthefinal smoothsolutionwouldbeG2 continuous.

(a) (b) (c)

Fig. 9. Featureremoval of ameshwith userdefinedboundarycurve.

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All shadedpicturesareflat shadedto enhancethefacetingeffect.Theconstructionalgorithmwasimplementedin Java1 � 3 andJava3D for WindowsNT runningonaPentiumIII with 500MHz.We noticedthatour fairing algorithmis very fastfor afairing approachbasedon intrinsics.Therunningtimesfor theexamplesshown inthis paperarepresentedin table1, they donot includethetime neededto build theprogressive meshrepresentation.An optimizedC implementationwould improvethesetimingswithoutdoubt.

8 Conclusion and future work

The techniquewe presentedin this papershows that it is possibleto usehigherorder intrinsic PDEsin modelingbasedon irregular meshes.Sincethe surfaceiscompletelydefinedby G1 boundaryconditionsanda PDE, the designerdoesnothaveto takesurfaceparameterizationor theactualmeshrepresentationinto accountandcanfreelydistributetheverticeson thesurface.

Oneof our futureplansis to integratethepresentedtechniquein a multiresolutionmeshmodelingtool asthosedescribedin (Kobbeltetal.,1998b).Althoughthenewconstructionalgorithmis obviouslyslower thanthesimplelinearfairingalgorithmusedthere,basedon theresultswe got from our currentimplementation,it seemspossibleto achieve interactive framerateswith a decentmeshcomplexity. In com-binationwith an inner fairnesscriterion thatenableslocal shapepreservation,thiswould leadto amuchsimplerdetailcoding.Becauseof theintrinsicsurfacedefini-tion, thiswouldalsosimplify ameshmodelingthatdoesnotfix theconnectivity orthemeshsizeduringthemodifications.

We will also explore other curvatureand updatealgorithmsand study differentintrinsicPDEsto introducenew designfreedoms.Furthermore,it will benecessaryto integratesurfaceconstraintsin theinteriorof thesurfacesuchasinterpolationofpoints,which might requirePDEsof higherorder.

Acknowledgements

We thankJensVorsatzandMario Botschfor their substantialcontributionsto thecreationof ourexamplesandalsoKolja Kahlerfor providing theprogressivemeshalgorithm.

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(a) (b)

(c) (d)

Fig. 10. Herewe smoothedthe faceof a bustmodelshown in a). The triangulationof thesolutionin thisexampleis discreteconformalto theinitial mesh.

References

Bloor, M. I. G., andWilson, M. J. (1990),Using partial differentialequationstogeneratefree-formsurfaces,Computer-AidedDesign22,202–212.

23

Brakke,K. A. (1992),TheSurfaceEvolver, ExperimentalMathematics1 (2), 141–165.

Burchard,H. G.,Ayers,J.A., Frey, W. H. andSapidis,N. S.(1994),Approximationwith AestheticConstraints,in: Sapidis,N. S., ed.,DesigningFair CurvesandSurfaces, SIAM, Philadelphia.

do Carmo,M. P. (1993),Dif ferentialGeometryof CurvesandSurfaces, Prentice-Hall, Inc EnglewoodClif fs, New Jersey.

Chopp,D. L., andSethian,J.A. (1999),Motion by IntrinsicLaplacianof Curvature,InterfacesandFreeBoundaries1, 1–18.

Desbrun,M., Meyer, M., Schroder, P. andBarr, A. H. (1999),Implicit fairing ofirregularmeshesusingdiffusionandcurvatureflow, SIGGRAPH’99 ConferenceProceedings,317–324.

Desbrun,M., Meyer, M., Schroder, P. andBarr, A. H. (2000),DiscreteDifferential-GeometryOperatorsin nD, submittedfor publication.

Duchamp,T., Certain,A., DeRose,T. andStuetzle,W. (1997),Hierarchicalcompu-tationof PL harmonicembeddings.Technicalreport,Universityof Washington.

Floater, M. S. (1997),Parametrizationandsmoothapproximationof surfacetrian-gulations,ComputerAided GeometricDesign14,231–250.

Golub, G. H. andVan Loan, C. F. (1989),Matrix Computations, JohnsHopkinsUniversityPress,Baltimore.

Greiner, G. (1994), Variationaldesignand fairing of spline surfaces.ComputerGraphicsForum13(3),143–154.

Guskov, I., Sweldens,W. andSchroder, P. (1999),MultiresolutionSignalProcess-ing for Meshes,SIGGRAPH’99 ConferenceProceedings,325–334.

Hoppe,H. (1996),Progressive meshes,SIGGRAPH’96 ConferenceProceedings,99–108.

Hsu,L., Kusner, R., andSullivan,J. (1992),Minimizing the SquaredMeanCur-vatureIntegral for Surfacesin SpaceForms,ExperimentalMathematics1 (3),191–207.

Kobbelt,L. (1997),Discretefairing, Proceedingsof theSeventhIMA Conferenceon theMathematicsof Surfaces,101–131.

Kobbelt,L., Campagna,S. andSeidel,H.-P. (1998a),A GeneralFramework forMeshDecimation,Proceedingsof theGraphicsInterfaceconference,43–50.

Kobbelt,L., Campagna,S.,Vorsatz,J.andSeidel,H.-P. (1998b),InteractiveMulti-ResolutionModeling on Arbitrary Meshes,SIGGRAPH’98 ConferencePro-ceedings,105–114.

Marciniak,K., andPutz,B. (1984),Approximationof spiralsby piecewisecurvesof fewestcirculararcsegments,Computer-AidedDesign16,87–90.

Meek,D. S. andWalton,D. J. (2000),On surfacenormalandGaussiancurvatureapproximationsgivendatasampledfrom asmoothsurface,ComputerAidedGe-ometricDesign17,521–543.

Moreton,H. P. andSequin,C. H. (1992),Functionaloptimizationfor fair surfacedesign,SIGGRAPH’92 ConferenceProceedings,167–176.

Ohtake, Y., Belyaev, A. G. and Bogaevski, I. A. (2000), PolyhedralSurfaceSmoothingwith SimultaneousMeshRegularization,GeometricModeling and

24

Processing2000Proceedings,229–237.Pinkall,U. andPolthier, K. (1993),Computingdiscreteminimal surfacesandtheir

conjugates,ExperimentalMathematics2 (1), 15–36.Schneider, R. and Kobbelt, L. (1999), DiscreteFairing of Curves and Surfaces

basedon linearCurvatureDistribution,Curve andSurfaceDesign- Saint-Malo’99 Proceedings,371–380.

Schneider, R. andKobbelt,L. (2000a),GeneratingFair Mesheswith G1 BoundaryConditions,GeometricModelingandProcessing2000Proceedings,251–261.

Schneider, R., Kobbelt,L. andSeidel,H.-P. (2000b),ImprovedBilaplacianMeshFairing,submittedfor publication.

Taubin, G. (1995a),A signal processingapproachto fair surface design,SIG-GRAPH’95 ConferenceProceedings,351–358.

Taubin,G. (1995b),Estimatingthetensorof Curvatureof asurfacefrom apolyhe-dralapproximation,ICCV ’95 Proceedings,902–907.

Weimer, H. andWarren,J. (1998),Subdivision Schemesfor Thin PlateSplines,ComputerGraphicsForum17,303–314.

Welch,W. andWitkin, A. (1994),Free-Form shapedesignusingtriangulatedsur-faces,SIGGRAPH’94 ConferenceProceedings,247–256.

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