Curved Folding Martin Kilian TU Vienna Evolute Simon Fl ory TU Vienna Evolute Zhonggui - PDF document

CurvedFoldingMartinKilianTUViennaEvoluteSimonFlTUViennaEvoluteZhongguiChenTUViennaZhejiangUniversityNiloyJ.MitraIITDelhiAllaShefferHelmutPottmannTUVienna Figure1:Topleft:ReconstructionofacarmodelbasedonafeltdesignbyGregoryEpps.Close-upsofthehoodandtherearwheelhouseareshownontheleft.Thefoldlinesarehighlightedonthecar'sdevelopment.Toprightandbottom:Architecturaldesign.Allshownsurfaces 1IntroductionDevelopablesurfacesappearnaturallywhenspatialobjectsareformedfromplanarsheetsofmaterialwithoutstretchingortear-ing.Papermodelssuchasorigamiartareprominentexam-ples.Thestrikingeleganceofmodelsfoldedfrompaper,suchasthosebyDavidHuffman[Wertheim2004],arisesparticularlyfromcreasesknownascurvedfolds.Suchfoldscanbegener- Figure2:ThecarmodelofFigure1anditsdevelopment(topright).Thepatchdecompositionintotorsalruledsurfacesisshownusingthefollowingcolorscheme:planesareshowninyellow,cylindersingreen,conesinred,andtangentsurfacesinblue.Sam-plerulingsareshownonsomepatchesofthewindshieldandthesidewindow.SuchasegmentationisessentialforNURBSsurfacettingandmanufacturing.surface.Developablesurfacesarecomposedofplanarpatchesandpatchesofruledsurfaceswiththespecialpropertythatallpointsofarulinghavethesametangentplane.Suchtorsalruledsurfacesconsistofpiecesofcylinders,cones,andtangentsurfaces,i.e.,theirrulingsareeitherparallel,passthroughacommonpoint,oraretan-genttoacurve(curveofregression),respectively.Whereasatorsalruledsurfacehasonlyonecontinuousfamilyofrulings,generalsmoothdevelopablesurfacesareusuallyamuchmorecomplicatedcombinationofpatches.Thepresenceofplanarpartsisthemainsourceofthishugevarietyofpossibilities.Thelevelofdifcultyisfurtherincreasedifoneadmitscreases,i.e.,curvedfolds(seeFig-ure2).geometricdesign,variouswaysoftreatingdevelopabilityhavebeenpursued:asaconstraintintensorproductB-splinesurfacesofdegree[ChuandSequin2002;Aumann2004],aimingonlyatapproximatedevelopability[PerezandSuarez2007],view-ingthesurfacesassetsoftheirtangentplanes[PottmannandWall-ner2001],subdividingstripsofplanarquads[Liuetal.2006],orcomputingwithtrianglemeshesandalocalconvexityconstraint[Frey2004;WangandTang2004;SubagandElber2006;Wang2008].BoandWang[2007]modelpaperstripsasrectifyingdevel-opablesofoneoftheirgeodesics.Digitalreconstructionoftorsalruledsurfacesemployingaplane-geometricapproachisthetopicof[Peternell2004].Meshparametrizationandsegmentationusingdevelopablesurfaceshasbeeninvestigatedin[Juliusetal.2005]and[Yamauchietal.2005].Roseetal.[2007]showhowtocomputedevelopablesur-facesfromboundarycurves,andpresentastrategyforselectinganoptimalsolution.Severalalgorithmshavebeenproposedfortheconstructionofpapercraftmodels[MitaniandSuzuki2004;Mas-sarwietal.2006;Shatzetal.2006]usingfoldsalonglineseg-ments.Inallthesepapers,trianglemeshesareusedtorepresentdevelopablesurfaces.Onlyafewcontributionsdealwiththedifcultanalysisandcompu-tationofcreasesindevelopablesurfaces.Mostofthemconcentrateonconicalcreases[Kergosienetal.1994;Cerdaetal.1999;Frey2004].Startingfromconicalfolds,Cerdaetal.[2004]investigatedgravity-induceddrapingofnaturallythinatsheets.Contributionsandoverview.Wepresentanoptimization-basedcomputationalframeworkforthedesignandreconstructionofgen-eraldevelopablesurfaceswithastrongfocusoncurvedfoldingap-plications.Ourmaincontributionsareasfollows:Weemployquadmesheswithplanarfacesasadiscretedifferen-tialgeometricrepresentationofdevelopablesurfaces,andforthisrepresentationintroducenewwaysofcomputingcurvaturesandbendingenergy.Moreover,wediscusscurvedfoldsfromthedis-creteperspective(Section2).InSection3,weintroducethecoreofourwork,anovelopti-mizationalgorithmwhichallowsustocomputedevelopablesur-faceswithcurvedfoldsthatareisometrictoagivenplanarsheet,whileatthesametimeachievingadditionalobjectivessuchasap-proximationofgivengeometricdata,aesthetics,andminimizationofbendingenergy.Section4presentsvariouswaysinwhichthebasicoptimizationalgorithmcanbeusedfordesign.Evensimpleapplicationsleadtonewresults,suchasmodelingdevelopablestripsofminimalbend-ingenergy.Anothermaincontributionofourworkisdigitalreconstructionofobjectsexhibitingdevelopablesurfaceswithcurvedfolds.Algo-rithmsforpreprocessingtheinputdatainordertomakethealgo-rithmofSection3applicabletodigitalreconstructionarediscussedinSection5.Combiningdigitalreconstruction,optimizationandrecentlyin-troducedalgorithmsforcomputinginshapespace[Kilianetal.2007]wehavearichtoolboxforgeometryprocessingwithcurvedfolds.ThisisdemonstratedbymeansofafewapplicationscenariosinSection6.Finally,wesummarizeourmainresults,andaddressdirectionsforfutureresearchwithinthelargelyunexploredareaofcurvedfolding.2DiscretedevelopablesurfacesDevelopablesurfaces.Asourbasicrepresentationofdevel-opablesurfacesweemployquad-dominantmesheswithplanarfaces,whichisalsotherepresentationofchoicefordiscretedif-ferentialgeometry[Sauer1970;BobenkoandSuris2005].Astripofplanarquadrilaterals(Figure3,left)isadiscretemodelofatorsalruledsurface.Sucha`PQstrip'canbetriviallyun-foldedintotheplanewithoutdistortions.Theedgeswheresucces-sivequadsjointogethergiveusthediscreterulings.Ingeneraltheyformtheedgelinesoftheregressionpolyline;:::;inspecialcasesthediscreterulingsareparallel,orpassthroughaxedpoint.Arenementprocesswhichmaintainsplanarityofquadsgenerates,inthelimit,atorsalruledsurface(Figure3,right).Itsrulingsarethelimitsofthediscreterulings,whichingeneralaretangenttotheregressioncurve,andinspecialcasesareparallel(cylinder),orpassthroughaxedpoint(cone).TherepresentationofdevelopablesurfacesasPQstripsprovidesvariousadvantagesovertrianglemeshes:(i)developabilityisguar-anteedbyplanarityoffacesandthedevelopmentiseasilyobtained,(ii)subdivisionappliedtoPQstripsprovidesasimpleandcompu-tationallyefcientmulti-scaleapproach[Liuetal.2006],(iii)theregressioncurve–whichissingularonthesurfaceandthusneedstobecontrolled–ispresentinadiscreteform,and(iv)thecurvaturebehaviorcanbeeasilyestimatedasshownnext. Curvaturesandbendingenergy.Therulingsonasmoothde-velopablesurfaceconstituteonefamilyofprincipalcurvaturelinescorrespondingtovanishingprincipalcurvature.Thesecondfamilyisgivenbytheorthogonaltrajectoriesoftherulingswhichinte-gratethedirectionsofnon-vanishingprincipalcurvatures.Weareinterestedinadiscretedenitionof,andtherelatedbendingenergydATherulingsonaPQstriparegivenbytheedgelines:=(cf.Figure4a).Asadiscreteprincipalcurvaturelinewetakeapoly-withverticeswhoseedgesareorthogonaltotheinnerbisectors.Inotherwords,eachedgeofintersectsconsecutiverulingsatthesameangle(thisdenitionisalsomotivatedbyananalogousdenitioninthecontextofcircularmeshes[BobenkoandSuris2005]).Wewanttoattachasurfacenormalvectortotheedgemidpoint,andtakethenormalvectoroftheplanespannedbyedges;Rforthatpurpose.TheunitvectorsformtheGaussianimageandthusitisnaturaltodenetheprincipalcurvature):= Thisdenitionhastheadvantagethatthedenominatorcanbecomputedintheplanardevelopmentofthestrip,whileonlythe:=requirestheembeddinginspace.Notethatthecurvaturealwayshasapositivesign,incontrasttousualdenitions.Asweuseditinitssquaredformonly,thisdoesnotmatter.UsingthenotationofFigure4thediscretebendingenergydAofaregionboundedbytwodiscreteprincipalcurvatureatdistance:=andtwobisectors(depictedbythebrownhighlightinFigure4a)isgivenasTheweightassociatedwiththerulingisgivenby(lnlnisthedenominatorofEquation(1).As,inthelimit,weget.ThusthebendingenergyofageneralPQ Figure3:APQstrip(left)isadiscretemodelofadevelopablesur-(right).Theintersectionsofedgesofadjacentplanarquadsgeneratetheregressionpolyline.Inthelimitofarene-mentprocess,thisregressionpolylinebecomestheregressioncurve.Polylines,whoseedgesintersectinnerbisectorsofconsecutivediscreterulingsatrightangles,arediscreteversionsofprincipalcurvaturelines,andserveforthedenitionofdiscretecurvatures.Theunitnormalstoplanarquadsaredenotedby Figure4:(a)TherulingsofaPQstriparegivenbytheedge:=.Theinnerbisectorofisdenotedby.Edgesofthepolylineintersectorthogonally.Ifisthemid-pointof,andisthenormaltotheplane:=,thentheprincipalcurvatureatisnaturallyde-nedbyEquation.(b)Toavoidinnitecurvaturesatconever-tices,computationtakesplaceonaslightlyshrunkenstrip.(c)Thetotalbendingenergyofaregion(brown)boundedbytwoprincipalcurvaturelinesisgivenbyEquationsstripcanbesimplyapproximatedbyasumofenergiesofthetype(2).Notethatwecannotdirectlyapplyknownformulaefordiscretebendingenergies[Desbrunetal.2005]sincethoseassumethatalledgelengthstendtozeroifonepassestothelimit.Thisisnotthecasefortherulingsinourapproach.Curvedfolds.Inthesmoothsetting,thefollowingfactaboutcurvedfoldsiswellknown(seee.g.[Huffman1976]):Ateachpointofafoldcurve,theosculatingplaneofabisectingplaneofthetangentplanesoneithersideofthefold.Thisfol-lowsimmediatelyfromtheidenticalgeodesiccurvaturesofthefoldcurvewithrespecttothetwoadjacentdevelopablesurfaces.Hence,giventhesurfaceononesideofafoldcurve,wecancompute(partof)theotherastheenvelopeofplanes,obtainedbyreectingthetangentplanesabouttheosculatingplanesofThisisdiscussedinsomedetailin[PottmannandWallner2001],butonendsonlythatpartofwhoserulingsmeet.Thus,theapproachisnotsufcientformostofourtaskswhere,inaddition,multiplefoldsmayappear,andthelocationsofsuchfoldcurvesonlybecomeknownintheprocessofoptimization.Incontrastto xxFigure5:TwoPQstripsmeetingatadiscretecurvedfold.ForagivenPQstrip,thediscreterulingoftheadjacentstripinapointliesonaquadraticconethesmoothsetting,inthediscretecasetherearemoredegreesoffreedominchoosingthesurfaceasdescribednext.Asadiscretemodelofagen-eraldevelopablesurfaceweuseaquad-dominantmeshwithplanarfaces,wherethesumofinneran-glesateachvertexisequaltoThismeansthatwehaveabijec-tivemappingtoaplanarmeshofthesamecombinatoricssuchthatcorrespondingfacesareisomet-SupposeacurvedfoldappearsasacommonpolygonoftwoPQ;DonthemodelGiventhestripononesideof ,weaskaboutthedegreesoffreedominchoosingtheadjacent.Clearly,wehavetochoosethediscreterulingsinthateachvertexexhibitstheanglesum.Let:=:=(Lettheanglesumof(seeFigure5).Then,thediscreterulingvectormustformtheanglesum)=2.With:=cos,thisreads:)=0Hence,therulingshavetolieonaquadraticcone.Notethattherulingofalsoliesonthisconesinceitsstraightextensionsatisesourrequirement,thoughitdoesnotdescribeasurfacewithacurvedfoldbutasmoothextension.Toobtain,wemaytakeonerulingontheconeandcomputefurtherrulingsatverticesbykeepingplanarityofconsecutiverulings.However,mostofthesesolutionswillnotbesuitablesincetheydonotdiscretizeasmoothsurface.Onehastotakerulingswhichyield`small'direc-tionalchangeswhenpassingfromoneconetothenextone.Thisnecessitatesanoptimizationapproachasdescribednext.3ThebasicoptimizationalgorithmThebasicoptimizationalgorithmoptimizesadis-cretedevelopablesurfaceanditsplanardevelopment.Tomaintainisometrybetweencorrespondingfacesof,weoriginallyletbeaquad-dominantsoupofplanarpolygonsspace.Thesepolygonsareisometrictothecorrespondingfacesintheplanarmesh,seeFigures6and7.Duringtheoptimization,thepolygonsoupwillbecomeameshviaaregistrationproce-durewhichbearssomesimilaritytothatusedinthePRIMOmeshdeformationtool[Botschetal.2006].However,ouroptimizationrequiresmoresophisticationsincewehavetosimultaneouslyopti-mizethedevelopmentwhilesatisfyingvariousotherconstraints.Optimizationstartswithaninitialsetofpairs;Pofisometricplanarpolygons(primarilyquadsinoursetting).Thefacesaplanarmesh,whileinspacethecorrespondingpolygonsareassumedtoroughlyrepresentadevelopableshape.Theyarenotyetpreciselyalignedalongedges.Thusisnotameshbutapolygonsoup.Later,inSections4and5,wedescribehowtocomputeinitialpositionsfordifferentapplications.Theunknowns.WeintroduceaCartesiancoordinatesystemintheplaneof,withoriginandbasisvectors.Eachfaceiscongruenttotherespectivefaceinspace.Foreachsuchface,theimageofundertheisometrictransformationisaCartesianframeintheplaneoftheface.If;parethecoordinatesofavertex,thenthecorrespondingvertexDuringtheoptimization,theframesundergoaspatialmotion,andthecoordinates;pcanalsovarysinceweallowthepolygonstochange.Welinearizethespatialmotionofanyfaceusinganinstanta-neousvelocityvectoreld:Thevelocityofapointcanberepre-sentedas):=,wherearevectorsin3-space. Figure6:Basicsetupfortheoptimizationwhenareferencesurfaceisused.Faceswiththesamecolorarecongruent. Figure7:Topleft:Initialpolygonsoup.Topright:Develop-.Bottomleft:aftersubdivisionandoptimization.Bottomafterthreeroundsofsubdivisionandoptimization. Figure8:Stabilityoftheproposedoptimizationstrategy.Afterar-ticialperturbationoffaces(left),roundsofoptimizationyieldanalmostalignedpolygonsoup(right).Thestabilityofthepro-cedureallowsustouseroughestimatesofrulingdirectionsandplanardevelopmenttoinitializethealgorithmThusavertexofthedisplacedquadfaceisgivenby:Thenewvertexpositionislinearintheunknownparametersofthevelocityeld,andalsolinearintheunknown;p.Weoptimizeoverthevelocityparametersandthecoordinates.Theproductsresultinnon-lineartermsifweinsistonsimultaneouslyoptimizingthem.Toavoidnonlinearoptimization,wealternatelyoptimizefordisplace-andforvertexcoordinates;p.Sinceourobjectivefunctionisquadraticinbothtypesofunknownsthisamountstoalternatelysolvingtwosparsesystemsoflinearequations.Applyingdisplacementscorrespondingtodestroystheexactisometricrelationbetweencorrespondingfaces.Itisthereforenecessarytofurthermodifytheverticesof.ThiscaneitherbedonebyrigidregistrationofthefacetotheestimatedvertexlocationsasproposedbyBotschetal.[2006],orbyusingahelicalmotionasdescribedin[Pottmannetal.2006]–weusetheformerapproach.Theobjectivefunction.Ourobjectivefunctionisdesignedtosimultaneouslyensurethatbecomesamesh,tstheinputdata,andsatisestheaestheticrequirementsoftheapplication. Ifavertexintheplanarmeshissharedbyfaces,thencor-respondstodifferentvertices;:::;ofthecorrespondingfacesin.Sincetheseverticesshouldagreeinthenalmesh,weuseavertexagreementtermoftheform:vertwherethesumextendsoverallcombinationspervertexandoverallverticesinFortoapproximateanunderlyingdatasurface,weincludeattingtermwhichisquadraticinthevertexcoordinates.Letdenotetheclosestpointin,andletdenotetheunitnormalattotheunderlyingsurface.Weusealinearcombina-tionofthesquareddistanceandthesquareddistancetothetangentplaneplane(m�mc)
nc]2asthedatattingterm.Whenttingcurves,especiallynearboundaries,weusetangentlinesin-steadoftangentplanes.Finally,weneedafairnesstermfair.ForeachpairofadjacentofthePQstrip,weusethediscretebendingen-ergyofthecorrespondingdevelopablesurface,asgivenbyEquations(2)and(3),asthefairnessterm.Thenormalofaquadisgivenby.Undersmalldisplace-ments,thisnormallinearlyvariesas.Givenapolyline;:::representingafoldline,i.e.,acreaseorasegmentofaboundarycurve,thecontributiontofairisasumofsquaredseconddifferences.FairnesstermsarealsoappliedtotherespectivepolylinesintheplanardomainThefairnesstermfairaloneisnotalwayssufcienttomaintainconvexquads,andtopreventipsintheplanarmesh,espe-ciallywhenthequadsbecomethinafterseveralstepsofsubdivi-sion.Henceweaddanothertermconvtoenforceconvexity.Weassumethattheorientationofeachfaceofcoincideswiththeorientationoftheplaneinducedbytheframe.Acornerofaplanarpolygonisconvexifandonlyiftheorientedareaofthetriangleispositive.Thistermalsopreventsippingoffaces.Thealgorithm.Combiningallindividualterms,ourbasicopti-mizationproblemreads,vertfairsubjecttoconvWealternatelyminimizetheobjectivefunctionovernewpositionsofverticesin,anddisplacementsoffacesinspace,i.e.,velocityvectorsforthecorrespondingfaceplanes.Observethattheweightsfair,whichonlydependontheplanarmesh,remainxedwhenoptimizingfordisplacementsoffacesinspaceandthesideconvisalsonotneeded.Hence,thespatialsub-problem E1E1E1E1E1E1E1E1E1E1E1E1E1E1E1E1E1E2E2E2E2E2E2E2E2E2E2E2E2E2E2E2E2E2 Figure9:Basicsetupforbendingenergyminimization.Westartwitharegulargrid(left).Afterprescribingpointlocationsandtangentplanesattheboundarythebasicoptimizationisapplied.(Right)Theresultafteroneroundofoptimization. Figure10:Resultsofbendingenergyminimizationfordifferentboundaryconditions.Givenuserconstraints,thenalmodelsareobtainedbyalternatelyoptimizingandsubdividing.amountstosolvingasparselinearsystem,andsubsequentapplica-tionofthecorrespondingrigidbodymotionperface.Optimizingthedevelopmentismoreinvolvedsincetheweightsinanonlinearwayasthegeometryofchanges.Additionallywehaveaquadratictermconvtomaintainconvexityasasidecon-straint.Withthemeshesscaledtotinsideaunitcube,wefound=1=10tobegoodvaluestostarttheoptimization.Givenaninitialmeshandapolygonsoupthatroughlyap-proximatesadevelopableshape,wealternatelyoptimizefor.Theoptimizationterminateswhenthevertexagreementtermfallsbelowagiventhreshold.Forthenextrenementlevel,wesub-dividethecurrentmesh,andmapthenewfacestospaceusingtherigidtransformationassociatedwiththefacesofatthecur-rentlevel.Therenementprocesssplitseachquadoftoformtwonewones.Splittingisperformedalongtheedgesthatdonotcorrespondtorulingdirections(seeFigure3,right).Theprocessisrepeateduntildesiredaccuracyisreached.4ApplicationstosurfacedesignInthissectionweemploythebasicoptimizationalgorithmtothedesignofobjectswithcurvedfolds.Developablesurfaceswithminimalbendingenergy.Asasimpleapplicationofourframework,withoutanyrelationtocurvedfoldingyet,weallowtheusertotakeaplanarstripofpaperandat-tachittosomepointsand/orlinesinspace.Theresultingshapeiscomputedusingabendingenergyminimization,aspopularlydoneforsplinecurvesanddoublecurvedsurfaces.OurapproachextendsthepapermodelingtoolofBoandWang[2007].Sincerulingdirectionsareunknown,weinitializeoptimizationfromasoupofcongruentquadrilateralsasshowninFigure9.Theusercanprescribenewlocationsfortheboundaryedgesaswellasthetangentplanesattheseedges,i.e.,theplanesoftheoutermostquads.WeobtaintheresultingshapebyminimizingvertfairFigure10showsseveralresultsobtainedusingourmodelingtool.Inallcases,thenalmaximalvertexdisagreementislowerthan(withthemodelsscaledtotaunitbox). Figure11:Modelingcurvedcreases.(Left)Creasecurvesarespeciedbytheuseronadevelopablesurface.(Right)Theresultingshapeobtainedwiththecurvedfoldsalongtheprescribedcurves.Bendinginthepresenceofacurvedfold.Ifasmoothdevel-opablesurfacealongwiththelocationofafuturefoldcurveonitisspecied,theshapeofthefoldeddevelopableisdeter-mined(uptothosepartswhoserulingsdonotintersectthefoldcurve).Thisisbecausetangentplanesonthetwosidesofthefoldarebisectedbytheosculatingplanesofthecurve(cf.Section2).However,nosuchuniquenesspropertyexistsforthediscretecase(seeFigure11).BymarkingthelocationofafoldonaPQstripwithnewvertices;:::ontheedges,wesegmenttheorig-inalstripintotwostrips,.Thereare,intheory,manypossiblestripssuchthatformacurvedfold.Weuseouroptimizationframeworktolteroutasolution.Moreover,minimizationofbendingenergyandfairnesstermsallowsustoalsocomputepartsofthesurfacewhoserulingsdonotintersectthefoldcurve.Foreachmarkedvertex,weapproximatethediscreteosculatingplaneofthepolyline;:::bytheplanespannedbytheedges(seeFigure5).Wealsoattachanosculatingplanetoeachedge,namelythebisectoroftheosculatingplaneatitsendpoints.Inordertoconstructthefaceofadjacenttothe,theplaneofcorrespondingfaceinisreectedabouttheosculatingplaneassociatedwiththatedge.Byintersect-ingneighboringmirroredplanesweestimatetherulingsof.Toensureavertexanglesumof,weprojecttheseestimatedrul-ingsontotheirrespectivecones,givenbyEquation(4).Fromtheseprojectedrulings,wegenerateamesh,whichmaycontainnon-planarfacesatthisstage.Thismeshisthenusedtoinitializeouroptimization,andalsoasthereferencesurfaceforthetermAtypicalmodelingresultobtainedusingthisprocessisshowninFigure11.5ApproximationalgorithmDesigninganobjectwithcurvedfoldswhenstartingfromscratchisnoteasy.Suchataskcanbedauntingevenforexperiencedusers,speciallyinpresenceofmultiplecurvedfolds.However,itismucheasierandintuitiveforadesignertobuildaroughshapeusingpa-perorsimilarmaterials.Themodelcanthenbescannedandap-proximatedusingourapproach.Subsequently,theusercaneditortweakthedigitalmodelusingtheproposeddeformationtools(seeFigure14).Duringthisprocesswealsoobtainaprecisesegmenta-tionofthemodelintoplanes,cylinders,cones,andtangentsurfaces(seee.g.Figure2).SuchaclassicationisusefulforNURBSttingandmanufacturing.Thereforeweaddressthefollowingproblem:Givenscanneddatarepresentinganalmostdevelopablesurface,tthedatawithanexactlydevelopablesurfacewhichmayexhibit(multiple)curvedfolds.InordertoinitializetheoptimizationframeworkdescribedinSec-tion3,werequirethefollowing:(1)Aplanardevelopmenttheinputdata,(2)estimatesoftherulingdirectionson,(3)aquad-dominantdecompositionofand(4)acorrespondingpoly-gonsoupwhichliesclosetoinspace.PlanardevelopmentofUsingtheconstraineddeformationtoolbyKilianetal.[2007],wederiveanas-isometric-as-possiblebetweenthedatameshandaplane,thusobtain-inganapproximatedevelopmentof.Thisgeneraltoolhan-dlesnear-isometricdeformationsunderconstraints–inourcase,theconstraintisthattheimagepointsmusthavezeronate.Suchaprocedure,unlikeparameterizationapproaches[Shef-feretal.2006],providesuswithasequenceofintermediatemeshesanditsplanardevelopment.Thisadditionalinformationisusefulfortrackingpersistentridgelinesorcurvesduringunfold-ingwhichareusedtoinitializecurvedfoldlocations.EstimatingrulingdirectionsonWerstestimateapproxi-materulingdirectionsonthegivendatameshasfollows:StageA:Ateachvertex,wecomputeageodesiccirclethesetofpointswhichareatconstantgeodesicdistance.Theradiusischosenastheminimumdistancetothemeshboundariesandallfeaturelines.Weuseridgelines[Ohtakeetal.2004]asinitialguessforcurvedfolds,andmarkthemasfeaturelines.Pointswithradiismallerthanathresholdareignored.Wecomputeascoreforpoints,as:):=[0denotetheunitnormalvectorsatrespectively.Inourexperimentsweuse.Typicallytherearetwostrongmaximaalongdiametricallyoppositepointsonthegeodesiccircle;thesepointslieonaruling.However,innearlyplanarregions,thevariationinthevalueofbeingsmall,wecannotdetectastablerulingdirection.Weexplicitlymarksuchregionsasplanar(seeFigure12).Laterwerenetheboundariesofsuchplanarregionsusingneighboringrulinginformation.StageB:Inthisstepweextendtherulings.Weusethefollowingfact:Foratorsalruledsurface,thesurfacenormalremainsconstantalongeachruling.Henceweextendtheestimatedrulingthrougha Figure12:Estimatingrulingdirections.InstageA,weguessrul-ingdirectionsusinggeodesiccircles,andalsoidentifycandidateplanarregions.InstagesBandC,theinitialguessesatrulingsareextended,andtheentirerulingcollectionisthinnedout.Bottomright:Thenalestimatedrulingsandtheregionswhichhavebeenestablishedasplanar(inyellow). untilthesurfacenormalsintheendpointsdeviatefromthemorethanapre-denedthreshold.Rulingsarealsoterminatediftheycomeclosetofeaturelinesorboundaries.Forpurposesoflaterpruning,weassignthenegativemeandeviationofsurfacenormalsalongtherulingfromasameasureofqualitytoeachextendedruling.StageC:Thesetofrulingsobtainedsofaristhinnedoutwhileretainingrulingswithhighqualitymeasure.Weuseagreedyap-proach:Therulingwithhighestqualitymeasureisretained,andtheonesintersectinganarrowbandarounditareremoved.Hereitisimportanttondtherightmeasureofproximityofrulings,becausethesurfacemayexhibitconicalpartswhererulingsintersectatacommonvertex.Thusourbandisshorterthantherulingandcen-teredinitsmidpoint(inFigure12(B+C)thesebandsaremarkedinredandslightlywidenedforbettervisibility).Allotherrulingswhichintersectthisbandareconsidered`close'andareremoved.Wecontinuetheprocessofpruninguntilwegetasetof(roughly)evenlyspacedrulingsonthesurface.Regionsmarkedasplanarareconrmedtobeplanariftheyareboundedbythreeormorerulingsorboundaryedges.QuaddominantdecompositionofAfterestimatingandpruningtherulings,wenowdealwithinitializingtheplanarde-velopmentWeusethedevelopmentmappingtomaptheestimatedrulingdi-rectionsfromthesurfacetotheplane.Fromthissetofmappedrulings,wegenerateacoarsequad-dominantmesh.Notethathereacorrectconnectivityismuchmoreimportantthantheac-tualcoordinatesofthevertices.Subsequentoptimizationretainstheconnectivityoftheinitialmeshwhileupdatingthevertexposi-Theavailableinputdataformeshgenerationaretheestimatedrul-ingsmappedtotheplane,theboundaryoftheplanarmeshandthelocationofridgelinesintheoriginalsurface,whichareusedascandidatecurvedfoldlocations(seeFigure13,left).First,allendpointsofrulingsaresnappedtotheclosestboundaryorridgeline—or,ifthelatteraretoofaraway,areclusteredinapoint.Additionallyshortridgelinesarecontractedtoasingleconepoint(seeFigure13,center).Dependingonthesnappingtarget,weroughlyclassifyanendpointasboundarypointfoldpoint,orconepoint,respectively.Inourexamples,wefrequentlyencoun-teredcombinationsoftwoofthese(e.g.acurvedfoldmightextendtotheboundary).Theextensionofrulingstoboundaryandridgelinesmightintroduceintersectionsclosetorulingendpoints.Suchintersectionsareresolvedbyswappingthecorrespondingendpointvertexcoordinates.Wegetapreliminarymeshbyconnectingrul-ingendpointsastheyaretraversedalongtheboundaryandridgelines,generatingmostlylongquadrilateralfaces. Figure13:Initialmeshlayout.Left:Agivencollectionofrulings(blue),ridgelines(brown),andmeshboundaries(gray).Center:Rulingendpointsaresnappedandclassiedasboundarypoints(gray),foldpoints(blue)andconepoints(brown).Right:Byin-sertinganddeletingrulings,avalidmeshconnectivitywithoutT-junctionsisobtained.Theplanarpartsoftheoriginalshapearemarkedinyellow.Theresultingmeshisnextmodiedbydeletingorinsertingrulingsbasedonthefollowingobservations:(i)Fromanypointonafold,tworulingsmustemanatetopreventanyT-junctionsonthefold.(ii)Planarregionsmustbeboundedeitherbyrulingsoraboundarycurve.(iii)Boundarycornerpointsshouldbeincludedtopreservetheshapeofthebasemesh.(iv)Facesadjacenttoconepointsmighthavemorethanfourvertices.Toensureanoptimalapproximationoftheseregions,suchfacesneedtobesplitintotrianglesorquadsforoursubdivisionstagetoapply.Ifafaceholdsmorethanasingleconepointandtheconnectinglineslieentirelyintheface,rulingsareinsertedconnectingtheconepoints.Ifnecessary,thefacesorig-inatingfromthissteparefurthersplitbyinsertingrulingsemanat-ingfromconepoints.Finally,weobtainaquaddominantplanar(seeFigure13,right).InitializationofthepolygonsoupInitializationofourop-timizationprocedureiscompletewhenapolygonsoup,corre-spondingtothedevelopmentandclosetotheoriginalshapeisfound.Wendafacecorrespondingtoafacebyapplyingtotheverticesof.Sincetheresultingvertices,ingeneral,donotformaplanarpolygonwhichisisometrictoweregisteracopyoftothesemappedverticestoinitializeNowwecanapplytheoptimizationalogrithmofSection3andob-tainameshwhichapproximatesthegivendataandhastheop-timizedversionofasitsprecisedevelopment.Inordertoef-cientlyachievehighapproximationquality,westartwithacoarseapproximationwhichissubsequentlyrened(bysplittingquadsinrulingdirection)andoptimizedagain.ResultsareshowninFigures1,2,7,14and16.6FurtherapplicationsanddiscussionAsillustratedbyFigure14,surfacereconstructioncannicelybecombinedwithdeformationtoolssuchas[Kilianetal.2007].Werstcomputeadigitalreconstructionofaphysicalmodelandthenvaryitsshapebyanas-isometric-aspossibledeformation.Thede-formationwillintroducedeviationsfromatruedevelopablesurface,butitturnsoutthatourreconstructionworksverywellonsuchde-formeddatasets.Notethatevenpreciselyisometricdeformationsingeneraldonotpreserverulingsandthereforerulingshavetobere-estimated.IntheexampleofFigure14itturnedoutthattheop-timizationworkedwellwiththeinitializationforthereconstructionofthephysicalmodel.Inothercases,onemayhavetore-initializeforintermediatepositionsinadeformationsequence.Weemphasizeherethatthedesignandreconstructionofobjectswithcurvedfoldsissimplysolvablebyaparameterizationmethod.Parameterizationwillnotyieldanyinformationaboutthepreciselocationoffolds,rulingsandtypesofruledpatches,norwillitmodifyadatasettobecomepreciselydevelopable.Thenatureofcurvedfolds.Digitalreconstructionofphysicalpapermodelsyieldsasegmentationintotorsalruledpatches.Thisprovidesinsightintothetypicalbehaviorofadevelopablesurfacenearcurvedfolds.Somefrequentlyoccurringsituationsarede-pictedinFigure2.Inthisway,ourworkcanfurthercontributebothtothetheoryofcurvedfoldingandtoapplications,e.g.,tothedevelopmentofinteractiveCADtoolsformodelingobjectswithcurvedfolds.Architecturalfreeformstructures.Developablesurfacesareprominentlyvisibleinarchitecturaldesign[Shelden2002;GlaeserandGruber2007;Pottmannetal.2007].Inparticular,FrankO.Gehryhasbeenusingthesesurfacesquiteextensively.Thepres-enceofrulingssimpliestheactualconstruction.Panelization,e.g. Figure14:Abendingsequencewhichexhibitsacurvedfold.Thelefthandmeshistheresultofapproximatinga3Dscanofapapermodel.Theothershapeshavebeencomputedbycombiningourreconstructionalgorithmwithanas-isometric-aspossibleshapemodicationofthereferencesurface,i.e.,thereferencedatasetinhaschanged,buttheremainingdataforoptimizationaretakenfromthelefthandmesh.bymetaltiles,iseasyduetodevelopability.Theaestheticcontin-uationofatilingoverageneralsharpedgeisadifcultproblem.However,atacurvedfoldthetilecontinuationisoptimal(cf.Fig-ures1and15)andthedesignofthetilingcanbedoneinthedevel-opment.Notethatoursegmentationintotorsalruledpatchesisofhighimportanceformanufacturingsucharchitecturalstructures.Industrialdesign.Theshapesshowninourpaperhopefullypro-videarstimpressionofthewideapplicabilityofcurvedfoldinginindustrialdesign.Suchapplicationsmayrequireahighqual-ityNURBSrepresentationwhichisveryeasytocomputefromoursegmentationintotorsalruledpatches.Theexactlocationsofrul-ings(whichcannotbeseeninthetrianglemeshofa3Dscanofaphysicalmodel)areimportantformanufacturingaswell.More-over,duetothefairnessmeasuresinouroptimizationframeworkweobtainaestheticallypleasingdigitalmodelswhilemaintainingthehardconstraintofapreciseplanardevelopment.Weperformedalargenumberofexperimentsondatasetsobtainedbyscanningmodelsbuiltfromfabricormateri-alswithasimilarstretchingbehavior.Itturnedoutthatthesemod-elshardlybehavelikedevelopablesurfaces,particularlyinregionswithdrasticfolds.Hence,wemustleavethetaskof(roughly)ap-proximatingsuchdatabyasingledevelopablesurfacewithcurvedfoldstofutureresearch.Thefullyautomaticgenerationoftheini-tialplanarmeshworkedwellforallconsideredmodels.Theonlyexceptionwasthecarmodel,asignicantlymorecomplexmodel,whereweinteractivelymodiedafewrulingdirectionstoensureasuitablemeshfortheadaptivesubdivisionweemploy.Implementationandruntimes.InourcurrentimplementationweuseCHOLMOD[DavisandHage2001]tosolveasparselinearsystemandtheKNITROoptimizationpackageforconstrainednon-linearoptimization.AverageruntimesforthemodelsofFigure16 Figure15:Architecturaldesignthatfeaturesrulingsaspartofthesupportstructure.are160secondsforrulingextraction(on50Kreferencemesh),20secondsmeshlayoutand140secondsforoptimization.Theobjec-tivefunctionwasreducedtoorderof.Inparticularthevertexagreementtermislessthan.Thettingweightwasreducedbyafactorof0.1aftereachstepofsubdivisiontofavorfairsolutionsurfacesinsteadofbestapproximatingones.ConclusionandFutureresearch.Wepresentedacomputa-tionalframeworkforthedesignanddigitalreconstructionofde-velopablesurfaceswithcurvedfolds.Ourworkcontributestothediscretedifferentialgeometryofdevelopablesurfaces,tothedis-cretegeometryofcurvedfolds,andtothegeometricoptimizationofsurfaceswithcurvedfolds.Moreover,weillustratedthepotentialofourdevelopmentsonanumberofexamplesmotivatedbyappli-cationsinarchitecture,industrialdesignandmanufacturing.Giventhelimitedamountofpriorresearchinthisarea,thereisstillalotofworktobedone.Openproblemsincludethereconstructionofmodelswhereahighapproximationerrorhastobeadmittedsuchasscannedfabric,acarefulanalysisandclassicationoftypicalruledpatcharrangementsatcurvedfolds,andthedevelopmentofnovelinteractivemodelingtoolsforcurvedfolding.Acknowledgments.ThisworkissupportedbytheAustrianSci-enceFund(FWF)undergrantsS92andP18865.Niloyisalsosup-portedbyaMicrosoftoutstandingyoungfacultyfellowship.WearegratefultoHeinzSchmiedhoferforhishelpwithbuildingandscanningthepapermodelsandrenderingthereconstructedmod-els.WealsothankMartinPeternellandJohannesWallnerfortheirthoughtfulcommentsonthesubject.ReferencesUMANN,G.2004.DegreeelevationanddevelopableBeziersurfaces.Comp.AidedGeom.Design21,661–670.,P.,ANDANG,W.2007.Geodesic-controlleddevelopablesurfacesformodelingpaperbending.Comp.GraphicsForum26,3,365–374.OBENKO,A.,AND,Y.,2005.Discretedifferentialgeometry.Consistencyasintegrability.Preprint,http://arxiv.org/abs/math.DG/OTSCH,M.,PAULY,M.,GROSS,M.,OBBELT,L.2006.Primo:coupledprismsforintuitivesurfacemodeling.InSymp.Geom.Process-,11–20.ERDA,E.,CHAIEB,S.,MELO,F.,ANDAHADEVAN,L.1999.Conicaldislocationsincrumpling.Nature401,46–49.ERDA,E.,MAHADEVAN,L.,ASINI,J.M.2004.TheelementsofProc.Nat.Acad.Sciences101,7,1806–1810.,C.H.,EQUIN,C.2002.DevelopableBezierpatches:proper-tiesanddesign.AidedDesign34,511–528.AVIS,T.A.,ANDAGE,W.W.2001.Multiple-rankmodicationsofasparsecholeskyfactorization.SIAMJournalonMatrixAnalysisandApplications22,4,997–1013. Figure16:Agalleryofdigi-talpapermodels.Modelswerecomputedwiththealgorithmde-scribedinSection5withscansofrealpapermodelsasreferencesurfaces.Reconstructedmod-elsexhibitcurvedandstraightfoldsandcanbeisometricallyunfoldedintotheplane.Severalspecialcaseslikeconesingular-ities(toprow–middle)andcon-vergingcurvedfolds(toprow–right)areshown.EMAINE,E.,ANDOURKE,J.2007.GeometricFoldingAlgorithms:Linkages,Origami,Polyhedra.CambridgeUniv.Press.ESBRUN,M.,POLTHIER,K.,ANDODER,P.2005.DiscreteDif-ferentialGeometry.SiggraphCourseNotes.ARMO,M.1976.DifferentialGeometryofCurvesandSurfacesREY,W.2004.Modelingbuckleddevelopablesurfacesbytriangulation.AidedDesign36,4,299–313.LAESER,G.,RUBER,F.2007.Developablesurfacesincontem-poraryarchitecture.J.ofMath.andtheArts1,1–15.UFFMAN,D.A.1976.Curvatureandcreases:aprimeronpaper.Trans.ComputersC-25,1010–1019.ULIUS,D.,KRAEVOY,V.,ANDHEFFER,A.2005.Dcharts:Quasidevelopablemeshsegmentation.ComputerGraphicsForum24,3,581–590.Proc.Eurographics2005.ERGOSIEN,Y.,GOTUDA,H.,ANDUNII,T.1994.Bendingandcreas-ingvirtualpaper.IEEEComp.Graph.Appl.14,1,40–48.ILIAN,M.,MITRA,N.J.,ANDOTTMANN,H.2007.Geometricmod-elinginshapespace.ACMTrans.Graphics26,3,64.,Y.,POTTMANN,H.,WALLNER,J.,YANG,Y.-L.,ANDANG,W.2006.Geometricmodelingwithconicalmeshesanddevelopablesur-faces.ACMTrans.Graphics25,3,681–689.ASSARWI,F.,GOTSMAN,C.,LBER,G.2006.Papercraftmodelsusinggeneralizedcylinders.InPacicGraph.,148–157.ITANI,J.,UZUKI,H.2004.Makingpapercrafttoysfrommeshesusingstripbasedapproximateunfolding.ACMTrans.Graphics23,3,HTAKE,Y.,BELYAEV,A.,ANDEIDEL,H.-P.2004.Ridge-valleylinesonmeshesviaimplicitsurfacetting.ACMTrans.Graph.23,3(Au-gust),609–612.EREZ,F.,ANDAREZ,J.A.2007.QuasidevelopableBsplinesurfacesinshiphulldesign.AidedDesign39,853–862.ETERNELL,M.2004.Developablesurfacettingtopointclouds.AidedGeom.Des.21,8,785–803.OTTMANN,H.,ANDALLNER,J.2001.ComputationalLineGeometrySpringer.OTTMANN,H.,HUANG,Q.-X.,YANG,Y.-L.,AND,S.-M.2006.Geometryandconvergenceanalysisofalgorithmsforregistrationof3DInt.J.ComputerVision67,3,277–296.OTTMANN,H.,ASPERL,A.,HOFER,M.,ILIAN,A.2007.Ar-chitecturalGeometry.BentleyInstitutePress.OSE,K.,SHEFFER,A.,WITHER,J.,C,M.-P.,HIBERT,B.2007.Developablesurfacesfromarbitrarysketchedboundaries.InSymp.GeometryProcessing.163–172.AUER,R.1970.Differenzengeometrie.Springer.HATZ,I.,T,A.,ANDEIFMAN,G.2006.PapercraftmodelsfromVis.Computer22,825–834.HEFFER,A.,PRAUN,E.,OSE,K.2006.Meshparameterizationmethodsandtheirapplications.Found.Trends.Comput.Graph.Vis.22,105–171.HELDEN,D.2002.DigitalsurfacerepresentationandtheconstructibilityofGehry'sarchitecture.PhDthesis,M.I.T.UBAG,J.,ANDLBER,G.2006.Piecewisedevelopablesurfaceapprox-imationofgeneralNURBSsurfaceswithglobalerrorbounds.InProc.GeometricModelingandProcessing.143–156.ANG,C.,ANDANG,K.2004.Achievingdevelopabilityofapolygonalsurfacebyminimumdeformation:astudyofglobalandlocaloptimiza-tionapproaches.Vis.Computer20,521–539.ANG,C.C.L.2008.Towardsattenablemeshsurfaces.Comput.AidedDes.40,1,109–122.ERTHEIM,M.2004.Cones,Curves,Shells,Towers:HeMadePaperJumptoLife.TheNewYorkTimes,June22AMAUCHI,H.,GUMHOLD,S.,ZAYER,R.,ANDEIDEL,H.-P.2005.MeshsegmentationdrivenbyGaussiancurvature.Vis.Computer21