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Constructing the Spline Atlas of a Free-Form Surface

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Title:
Constructing the Spline Atlas of a Free-Form Surface
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Sarov, Martin G
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[Gainesville, Fla.]
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Computer Engineering
Computer and Information Science and Engineering
Committee Chair:
PETERS,JORG
Committee Co-Chair:
UNGOR,ALPER
Committee Members:
VEMURI,BABA C
KUMAR,ASHOK V

Subjects

Subjects / Keywords:
atlas -- b-spline -- bezier -- bicubic -- cagd -- continuity -- degree-reduction -- design -- differentiable -- extraordinary-point -- fea -- fem -- free-form -- g-spline -- genus -- geometric -- iga -- knot-insertion -- mesh -- nested -- parametric -- patch -- polycube -- quad -- refinable -- reparameterization -- spline -- surface -- t-junction -- tensor-product -- trimming
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Computer Engineering thesis, Ph.D.

Notes

Abstract:
High-end shape design of product-defining surfaces and engineering analysis of the resulting objects employ vastly different representations. The resulting gap between design and analysis impedes an effective feedback loop that is necessary for optimal overall design. This work bridges the gap by linking high-end design with a high-order representation, called G-spline, that provides not only good shape but also consistent parameterizations for analysis. The contribution is two-fold. First, refinability of the space of functions that form the G-spline atlas is addressed. Exact refinablility, i.e. re-representation with more degrees of freedom that may be adjusted to improve the outcome, is needed to adaptively compute on surfaces. This work presents a construction, called Refinable Polycube G-splines, that forms a 2-manifold controlled by a mesh that has quadrilateral faces and such that at most six quads meet at each vertex. This replicates the layout of so-called polycubes, a stacking of cubes to approximate or enclose geometry. Refinable Polycube G-splines are piecewise bi-cubic and almost everywhere tangent-continuous (G1). They are special in that the change of variables between four-sided surface pieces is rational linear, i.e. as close as possible for a general free-form surface to being everywhere C1 continuous. Refinable Polycube G-splines can be constructed in two different ways. One construction interprets the quad mesh vertices in the fashion of C2 bi-cubic splines -- this provides for good shape; the other interprets the 2-by-2 inner Bezier coefficients of each bi-cubic as C1 bi-cubic B-spline coefficients -- this offers four degrees of freedom per patch and enables adaptive refinement so that the resulting G-spline spaces are nested, i.e. any G-spline surface can be exactly re-represented at different levels of refinement. Second, a practical approach for constructing the global, uniform, differentiable, and graded atlases of real-world artifacts is developed. This work focuses on artifacts designed by stylists as collections of separate and heterogeneous trimmed design surfaces. Here spline surfaces closely approximate the untrimmed portions of primary surface pieces and embed them in one, by default smoothly-connected, G-spline atlas. This yields a bi-cubic tensor-product spline wherever the layout of the quadrilaterals is a regular tensor-product grid and a bi-quartic spline whenever the layout is irregular, i.e. three or more than four quadrilaterals meet. By transforming heterogeneous collections of unrelated patches output by industrial stylists, this work moves closer to providing designers with the tools to directly work with an analysis-friendly representation -- while preserving the interfaces they are accustomed to. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
Local:
Adviser: PETERS,JORG.
Local:
Co-adviser: UNGOR,ALPER.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-11-30
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by Martin G Sarov.

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11/30/2017
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CONSTRUCTINGTHESPLINEATLASOFAFREE-FORMSURFACEByMARTING.SAROVADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2017

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c2017MartinG.Sarov

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Tomylovedones

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ACKNOWLEDGMENTSIwouldliketothankmyadvisorDr.JorgPetersforhisprovidentguidanceandpatienceandforneverleavingmyquestionsunanswered.IwouldalsoliketothankmysupervisorycommitteemembersDr.BabaVemuri,Dr.AlperUngor,andDr.AshokKumarwhofurthermotivatedmetopursuequalityinmywork.IamalsogratefulformySurfLabcolleaguesfromwhomandwithwhomIacquiredtheskillsandtheknow-hownecessarytoseemyacademicgoalstofruition,mostnotably:SalehDindar,RuijinWu,ShayanJaved,YoungInYeo,Dang-ManhNguyen,andThienNguyen.Finally,Iwouldliketothankmyfamilywhosesacriceshaveshapedmydeterminationandwhosesupporthasgivenmestrength. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 2BACKGROUND ................................... 13 2.1SplineBasics .................................. 13 2.1.1BezierCurves .............................. 13 2.1.1.1DenitionofBeziercurves .................. 13 2.1.1.2Derivatives .......................... 14 2.1.1.3DeCasteljau'salgorithmandsubdivision .......... 14 2.1.2B-splineCurves ............................. 15 2.1.2.1DenitionofB-splinecurves ................. 15 2.1.2.2Knotinsertion ........................ 16 2.1.3Tensor-productSurfaces ........................ 17 2.1.3.1Denitionoftensor-productsurfaces ............ 17 2.1.3.2Tensor-productB-splinetoBezierconversion ........ 17 2.2GeometricContinuity .............................. 18 2.2.1DenitionofGeometricContinuity .................. 18 2.2.2GeometricContinuityofEdge-adjacentPatches ........... 18 2.3Polycubes .................................... 19 2.3.1DenitionofPolycubes ......................... 19 2.3.2PolycubeVertexClassication ..................... 20 3REFINABLEPOLYCUBEG-SPLINES ...................... 22 3.1Motivation .................................... 22 3.2RestrictedQuad-layoutsCompatiblewithRationalLinearTransitionMaps 24 3.2.1RestrictedQuad-layoutand[3;4;5;6]Polyhedron .......... 24 3.2.2GeometricContinuityofStructurallySymmetricRationalLinearTransitionMaps ............................. 27 3.2.3IndexingtheSurfaceQuads ...................... 29 3.2.4GeometricSmoothnessConstraints .................. 29 3.3G1PolycubeSurfaces .............................. 32 3.3.1LabelingBoundaryEdgesbyTheirApparentValence ........ 32 3.3.2PGS:ConstructingaSmooth,PiecewiseBi-3SurfacefromaPolycube 34 3.3.2.1Initialization ......................... 34 3.3.2.2PGS(polycubeG-spline)algorithm ............. 35 5

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3.3.3PropertiesoftheSurfaces ........................ 38 3.3.4ExamplesofPGSSurfaces ....................... 39 3.4Renement ................................... 40 3.4.1InvariantReconstructionofBoundaryCurves ............. 41 3.4.2ER(EdgeRecovery)Algorithm .................... 43 3.4.3PGSERAlgorithm ............................ 44 3.5Discussion .................................... 45 4CONSTRUCTINGAWATERTIGHTSPLINEATLASFROMHETEROGE-NEOUSTRIMMEDSURFACES .......................... 48 4.1Motivation .................................... 48 4.2ConstructingaSplineAtlasfromaCollectionofTrimmedPatches ..... 52 4.2.1HomogenizingthePatchRepresentation ................ 52 4.2.1.1Degree-matchingofpatches ................. 53 4.2.1.2Establishingpatchadjacenciesandgradation ....... 58 4.2.2CreatingaGlobalB-splineScaold .................. 61 4.2.3FinalSmoothSurfaceConstruction .................. 65 4.3Discussion .................................... 67 APPENDIX AEDGELABELINGHEURISTIC .......................... 70 A.1Preliminaries .................................. 70 A.2Heuristic ..................................... 71 BBI5PGS`=1ALGORITHM ............................ 73 B.1GeometricSmoothnessConstraints ...................... 73 B.2Algorithm .................................... 74 CALGORITHMSFORGENERATINGMATRICESSANDK .......... 78 C.1SubdivisionMatrixS .............................. 78 C.2B-splinetoBezierConversionMatrixK .................... 79 DEXAMPLESOFPGSANDPGSERSURFACES ................. 80 EEXAMPLESOFSURFACESPRODUCEDBY[1] ................ 82 REFERENCES ....................................... 83 BIOGRAPHICALSKETCH ................................ 90 6

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LISTOFFIGURES Figure page 1-1High-enddesignemphasizesshapeoverparameterization. ............. 11 2-1Evaluatingp(t)att=0:5. .............................. 15 2-2Notationforthetensor-productB-splinetoBezierconversionrules. ....... 17 2-3Polycubesurfacecongurationsandtheirvalences. ................ 20 3-1Multi-resolutionofPolycubeG1-splines. ...................... 23 3-2Figure-8from[2]. ................................... 25 3-3Thelayoutinducedbyonce-subdividedpolycubequadsallowsmodelingsmoothobjectsofanygenusgreaterthanoneusingonebi-cubicpolynomialpiecepersub-quad. ....................................... 27 3-4LabelingofquadandtheBB-netofcontrolpointsassociatedwithonesub-quad. 30 3-5IndicesofBB-coecientsoftheG1stripalongah3;6iedge. ........... 31 3-6Apparentcongurationoftangentsemanatingfromverticesofvalencen=5;4basedonsymmetricallydistributingn=6tangentsandremovingtangents. ... 34 3-7B-spline-likecontrolpointsc(green)anddependentBB-coecientsobtainedbyC1constraints(gray)orG1constraints(G1)inred. ................ 35 3-8StepsofPGS(foreachlabelinf6,4,3g). ...................... 36 3-9Highlightlinemismatchvisibleunderzoom(1/100thoftheedgelength). ... 39 3-10Hierarchicaledit. ................................... 40 3-11AveragesofpolycubesplinecoecientsusedintheEdgeRecoveryalgorithm. 42 3-12SurfaceswiththeircorrespondingBB-netsandhighlightlinesaftereachmajorstepofPGSER. .................................... 45 4-1Evaluatingsurfacequalitybyreectionandhighlightlines. ............ 48 4-2Trimmingexpained. ................................. 49 4-3Heterogeneousparameterizationincardesign. ................... 50 4-4Thedegreeofadp=5BBcurveisreducedtodt=3,naiveapproach ...... 56 4-5Thedegreeofadp=5BBcurveisreducedtodt=3,ourapproach ....... 57 7

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4-6Comparisonbetweenindustrialinputcontainingpatchesofdegree3through9andpatchesofuniformbi-degreedt=3producedbyourdegreereductiontech-nique .......................................... 58 4-7Top-leveloctreecontainingtrimcurvesamplesinR3(MiniCoopercarhoodmodel)usedforestablishingpatchadjacencies ................... 59 4-8Achievingaconsistentpatchparameterization. ................... 60 4-9Rasterizationoftrimcurvesinthepatchdomainproducesinclusionregionsthatcanbeusedtoidentifycoecientsneededforthenalmeshconstruction. ... 62 4-10CompletingB-meshconnectionsatpatchtransitions. ............... 64 4-11FinalC1BBsurfaceproducedbyapplying[1]toourG-mesh. .......... 66 4-12SurfaceeectwhenT-facesarepartofameshfeature. .............. 68 A-1Dualassignmentoflabelstoedgesincidenttoa4valentvertexhavingdirectneighborswithvalencesm,n,m0,andn0,wherem6=nandm06=n0. ...... 70 D-1SmoothPGSsurfaces(withoutrenement)frompolycubecongurationsalongwiththeircorrespondingBB-controlnetsandhighlightlineplots. ........ 80 D-2PGSERsurfacesofmoderate(gold)andhigh(cyan)patchcount. ......... 81 E-1GalleryofG-splinesurfacesproducedby[1]. .................... 82 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCONSTRUCTINGTHESPLINEATLASOFAFREE-FORMSURFACEByMartinG.SarovMay2017Chair:JorgPetersMajor:ComputerEngineeringHigh-endshapedesignofproduct-deningsurfacesandengineeringanalysisoftheresultingobjectsemployvastlydierentrepresentations.Theresultinggapbetweendesignandanalysisimpedesaneectivefeedbackloopthatisnecessaryforoptimaloveralldesign.Thisworkbridgesthegapbylinkinghigh-enddesignwithahigh-orderrepresentation,calledG-spline,thatprovidesnotonlygoodshapebutalsoconsistentparameterizationsforanalysis.Thecontributionistwo-fold.First,renabilityofthespaceoffunctionsthatformtheG-splineatlasisaddressed.Exactrenablility,i.e.re-representationwithmoredegreesoffreedomthatmaybeadjustedtoimprovetheoutcome,isneededtoadaptivelycomputeonsurfaces.Thisworkpresentsaconstruction,calledRenablePolycubeG-splines,thatformsa2-manifoldcontrolledbyameshthathasquadrilateralfacesandsuchthatatmostsixquadsmeetateachvertex.Thisreplicatesthelayoutofso-calledpolycubes,astackingofcubestoapproximateorenclosegeometry.RenablePolycubeG-splinesarepiecewisebi-cubicandalmosteverywheretangent-continuous(G1).Theyarespecialinthatthechangeofvariablesbetweenfour-sidedsurfacepiecesisrationallinear,i.e.ascloseaspossibleforageneralfree-formsurfacetobeingeverywhereC1continuous.RenablePolycubeG-splinescanbeconstructedintwodierentways.OneconstructioninterpretsthequadmeshverticesinthefashionofC2bi-cubicsplines{thisprovidesforgoodshape;theotherinterpretsthe22innerBeziercoecientsofeachbi-cubicasC1bi-cubicB-spline 9

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coecients{thisoersfourdegreesoffreedomperpatchandenablesadaptiverenementsothattheresultingG-splinespacesarenested,i.e.anyG-splinesurfacecanbeexactlyre-representedatdierentlevelsofrenement.Second,apracticalapproachforconstructingtheglobal,uniform,dierentiable,andgradedatlasesofreal-worldartifactsisdeveloped.Thisworkfocusesonartifactsdesignedbystylistsascollectionsofseparateandheterogeneoustrimmeddesignsurfaces.Heresplinesurfacescloselyapproximatetheuntrimmedportionsofprimarysurfacepiecesandembedtheminone,bydefaultsmoothly-connected,G-splineatlas.Thisyieldsabi-cubictensor-productsplinewhereverthelayoutofthequadrilateralsisaregulartensor-productgridandabi-quarticsplinewheneverthelayoutisirregular,i.e.threeormorethanfourquadrilateralsmeet.Bytransformingheterogeneouscollectionsofunrelatedpatchesoutputbyindustrialstylists,thisworkmovesclosertoprovidingdesignerswiththetoolstodirectlyworkwithananalysis-friendlyrepresentation{whilepreservingtheinterfacestheyareaccustomedto. 10

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CHAPTER1INTRODUCTIONSmoothgradedparameterizationsofthecomputationaldomainmanifoldareindis-pensableforthestabilityandconvergenceofsolversforhigher-orderdierentialequations.Bycontrast,stylists,sayinthecarindustry,focusexclusivelyonthevisibleshapeofoutersurfacesdisregardingrelativeorientation,tessellation,mathematicalguaranteesofcontinu-ityorsuitabilityforvolumetrictessellation(Fig. 1-1 ).Theresultingheterogeneityofsize,parameterlineorientationandpolynomialdegreeofouter`classA'surfaceshasgivenrisetowholeresearchcommunitiesandsoftwaresuitesdevotedtothetrickytaskofmeshing,`healing',andre-approximatinghigh-enddesignsurfaceswiththegoalofpreventingcostlydownstreamstructuralandmanufacturingchallenges.Suchre-approximationisnotonlycomplexbutalsoitdisruptsthedesign-and-simulationcycleatthecore.Interactivityofthiscycleiscriticaltosteerdesignearlyontowardsoverallmorerobustsolutions,allowmoreinnovativesolutions,andcounteracttheincreasedspecializationindesignversusanalysis.Inthisworkwedevelopandevaluatenewtechniquesforpreparingsurfacesforhigher-ordersimulation{throughbetterparameterizationofthegeometricdesign.Hereindustrycompatiblepiecewisetensor-productsplinesjoinedwithgeometriccontinuity,a.k.a.G-splines,playakeyrole.G-splinesarecapableofproducingsmoothconsistent A B CFigure1-1. High-enddesignemphasizesshapeoverparameterization.A)Surfacelayout.B)Trimmedpatches:controlnets.C)Un-trimmedpatches:controlnets. 11

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parameterizationsalsointhepresenceofirregularitiesinotherwisequadrilateralmeshes,i.e.theyprovideasmoothsplinespacealsowhere3ormorethan4quadrilateralsmeet(Fig. 3-3 ,Fig. 4-1D )or,dually,mesheswithmulti-sidedfacetsandvalence4everywhere.Inthefollowing,Section 2 ,webrieyreviewthefundamentalconceptsusedinourworkincludingsplinebasicsandgeometriccontinuityamongothers.Then,inSection 3 ,wedescribeaspecialclassofG-splines,namelyourRenablePolycubeG-splines,capableofproducingalmosteverywheretangentcontinuous(G1)surfacesoflowestdegreebasedonasimple(rationallinear)reparameterization.Lastly,inSection 4 ,wedelineateourapproachforconstructingaglobal,uniform,dierentiable,andgradedatlasofartifactsoriginallydesignedbystylistsasacollectionofseparateandheterogeneoustrimmeddesignsurfaces. 12

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CHAPTER2BACKGROUND 2.1SplineBasicsBelowwebrieydescribethefundamentalconceptsofsplinetheorythatarerelevanttothiswork.Thetextbooksandsurveypapers[ 3 { 10 ]covermanyotherinterestingaspectsofthisbeautifultheory. 2.1.1BezierCurves 2.1.1.1DenitionofBeziercurvesTheBezierformisageometricrepresentationofpolynomials.ABeziercurveisaparametriccurve,eachofwhosecomponentsareBezierfunctions.AdegreedBezierfunctionisdeterminedbyd+1coecients.Thepiecewiselinearinterpolantforagivenorderingofthecontrolpointsisknownasthecontrolpolygon.ThedegreedBezierpolynomialpwithcontrolpointsc=(c0;c1;:::;cd)isdenedbyp(t):=dXi=0ciBdi(t)wheretheithBezierbasisfunctionofdegreed,alsoknownasBernsteinpolynomialbasis,isBdi(t):=di(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t)d)]TJ /F7 7.9701 Tf 6.587 0 Td[(iti:ABezierfunctionistypicallysaidtobeamapfromthedomain[0;1],particularlyasitinterpolatestherstandlastcontrolpointsattheparametervalues0and1,respectively.However,thefunctioniswelldened{andcanbeevaluated{overallt2R.Thed+1basesofdegreedarenon-negativeover[0;1]andformapartitionofunitysincedXi=0Bdi(t)=dXi=0di(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(t)d)]TJ /F7 7.9701 Tf 6.587 0 Td[(iti=(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t+t)d=1:Hence,p(t)isaconvexcombinationofitscontrolpoints,givingtheBezierformthefollowingproperties: 13

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1. Theconvexhullproperty:ABeziercurvealwaysliesintheconvexhullofitscontrolpoints. 2. Aneinvariance:ApplyingananetransformationtoacontrolpolyhedronappliesittothecorrespondingBeziercurveaswell.Ageometricconsequenceoftheabovepropertiesisthatthecontrolpolyhedronisanexaggerationofthecurve. 2.1.1.2DerivativesDerivativesoftheBezierformalsohaveaverygeometricinterpretation.ThecontrolpointsofthederivativeofaBeziercurve,writtenintermsofaBeziercurveofalowerdegree,aremerelydierencesofconsecutivecontrolpoints,scaledbythedegree.Inotherwords,p0(t)=dd)]TJ /F4 7.9701 Tf 6.587 0 Td[(1Xi=0(ci+1)]TJ /F6 11.9552 Tf 11.955 0 Td[(ci)Bd)]TJ /F4 7.9701 Tf 6.586 0 Td[(1i(t):Byinductiononthenumberofderivatives,thekthderivativecanbewrittenas p0(t)=d! (d)]TJ /F3 11.9552 Tf 11.956 0 Td[(k)!d)]TJ /F7 7.9701 Tf 6.587 0 Td[(kXi=0Dki(c)Bd)]TJ /F7 7.9701 Tf 6.587 0 Td[(ki(t);(2{1)wherethelinearoperatorDki(c):=kXj=0()]TJ /F1 11.9552 Tf 9.298 0 Td[(1)k)]TJ /F7 7.9701 Tf 6.587 0 Td[(jkjci+jcomputesthekthforwarddierencebeginningatindexi.Forinstance,D2i(c)=ci)]TJ /F1 11.9552 Tf -432.768 -23.908 Td[(2ci+1+ci+2,andD3i(c)=)]TJ /F6 11.9552 Tf 9.299 0 Td[(ci+3ci+1)]TJ /F1 11.9552 Tf 11.813 0 Td[(3ci+2+ci+3.SincetheBezierforminterpolatesitsendpoints,Equation 2{1 impliesthatthekthderivativeatthebeginning(end)ofaBeziercurveisdeterminedsolelybytherst(last)k+1controlpointsofthecurve. 2.1.1.3DeCasteljau'salgorithmandsubdivisionTheDeCasteljau'sAlgorithmisarecursivemethodforevaluationofBeziercurvesdenedbythefollowingrecurrencerelation:c(0)i:=ci;wherei=0;:::;dc(j)i:=c(j)]TJ /F4 7.9701 Tf 6.587 0 Td[(1)i(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)+c(j)]TJ /F4 7.9701 Tf 6.587 0 Td[(1)i+1t0;wherei=0;:::;n)]TJ /F3 11.9552 Tf 11.955 0 Td[(j;j=1;:::;d 14

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c(0)1c(1)1c(0)2c(1)0c(2)0c(3)0c(2)1c(1)2c(0)0c(0)3Figure2-1. Evaluatingp(t)att=0:5. Therefore,evaluationofp(t)att=t0(Fig. 2-1 )requiresdstepsofthealgorithmandisgivenbyp(t0)=c(d)0.Averydesirableside-eectofDeCasteljau'sAlgorithmisthatitrepresentsanintuitivewaytosubdividethegivencurvep(t)intotwopieces,p0(t)andp1(t),ofdegreedbysimplyreadingotheauxiliaryc(j)ivalues:controlpolygonofp0(t):c(0)0;c(1)0;:::;c(d)0controlpolygonofp1(t):c(d)0;c(d)]TJ /F4 7.9701 Tf 6.586 0 Td[(1)1;:::;c(0)d 2.1.2B-splineCurves 2.1.2.1DenitionofB-splinecurvesAB-splinecurvepofdegreed,havingcontrolpointsci=[c0;c1;:::;cn],isapiecewisepolynomialcurverepresentedbyalinearcombinationofB-splinebasisfunctionsNd(t)andisdenedbyp(t):=nXi=0ciNdi(t): 15

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TheithbasisfunctionofdegreedisrecursivelydenedasN0i(t):=8>><>>:1ift2[ti;ti+1]0otherwise,Ndi(t):=t)]TJ /F3 11.9552 Tf 11.955 0 Td[(ti ti+d)]TJ /F3 11.9552 Tf 11.955 0 Td[(tiNd)]TJ /F4 7.9701 Tf 6.586 0 Td[(1i(t)+ti+d+1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t ti+d+1)]TJ /F3 11.9552 Tf 11.955 0 Td[(ti+1Nd)]TJ /F4 7.9701 Tf 6.587 0 Td[(1i+1(t);wheretiisreferredtoasaknotanditbelongstothenon-decreasingsequence[t0;t1;:::;tk)]TJ /F4 7.9701 Tf 6.587 0 Td[(1],k=n+d+1delineatingthedomain.AnimportantrealizationhereisthatBeziercurvesareactuallyveryspecialcasesofB-splineoneshavingaknotsequenceofthetype[ti;ti+1]wherex=x:=(xi)1imjx1=x2==xmiftheargumentxisrepeatedmtimes. 2.1.2.2KnotinsertionGivenacurvep(t)overtheknotsequence[t0;t1;:::;tk)]TJ /F4 7.9701 Tf 6.586 0 Td[(1]asdenedabove,insertaknott,sothatthenewcurvep(t):=Pni=0ciNdi(t)over[t0;t1;:::;tl;t;tl+1;:::],wheretlttl+1,isidenticaltop(t).Becauseofthefundamentalequalityk=n+d+1,afterinsertingaknot,eitherthenumberofcontrolpointsnorthedegreeofthecurvedmustalsobeincreasedbyone.Bohm'sAlgorithm[ 11 ]providesthesimplerulefordeterminingthenewcontrolpolygoncisci:=(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(i)ci)]TJ /F4 7.9701 Tf 6.586 0 Td[(1+ici;wherei:=8>>>>>><>>>>>>:1il)]TJ /F3 11.9552 Tf 11.955 0 Td[(k+10il+1t)]TJ /F7 7.9701 Tf 6.587 0 Td[(ti tl+k)]TJ /F13 5.9776 Tf 5.756 0 Td[(1)]TJ /F7 7.9701 Tf 6.586 .001 Td[(til)]TJ /F3 11.9552 Tf 11.955 0 Td[(k+2il:B-splineknotinsertionisafundamentaloperation.Itcouldbeusednotonlyforincreas-ingthedegreesoffreedomwhenmodelingwithB-splines,butalsoforconvertingacurvefromitsB-splinetoitsBezierrepresentation. 16

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s02s12s22c11s01s11c00c10s21s00s10s20Figure2-2. Notationforthetensor-productB-splinetoBezierconversionrules. 2.1.3Tensor-productSurfaces 2.1.3.1Denitionoftensor-productsurfacesTensor-productbases(Bezier,B-spline,etc)areproductsoftwounivariatebases.Thesplinepatchpofdegree(d0;d1)correspondingtothecontrolpointsc=(ci:j)0id0;0jd1isdenedbyp(s;t):=d0Xi=0d1Xj=0ci;jBd0i(s)Bd1j(t):Tensor-productsurfacesinheritallthepropertiesoftherespectiveunivariatecurves,includingthederivativeformulaswithrespecttoeachindividualvariable: 1. Derivativeswithrespecttotcorrespondtodierencingcalongthejdirectionandscalingitbyd1. 2. AlthoughthesurfacepatchisdenedonallofR2,thedomainistypicallychosentobe[0;1][0;1]sincethebasesarenon-negativeoverthisdomain,andthefourcornercontrolpointsofthepatchareinterpolatedatthefourcornersofthisdomain. 2.1.3.2Tensor-productB-splinetoBezierconversionAcentralpracticalissueforB-splinesishowtondthecontrolpointsfortheBeziercurvesthatmakeupaB-spline.AswediscussedinSection 2.1.2 ,BeziercurvesarejustaspecialcaseofB-splineones.Wealsodescribedasimplealgorithmforknotinsertion 17

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whichpreservesthegeometryoftheoriginalcurve.Thus,onecouldconvertacurvefromitsB-splinetoitsBezierrepresentationbymerelyinsertingknotsuntileachoriginalknothasamultiplicityequaltothepolynomialdegreed+1[ 12 ],resultinginasmanyBeziercurvesegmentsasthereareuniqueknotintervals.Tensoredrules(Fig. 2-2 )convertatensor-productuniformsplinewithB-splinecoecientssijtoBernstein-Bezierform:c00=)]TJ /F1 11.9552 Tf 5.48 -9.683 Td[(16s11+4(s21+s12+s01+s10)+(s22+s02+s00+s20)=36;c10=)]TJ /F1 11.9552 Tf 5.48 -9.684 Td[(8s11+2(s10+s12)+4s21+(s22+s20)=18;c11=)]TJ /F1 11.9552 Tf 5.48 -9.684 Td[(4s11+2(s21+s12)+s22=9:Theremainingrulesintuitivelyfollowfromsymmetry. 2.2GeometricContinuityWestartwiththedenitionsofCkandGkcontinuity.Geometriccontinuityisarelaxationofparametrization,andnotarelaxationofsmoothness. 2.2.1DenitionofGeometricContinuity Denition1(Ck). AfunctionissaidtobeCkifitisk-timescontinuouslydierentiable.TwoCkfunctionsegmentsaresaidtojoinCkifatallpointsalongtheirsharedboundary,theyhavethesameithderivativeforalli20;1;;k. Denition2(Gk). TwoGkfunctionsegmentsaresaidtojoinGkifatallpointsalongtheirsharedboundary,theyhavethesameithderivativeforalli20;1;;kafterasuitablereparametrization,.InthecaseofG1continuity,twopatcheswithacommonboundarycurvehaveacontinuouslyvaryingtangentplanealongthatboundarycurve.TheconceptofG1continuityisageneralizationofC1continuity,butnottheotherwayaround. 2.2.2GeometricContinuityofEdge-adjacentPatchesWheneverapointisenclosedbyn3patchesadditionalconstraintsariseonboththereparameterization,,andthegeometrymap,bi,wherei=0;:::;n,(theimageof 18

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whichinR3isapatch).Thereasonisthatifoneweretostartwithonepatchandaddedonepatchatatime,thelastpatchwouldhavetomatchpairwisesmoothnessconstraintsacrosstwoofitsedges.Thiscircularinterdependenceleadstothe[ 6 ]: compositionconstraintson:theTaylorexpansionuptokthorderofthecomposi-tionofallreparameterizationsi;i+1,i=0;:::;n,mustagreewiththeexpansionoftheidentitymap.Anexamplefrom[ 6 ]clariesthisfurther:assumingk=1,n=3,and(0;t)=(t;0),then1;22;33;1j0=0;D1;2D2;3D3;1j0=Didj0Thereparameterizationshouldbecarefullychosensothatittreatsanytwosplinesurfacepiecesalike,e.g.withoutpreferenceforonethathasalowerindexorisgeometricallyclosertosomelocation.Suchastructurallysymmetricensuresthatboth@bi=@(bi+1)and@bi+1=@(bi)hold(Section 3.2.2 discussesourchoiceof). vertex-enclosureconstraintsonbi:Ifniseven,thecirculantmatrixofGkcon-straints(asdenedin[ 6 ])isrank-decient.Therefore,adjustingtherighthandsideisonewayofsolvingthevertex-enclosureproblem.Ifnisoddthenthecirculantconstraintmatrixisoffullrankandvertex-enclosuredoesnotimposeadditionalconstraintsonhigherorderderivative.Ifn=4andthetangentsemanatingfromthevertexformanXthentheunder-constrainedproblemalwayshasasolution(Step 2 ofPGS,Section 3.3.2 ,representsanexampleinpractice). 2.3Polycubes 2.3.1DenitionofPolycubesInthecontextofseamlesstexturemapping,Tarinietal.[ 13 ]pioneeredtheuseofunionsofcubesformingasolid,apolycube.Mappingeachvertexofaclosedpolygonalsurfacetoajudiciously-chosenpolycube,yieldsadomainfortexturecoordinatesthatcan 19

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reducedistortioncomparedtomappingfromtheplane.Thetrickyconstructionofgoodpolycubeshasbeenaddressedinaseriesofpapers,notably[ 14 { 18 ],withWangetal.[ 14 ]usingthepolycubetotstandardsplinesintheregularmeshregionsandllingmulti-sidedholeswithsimplexsplines.Wanetal.[ 19 ]optimizepolycubedomainstominimizethenumberofcubeswhilepreservingthegenusoftheoriginal3Dshape.Forourpurposeauniformcubepartitionsuces. 2.3.2PolycubeVertexClassicationWeconsideravertexsurroundedbyaneighborhood2of222cubesalignedwithorthogonaldirectionsdi,i=1;2;3.Thecubesareconsideredeitherfullorempty. Observation1(polycubevalences). Thevalencenofapolycubevertexisn:=3+min(;),whereisthenumberofdirectionsdjforwhich2containscubesonbothsidesoftheplaneorthogonaltodj;andisthenumberofdirectionsforwhich2containsemptycubesonbothsidesoftheplaneorthogonaltodj. Figure2-3. Polycubesurfacecongurationsandtheirvalences. Wewillconstructsmoothapproximationsoftheouterquad-meshofcollectionsofcubesthatareconnectedbyasequenceofsharedfacets.Non-manifoldcongurations,e.g.when2isemptyexceptfortwocubestouchingatexactlyonepoint,willbetreatedas 20

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belongingtoseparatesurfaces.Analogously,ifthereareexactlytwodiagonallyoppositeemptycubes,twosurfacesaret,onetothe`front'andonetothe`back'.Thus,wecanrestrictattentiontothesevencongurationsshowninFig. 2-3 .Theonlyadmissiblevalencesforpolycubequadsurfacesthenare3,4,5,or6. 21

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CHAPTER3REFINABLEPOLYCUBEG-SPLINES 3.1MotivationAmajorchallengewhenconstructingasplinesurfacefromdatasuchasapointcloudoritstriangulation,istodeterminethepatchlayout.Oneapproachistoapplyanoct-tree-likepartitiontocapturethedataasacollectionofcubesformingapolycube[ 13 19 ].Suchpolycubeapproximationsaddressthechallengesofaxisorientationandassociatingthedatapointswith(u;v)parametersonthepolycubedomain.Startingwithapolycubeapproximation,thischapteraddressesthesecondchallengeofconvertingthequadrilateraloutersurfaceofapolycubeintoasmoothbi-cubicsplinesurfacewiththesameconnectivity.Weobtainasurfacethat isalmosteverywhereG1continuouswitharationallinearreparameterization; has,justlikeregularC1bi-cubicsplines,fourdegreesoffreedom(d.o.f.)perpatch,essentiallytheinnerBeziercoecients; aordsarenable,nestedsequenceofspaces,sothatthenextnerleveloersfourtimesasmanygeometricd.o.f..Thelastbullet,derivinganestedsequenceofspaceswithG-transitions,isthemainnewcontributionofthischapter.Theconstruction,abbreviatedPGSinthefollowing,improvesonasimilarconstruction,[ 20 ],byrelaxingtheconstraintthatallprimaryvertices,beforerenement,beexclusivelyofvalence3or6.Addingverticesofvalence4and5accountsforthevertexcongurationsoccurringinapolycubequadmesh(Fig. 2-3 ).Rationallinearreparameterization(whenenforcingG1continuity)preservesamaximalnumberofuniformly-distributedadditionalfreeparametersateachrenementstep.Higher-orderreparameterizationsimplymoreconstraintsthatreducethed.o.f.count,reduceexibilityofthesurfacewherepatchesjoinandhencelessentheabilitytocomputeonthesurface[ 21 ].Thed.o.f.ofthenewbi-cubicconstructionhaveauniformlayout:theyareakintothesplinecoecientscijofbi-cubicB-splineswithdoubleknots, 22

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i.e.areco-locatedwiththeinnerfourBeziercoecientsbij;0
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acarefulalternativeinitializationoftheboundarycurvesviaanalgorithmforEdgeRecoveryinSection 3.4 .Unlike[ 20 ],whichbuiltrenementinthespiritofHierarchicalB-splines[ 8 ]byaddingupsurfacesofseverallevels,PGSwithEdgeRecoveryprovidestruenestedrenement.Thatis,agivensurfacecanexactlybere-representedatanerlevelasillustratedinFig. 3-1B andtheadditionald.o.f.oftheadaptiverenementallowmodelingfeaturesorfunctionsonthesurfacesatmultipleresolutionsFig. 3-1A 3.2RestrictedQuad-layoutsCompatiblewithRationalLinearTransitionMapsBelow,inSection 3.2.1 ,wedenetherestrictedpatchlayoutandaclassofpolyhedra,the[3;4;5;6]polyhedra,thatcanbeendowedwithapiecewisetensor-productsplinesurfacetosatisfyarestrictedpatchlayout.InSection 3.2.2 wereviewthedenitionofandresultsconcerningthegeometriccontinuityofstructurallysymmetricrationallineartransitionmaps.Finally,inSection 3.2.3 ,weshowhowweindexthetensor-productpatchesandtheircoecients. 3.2.1RestrictedQuad-layoutand[3;4;5;6]PolyhedronEuler'sformula,v)]TJ /F3 11.9552 Tf 13.002 0 Td[(e+f=,characterizesclosedpolyhedrabyrelatingthenumbersofverticesv,edgeseandfacesfofagivenpolyhedrontotheEulercharacteristic(Euler-Poincarecharacteristic)=2)]TJ /F1 11.9552 Tf 12.999 0 Td[(2g.Heregisthetopologicalgenus,i.e.thenumberofhandles.Forexample,Fig. 3-3 showsasurfaceofgenus2builtexclusivelyfromquadswhoseverticeshavevalencen=3;6;6and6.Wenotethatsincewecanassociatewitheach4-valentvertexfourhalf-edgesandfourquartersofattachedquadsthenetcontributionof4-valentverticestotheEulercountis1)]TJ /F1 11.9552 Tf 12.232 0 Td[(4=2+4=4=0.Thenaddingorremovingcheckerboardarrangementsofquadrilateralsdoesnotaectthelefthandsidev)]TJ /F3 11.9552 Tf 11.955 0 Td[(e+fofEuler'sformula. 24

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Lemma4of[ 22 ]provesthatstructurallysymmetricgeometricallysmoothsurfacesconstructionsmustrelatesomeabuttingpatcheswithquadraticorhigher-degreerepa-rameterizations{unlesstheyhavearestrictedlayout.Restrictedlayoutrelatesthesmoothnessofthesurfacetotheunderlying(polyhedral)layoutofthepatchesasfollows. Denition3(restrictedpatchlayout). Asurfacehasrestrictedpatchlayoutif,wherevertwoverticesareconnectedbyacurveacrosswhichtwosplinesmatchaccordingto(G1)with!linearandnotconstant,theverticeshaveanequalnumberofneighbors.NotethatthetwosplinesinDenition 3 mayconsistofmultiplepolynomialpieces. A BFigure3-2. Figure-8from[ 2 ].A)Identicationofedgesofthe4quadrilaterals.B)EmbeddinginR3with2g)]TJ /F1 11.9552 Tf 11.955 0 Td[(2=2cornerpointsofvalence8. [ 2 ]showedthatrestrictedpatchlayoutsexistandpresentedaconstructionofsplinesurfacesonpartitionsconsistingexclusivelyofquadrilateralsallofwhosecornervertices 25

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are8-valent(Fig. 3-2 ;toobtainadditionalgeometricdegreesoffreedomwithoutaectingtheEulercount,quadrilateralscanbepartitioned).Atthetime,thisbaselayoutappearedtobeonlyquad-layoutyieldingarestrictedpatchlayout.[ 2 ]alsoprovedthattheprojectivelinear(rationallinear)reparameterizationforconstructingsmoothsurfacesofgenusg>0fromtensor-productsplinesisessentiallyuniqueandresultsinalinearrelationofderivativesforadjacentpatches.TogetherwithLemma4of[ 22 ]thisimpliesthatconstructingmanifoldswitharationallinearreparameterizationrequiresarestrictedlayoutofthequadrilateralsthatisachievedbythefollowing. Denition4. ([n0;:::;n3]quad,[n0;:::;nm]polyhedron,restrictedpolyhedron)A[n0;:::;n3]quadisaquadrilateralfacetwhoseverticeshavevalencen2fn0;:::;n3g.Theedgesofa[n0;:::;n3]quadarelabeledhn;miedgesaccordingtotheapparentvalenceoftheirendpoints.Apolyhedronisa[n0;:::;nm]polyhedronifitconsistsentirelyof[bn0;:::;bn32fn0;:::;nmg]quads.Apolyhedronisarestrictedpolyhedronifitcanbeobtainedfroma[n0;:::;nm]polyhedronbysplittingits[bn0;:::;bn32fn0;:::;nmg]quads. Denition5. (polycubequad)Aquadrilateralfacetisapolycubequadifitisa[3;4;5;6]quad.Inthefollowing,wewillsplitthepolycubequadsonlyinauniform,binaryfashionandthe[3;4;5;6]polyhedronisendowedwithasurfaceconstructionthatsatisestherestrictedpatchlayoutbyrelatingconstantlythederivativesofpatchesabuttingacrosscurvesconnectingverticesofapparentvalence3and6(edgeslabeledh6;3iorh3;6i)andlinearlyacrossedgeslabeledh3;3iandh6;6i(Fig. 3-6 showsouredgelabelingconvention).Howeverotherpartitionsmaybeusefulforpracticalandartisticpurposes[ 23 24 ].A[3;6]quadwithn3verticesofvalencen=3contributes(n3)]TJ /F1 11.9552 Tf 12.057 0 Td[(2)=6tothev)]TJ /F3 11.9552 Tf 12.057 0 Td[(e+fcountofEuler'sformula.Forexample,usingexclusively[3;6]quadswithasinglevertexofvalencen=3,Fig. 3-3 consistsof12once-subdivided[3;6]quadsthathave4verticesofvalence3and6verticesofvalence6. 26

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A BFigure3-3. Thelayoutinducedbyonce-subdividedpolycubequadsallowsmodelingsmoothobjectsofanygenusgreaterthanoneusingonebi-cubicpolynomialpiecepersub-quad.A)3-neighborhood,level`=1[3;6]quads.B)Adipyramid-likeobjectofgenusg=2constructedusing26[3;6]quads. 3.2.2GeometricContinuityofStructurallySymmetricRationalLinearTransitionMapsWhilethesmoothnessoftheresultingsurfacecanbeexpressedinthelanguageofdierentialgeometry,i.e.intermsofoverlappingcharts[ 25 ]itsuces,andisoftenmoreecient,toexpresssmoothnessasagreementofone-sidedjets,i.e.equivalenceclassesofTaylorexpansions,alongthecurvewheretwosurfacepiecesaregluedtogether.Weusethisapproachinthefollowing. Denition6(geometriccontinuity,G1). Twosurfacepiecesbandbthatjoinalongacurve(t):=b(t;0)=b(0;t)joinG1ifthereexistsasuitablyorientedandnon-singularreparameterization:R2!R2,sothatthejets@kband@k(b),agreeateverypoint(t)fork=1.Thatis,adjacentsurfacepiecesarerelatedbyreparameterization,sothat,uptorstorder,b=b.Althoughisjustachangeofvariables,itschoiceiscrucialforthepropertiesofthesurface. 27

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Denition7(structurallysymmetricreparameterization). Areparameterization:R2!R2andthecorrespondingG1surfaceconstructionisstructurallysymmetric(orunbiased)ifboth@b=@(b)and@b=@(b).[ 2 ]showedthatastructurallysymmetricGktransitionwitharationallinearreparam-eterizationmusthavetheform@i(brb)(t;0)=@i(brb)(t;0);i=0;:::;k;rb(u;v):=1 s0+v264)]TJ /F3 11.9552 Tf 9.298 0 Td[(vsu+c0v375;rb(u;v):=1 s0)]TJ /F3 11.9552 Tf 11.955 0 Td[(v264s0u)]TJ /F18 11.9552 Tf 11.955 0 Td[(c0vv375;c0:=cos2 n0;s0:=sin2 n0;c1:=cos2 n1;s1:=sin2 n1;wheren0isthevalenceatb(0;0)and2Risdeterminedbythevalencen1atb(1;0).Alternatively, (u;v):=rbr)]TJ /F4 7.9701 Tf 6.586 0 Td[(1b=1 1+2v(c0+c1)264)]TJ /F3 11.9552 Tf 9.299 0 Td[(vu+2c0v375:connectsthedomainstoformacommondomainacrossthepre-imageof(t)andcompo-sitionuptovalencenmanymapsoftyperbdeneacommonpre-imageorchartofthepatchessurroundingavertex.Wenotethatbandbshare(t)andhencethederiva-tivesalongthecommonedgeagree:@1b(t;0)=@1b(0;t).Weneedonlycheckthat,foralinearfunction!(t),thepartialderivativeintheseconddirectionmatch: @2b(t;0)+@2b(0;t)=!(t)@1b(t;0):( G1)Therationallinearreparameterizationimpliesthat!(t)whichequalsapartialderivativeofthesecondcoordinateislinear:!(t):=@2[2](t;0)=(1)]TJ /F3 11.9552 Tf 11.956 0 Td[(t)2c0+t2()]TJ /F18 11.9552 Tf 9.298 0 Td[(c1): 28

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Inparticular,whentheapparentvalencesoftheedgeendpointsaren0=3andn1=6,then!(t)=1,i.e.thepatchderivativesareinaconstantrelationandthepatchesarejoinedparametricallyC1. 3.2.3IndexingtheSurfaceQuadsWepartitionthepolycubequadsuniformlysothatthepartitionathierarchylevel`yields2`2`sub-quadsindexedbythesuperscript(k;l)21:2`(Fig. 3-4 )whereweusedtheabbreviationa:b:=8>><>>:a;a+1;:::;b;ifab;a;a)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;:::;b;ifa>b:Ingeneral`canbechosentoprovidesucientexibilityformodeling.Whenfocusingonavertex(correspondingtotacubecorner),weassociatethesuperscript1;1withthesub-quadssurroundingthepoint.Eachsub-quad(k;l)ofa(top-level)quadwillbeassociatedwithonepolynomialpatchb;klthatwerepresentintensor-productBernstein-Bezier(BB)formofbi-degreed=3overtheunitsquare(u;v)2[0::1]2:b(u;v):=dXi=0dXj=0bijBdi(u)Bdj(v); (3{1)wherebijaretheBB-coecientsandBdk(t):=)]TJ /F7 7.9701 Tf 5.612 -4.379 Td[(dk(1)]TJ /F3 11.9552 Tf 12.203 0 Td[(t)d)]TJ /F7 7.9701 Tf 6.587 0 Td[(ktkisthekthBernstein-Bezierpolynomialofdegreed.TheBB-coecientsi;jofpatchinsub-quadk;listhendenotedby b;klij;i;j20:d:(3{2) 3.2.4GeometricSmoothnessConstraints Lemma1. Twopatchesp:=b;k1andq:=b;1kofdegreebi-3,withBB-coecientspijandqij,thatsharetheboundarycurvep(t;0)=q(0;t)joinG1withlinearscaling(G1)if 29

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b;42kk11k1Figure3-4. LabelingofquadandtheBB-netofcontrolpointsassociatedwithonesub-quad. andonlyifp01+q10=!0p10+(2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!0)p00; (3{3)p11+q11=(!0;!1); (3{4)p21+q12=(!0;!1); (3{5)p31+q13=(2+!1)p30)]TJ /F3 11.9552 Tf 11.955 0 Td[(!1p20: (3{6)(!0;!1):=1 3)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(2!0p20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!1p00+(6)]TJ /F1 11.9552 Tf 11.955 0 Td[(2!0+!1)p10(!0;!1):=1 3)]TJ /F3 11.9552 Tf 5.479 -9.683 Td[(!0p30)]TJ /F1 11.9552 Tf 11.955 0 Td[(2!1p10+(6)]TJ /F3 11.9552 Tf 11.955 0 Td[(!0+2!1)p20: Proof. Equation(G1)isequivalenttomatchingfourBB-coecients.Equations( 3{3 )and( 3{6 )enforce(G1)att=0andt=1,respectively.Toverifytheremainingobservethat 30

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thefourcoecientsof)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[((1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t)!0+t!1(@1pk)(t;0)are!0(p10)]TJ /F6 11.9552 Tf 11.955 0 Td[(p00);2!0(p20)]TJ /F6 11.9552 Tf 11.956 0 Td[(p10)+!1(p10)]TJ /F6 11.9552 Tf 11.956 0 Td[(p00); (3{7)!0(p30)]TJ /F6 11.9552 Tf 11.955 0 Td[(p20)+2!1(p20)]TJ /F6 11.9552 Tf 11.955 0 Td[(p10);!1(p30)]TJ /F6 11.9552 Tf 11.955 0 Td[(p20) (3{8)andthecoecientsof(@2q)(0;t)+(@2p)(t;0)arep01+q10)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p00;3(p11+q11)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p10); (3{9)3(p21+q12)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p20);p31+q13)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p30: (3{10) Wewilladdsuperscripts;kand;ktoindicatepatchandsub-patchalonganedgeasweconsidertheG1stripofBB-coecientsb;k1ij;b;k1ij;i=0;1;j=0;1;2;3;k=1:2`involvedintheG1joinbetweentwopatchesbandb(Fig. 3-5 ). b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;121b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;131b;k111b;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(1;110b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;120b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;130b;k110b;k120b;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(121b;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(131b;1k11Figure3-5. IndicesofBB-coecientsoftheG1stripalongah3;6iedge. Lemma2. (Fig. 3-5 showstheindexing)Letb;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(1;1andb;1;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1joinG1accordingto(G1)andb;k1andb;1kjoinG1accordingto(G1).Thenbothb;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;1andb;k1joinC1 31

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uptorstorderandb;1;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1andb;1kjoinC1uptorstorderifandonlyifb;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;131=(b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;121+b;k111)=2;b;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(131=(b;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(121+b;1k11)=2; (3{11)and(!k)]TJ /F4 7.9701 Tf 6.586 0 Td[(10+!k1)()]TJ /F5 11.9552 Tf 9.298 0 Td[(b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;110+2b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;120)]TJ /F1 11.9552 Tf 11.955 0 Td[(2b;k110+b;k;120)=0: (3{12) Proof. Settingb;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;131andb;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(131accordingto( 3{11 )enforcestheC1constraintsawayfromthecentralpointsharedbyallfourpatches.SymbolicallysolvingthefourG1-constraintsoftype( 3{5 )and( 3{6 )forb;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1andb;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(1,respectively( 3{3 )and( 3{4 )forb;kandb;k,thenenforcesb;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(130=(b;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(120+b;1k10)=2ifandonlyif( 3{12 )holds.Hereweusethat,in(G1),!k)]TJ /F4 7.9701 Tf 6.586 0 Td[(11=!k0=(!k)]TJ /F4 7.9701 Tf 6.587 0 Td[(10+!k1)=2duetotherationallinearreparameterization. Wenotethatwhentheboundaryislabeledash6;6iorh3;3iandk=2`=2(themidpointofthecurve)thenourconstructionwillautomaticallyyield!k)]TJ /F4 7.9701 Tf 6.586 0 Td[(10+!k1=0andtherstfactorof( 3{12 )vanishes.Iftheedgeislabeledash3;6i,wecandividetheexpressionin( 3{12 )by(!k)]TJ /F4 7.9701 Tf 6.586 0 Td[(10+!k1)6=0andobtaintheconstraintthatthepiecesof(t)mustbeC2connectedsinceitrepresentsthedierenceoftwoseconddierencesattheircommonpoint. 3.3G1PolycubeSurfaces 3.3.1LabelingBoundaryEdgesbyTheirApparentValenceAsimpleinsightisthata5-valentvertexcanbemadetoappeartobe6-valentvertexfromthreedirectionsandtobe4-valentfromtheremainingtwoasfollows.Let,withoutlossofgenerality,b;1100betheBB-coecientsharednsurfacepatches(Fig. 3-5 ),andchooselocalcoordinatessuchthatb;1100istheorigin.Wesaythatthevertexofvalence 32

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nappearstobeofvalencepforanedgeebetweenpatchesand+1,ifthetangentcoecientsb;1110,2f)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;;+1g(modulon)satisfy2cpb;1110=b+1;1110+b)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;1110;cp:=cos2 p: (3{13)Specically,whenn=5,wetemporarilyinsertadummyedge(dashededgeinFig. 3-6 a)toobtaintangentcoecientsb;1110,20:5andprojecttheseontoananeimageofacirclewithcenterb;1100.Removingthedummytangent,weareleftwithn=5tangentcoecientsb;1110,20:4andopeninganglesof[1;1;1;1;2]60.Threeoftheseedges,e1,e2,e3,satisfy( 3{14 ),hencereceivethelabel6,whiletheothertwo,e0,e4satisfy( 3{15 ),andreceivethelabel4:2c6b;1110=b;1110=b+1;1110+b)]TJ /F4 7.9701 Tf 6.587 -.001 Td[(1;1110; (3{14)2c4b;1110=0=b+1;1110+b)]TJ /F4 7.9701 Tf 6.586 0 Td[(1;1110;cn:=cos2 n: (3{15)Inapolycubeconguration(cf.thethirdcolumnofFig. 2-3 ),thesecondofthethreeedgeswithlabel6,e2,isbydefaultassignedtobetheedgetowardsthesingleneighborthatisnottheplanesharedbytheotherfour.Analogously,theedgesofa4-valentvertexcanbeassignedlabels3;4;6;4asshowninFig. 3-6 bsothatthevertexisapartofalongeredgebetweenavertexofvalence6andavertexofvalence3(andtherebyavoidsedgesconnectingvalence4andvalencem6=4).Foredgesequenceswithlabelshm;4ih4;niwherem6=n,wewillonlyguaranteeC0(C1)]TJ /F7 7.9701 Tf 6.587 0 Td[(inactuality,Section 3.3.3 ),notG1orC1continuity.Appendix A speciesapre-processingEdgeLabelingalgorithmthatpre-labelstheinputquadmeshtominimizethenumberofsuchedgepairs. 33

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e1e0e0e2e1e3e3e4e2Figure3-6. Apparentcongurationoftangentsemanatingfromverticesofvalencen=5;4basedonsymmetricallydistributingn=6tangentsandremovingtangents. 3.3.2PGS:ConstructingaSmooth,PiecewiseBi-3SurfacefromaPolycube 3.3.2.1InitializationForanhn;miedge,weabbreviate!kij:=!(k 2`);!(t):=2(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t)cn)]TJ /F3 11.9552 Tf 11.955 0 Td[(tcm;cn:=cos(2 n);anddene1bearowvectorofones,bijthevectorwithentriesb;11ijaccordingtocounterclockwise(ccw)ordering,andb+ijthevectorsamevectorexceptwithentriesshiftedccwbyone.Weinitializeasfollows.EdgeLabelinglabelsalledgessothatthelabelsare3,4,or6.OneCatmull-Clarksubdivisionstepisappliedtotheinputmesh.Interpretingtheresultingquadmeshpointsasbi-3B-splinecontrolpointsgeneratesdefaultdegreesoffreedom(d.o.f.)thatformacollectionofB-spline-likecontrolpointscfortheG)]TJ /F1 11.9552 Tf 9.299 0 Td[(splines.Specically,therulesforconversionfromB-splinetoBB-formdenea(collectionofnon-boundary)BB-coecientsb:=fb;klij;i;j2f1;2gg: 34

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biciA BFigure3-7. B-spline-likecontrolpointsc(green)anddependentBB-coecientsobtainedbyC1constraints(gray)orG1constraints(G1)inred.A)Analogouscurvecase.B)G1splined.o.f.:duetotheinitializationbyoneCatmull-Clarkstep,therearefourpatchesperpolycubequadandeachhas22controlpointsc. (Theconversionruleisb;kl11=4v00+2v10+2v01+v11)=9forfourB-splinecontrolpointsvijformingthequad.)Foraregulargridlayoutthepointsinbcoincidewith(buthavedierentbasisfunctionsthan)thecoecientsofC1bicubicsplineswithdoubleknots.Weinitializec:=b.Thereareexactly22suchcontrolpointsperquadoftheCatmull-Clarkrenedpolycubemesh(Fig. 3-7 ).WecannowstatethePGSalgorithm. 3.3.2.2PGS(polycubeG-spline)algorithmInput:4NB-spline-likecontrolpointsc,4foreachoftheNfacetsofapolycubequadmesh;arenementlevel`.Output:Asurfaceconsistingofbi-cubictensor-productsplinesurfacesbthatjoinG1{exceptalongedgeslabeledh4;niwherethesurfaceisonlyguaranteedtobeC0(C1)]TJ /F7 7.9701 Tf 6.587 0 Td[().Overview:Visitingeachsub-patchb;k1,b;k1correspondingtotheunsubdivided(polycube)quadedge,westartfromeachh6;miedgeandworkalongtheoriginaledgetothemiddleindex,k=1:`where`:=2`)]TJ /F4 7.9701 Tf 6.586 0 Td[(1ishalf-waystoitsneighbor.IntheprocessweadjusttheBB-coecientstoformaG1strip.Thenwecompletetheinputhalfedges 35

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Steps1,2Steps3,4Step5b11b21b31bk)]TJ /F4 7.9701 Tf 6.587 0 Td[(131b00b10b20b30bk)]TJ /F4 7.9701 Tf 6.587 0 Td[(130Figure3-8. StepsofPGS(foreachlabelinf6,4,3g). startingfromoutgoinglabels4andnallywecompletetheremainingoriginaledgesfromthemiddletowardstheverticesofvalence3.Algorithm: 0. Setallinnercoecientsb;ijkl:=c;ijklforallsubpatcheswithindices;ijand0
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vertex,applytheprojectionmatrixPtothetangentcoecientsb10 Pb10;P:=(1+C)=n2Rnn (3{18)1(i;j):=1;C(i;j):=2cj)]TJ /F7 7.9701 Tf 6.587 0 Td[(i:(Forn=5,Pensuresthatedgese0,ande3respectivelye1ande4arecollinear.)Forn=5ignorethedummyentryinb10.Forn=4,set(cf.( 3{4 )with!0=0)b10:=3(b11+b+11)+!1b00 6+!1: (3{19) 2. Letr:=;1(!10;!11)andr2Rn3thevectorwithentriesrincounterclockwiseorder.Ifn=6,adjustthesecondderivativesatthevertexb20 b20)]TJ /F1 11.9552 Tf 13.151 8.088 Td[(()]TJ /F1 11.9552 Tf 9.298 0 Td[(1) 4a;a:=n)]TJ /F4 7.9701 Tf 6.587 0 Td[(1X=0()]TJ /F1 11.9552 Tf 9.299 0 Td[(1)r; (3{20)Ifn6=4,formthenncirculantmatrixMM(i;j):=(1ifi=jori=j)]TJ /F1 11.9552 Tf 11.955 0 Td[(1modn0else:anddenotebyM)]TJ /F1 11.9552 Tf 10.986 -4.338 Td[(thepseudo-inverseofMwhenn=6andM)]TJ /F4 7.9701 Tf 6.586 0 Td[(1otherwise.Setb11:=M)]TJ /F6 11.9552 Tf 7.085 -4.936 Td[(r: (3{21) 3. Foreachh6;miedge,where!6=0,makethecurvesC2uptothemiddleoftheedge.Fork=1:`b;k+1;110:=b;k120+1 2(b;k+1;120)]TJ /F5 11.9552 Tf 11.956 0 Td[(b;k110): (3{22)Foreachh3;miedgestartingfromthemidpoint,makethecurvesC2.Fork=`:1(notetheordering!)b;k120:=b;k+1;110)]TJ /F1 11.9552 Tf 13.151 8.087 Td[(1 2(b;k+1;120)]TJ /F5 11.9552 Tf 11.955 0 Td[(b;k110): (3{23)Observethath3;6iedgesareexecutedonlyafterh6;3iedges. 37

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4. Fork=1:`,i.e.uptothemiddleoftheedge,minimallyperturbthetransversalderivativestoenforceG1continuity:b;k111 b;k111+1 2)]TJ /F3 11.9552 Tf 4.948 -9.684 Td[(;k(!k0;!k1))]TJ /F5 11.9552 Tf 11.955 0 Td[(b;k111)]TJ /F5 11.9552 Tf 11.955 0 Td[(b;1k11; (3{24)b;k121 b;k121+1 2)]TJ /F3 11.9552 Tf 4.948 -9.684 Td[(;k(!k0;!k1))]TJ /F5 11.9552 Tf 11.955 0 Td[(b;k111)]TJ /F5 11.9552 Tf 11.955 0 Td[(b;1k11; 5. Fork=1:`averageuptothemidpoint(toenforceC1continuitywithinthepolycubequadsandtheneighborof):b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;130:=(b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;120+b;k110)=2;b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;131:=(b;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1;121+b;k111)=2; (3{25)b;1;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(131:=(b;1;k)]TJ /F4 7.9701 Tf 6.587 0 Td[(121+b;1k11)=2:ThiscompletesthePGSalgorithm. 3.3.3PropertiesoftheSurfacesSince!isconstantforh3;6iedges,thecorrespondingpairsoftensor-productsplinepatchesjoiningacrossah3;6iedgeareparametricallyC1connected.Acrossequallylabelededges(h3;3i,h4;4i,h6;6i)thepatchesareG1connected.Accordingto[ 22 ],wecannothopeforsmoothnesswhen!islinearandwehaveopposinghn;4iandh4;miedgesformingalabelsequencehn;4i,h4;miwithn6=m.Thesubtlechallengeisthatifn6=mthenthelengthsofthetangentsofthetransversalcurvewithcoecientsb;113i,i=0:3,b;1130)]TJ /F5 11.9552 Tf 11.955 0 Td[(b;1120=b+1;1110)]TJ /F5 11.9552 Tf 11.955 0 Td[(b+1;1130;butb;1131)]TJ /F5 11.9552 Tf 11.955 0 Td[(b;1121=(b+1;1111)]TJ /F5 11.9552 Tf 11.955 0 Td[(b+1;1131);where6=1:donotagreewhen!islinear.Nevertheless,sequenceshn;4i,h4;miwithn6=m,oftendonotvisiblydisruptsmoothness:highlightlinediscontinuitiesareverysmallwhendetected(Fig. 3-9 ).Also,whenthepartialderivativesofthetransversalcurveareallcollinear,thenthesurfaceissmoothsincethedeterminantalongthetransversalboundarydet[(@1b;k1)(t;0);(@2b;k1)(t;0);(@1b;k+1;1)(t;0)]vanishes.Inpreparationforrenementwenotethefollowing. 38

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Figure3-9. Highlightlinemismatchvisibleunderzoom(1/100thoftheedgelength). Lemma3. Theplacementofthevertexcoecientsb;1100byPGSisidenticalto[ 20 ]whereb1100:=1 n1Pb1110. Proof. Denotebybijthevectorofb;11ijsurroundingthevalencenvertexandbyb+ijthevectorbijwiththeindicesincrementedbyone.TheinitializationbyB-splinetoBB-formconversiondenesb10:=(b11+b+11)=2.Since1C=[0;:::;0];and11=n1;theclaimfollowsfrom1Pb10=11b10=n+1Cb10=n=1b10 (3{26)=1(b11+b+111)=2=1b11: 3.3.4ExamplesofPGSSurfacesExamplesofsurfacesobtainedfromvariouspolycubecongurationsareshowninFig. D-1 andFig. D-2 .Inparticular,thetestcasesinFig. D-1 includeverticesofvalence5surroundedbyverticesofvalence3,4,5,6labeledsothatthesurfacesaresmoothalongtheseedges. 39

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3.4RenementAgivensurfaceisunchangedwhensplittingthepatchesviadeCasteljau'salgorithm.Buttheresultingner-levelBeziercoecientsareclearlynotalld.o.f.oftheG1polycubesplinespace.Equallyobvious,partitioningthepolycubequadstoserveasinputwouldleadtodierentsurfaces{forexamplethepatchescorrespondingtointeriorpolycubefacetswouldbeat.OnecanaddsurfacesatdierentlevelsofrenementasinFig. 3-10 (akintoHierarchicalB-splines[ 8 ]),butforapplicationsitismoreconvenienttohaveanadaptivebutuniformrepresentationofthed.o.f.withouthavingtoaddupfunctionsofalllevels. A BFigure3-10. Hierarchicaledit:ata6-valentvertexanebasicfunction(atlevel`=2)isaddedtoalevel`=1(coarse)basicfunction(showningold)inordertocreateacombined(red)functionthatissmoothalsoacrossthe6-6edgeconnectingtwoverticesofvalencen=6.A)Coarsebasicfunctionwithneaddition.Theupperrightinsetdisplayshighlightlinesonthemulti-levelsurface.B)Level`=1basicfunction+level`=2basicfunction=hierarchicaledit. 40

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Wewouldlikeasubdivisionalgorithmforthed.o.f.wherethenextlevelisobtainedasalocalconvexcombinationofthed.o.f.ofthepreviouslevel.Canone,saybybinarypartitionasintheCatmull-Clarkalgorithm,deriveanewB-spline-likecontrolnetsothatapplicationofPGSyieldsthesamesurface?Wethinkthisisnotpossible,alreadyduetotheinitial22splitacrosswhichthepiecesonlyjoinC1{whereasthemodelfortheinitializationofPGSisconversionofaC2bicubicB-splinesurfacetoBezierform.HoweverarenementcanbeobtainedinanalogywithbicubicB-splineswithdoubleknotsandobservingthat,forC1bi-3splines,theB-splinecontrolpointsandtheinnercontrolpointsb:=fb;klij;0
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c;k;121c;k+1;111a;k20aka;k+110c;1;k12c;1;k+111Figure3-11. AveragesofpolycubesplinecoecientsusedintheEdgeRecoveryalgorithm. Theconstructionofboundarycurvesintheinteriorandforallh4;4iedgesistrivial,namelyaveragingc=btoenforceC1transitions.ThechallengeisthattheG1con-straints( 3{3 ){( 3{6 )linkboundarycoecientsallalongtheboundaries,resultinginaglobalsystem.Thekeytoalocalconstructionis( 3{12 )ofLemma 2 .When!(k=2`)=0thenwecanlocallyapplyC1constraintstoseethatb;k;130=(b;k;121+b;1;k21+b;k;111+b;1;k11)=4 (3{28)andwecanunravelthelinkedconstraintsfromthemiddlek=`)]TJ /F1 11.9552 Tf 12.391 0 Td[(1ofthesubdividedboundarycurvebacktothevertexwithoutgoinglabelm6=4.Similarly,when!(k=2`)6=0,theboundarycurveshavetobeC2.TheseC2constraintsonthecurvetogetherwithLemma 3 yielda33systemofequationsattheendpoints.Thissystemiseasilysolved( 3{29 )andformsthestarttolocallysolvefortheboundarycoecients.TheresultisthefollowingalgorithmforEdgeRecovery. 42

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3.4.2ER(EdgeRecovery)AlgorithmWeabbreviate(Fig. 3-11 )a;k20:=(c;k;121+c;1;k12)=2;a;k+110:=(c;k+1;111+c;1;k+111)=2;ak:=(a;k+110+a;k20)=2:Input:Splinecoecientsc;k;`ij,i;j2f1;2gofapartitionedpolycubequadmesh,fourpersub-quad.Output:CubicboundarycurvessuitableforPGS.Algorithm: 1. InitializethevertexBB-coecientsb;1100:=1 nn)]TJ /F4 7.9701 Tf 6.586 0 Td[(1X=0b;1111:( 10') 2. Where!k1=0(notethatj!k0j1),fork=kb;k;130:=ak;( 6')b;k;120:=1 6)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k0(6a;k20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k0b;k;130);( 5')b;k;110:=1 6)]TJ /F1 11.9552 Tf 11.955 0 Td[(2!k0(6a;k10)]TJ /F1 11.9552 Tf 11.955 0 Td[(2!k0b;k;120);( 4')andfork:=k)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;:::;1(when!k16=0)b;k;130:=1 2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k1(ak)]TJ /F3 11.9552 Tf 11.956 0 Td[(!k1b;k+1;110);( 3k+')b;k;120:=1 )]TJ /F3 11.9552 Tf 9.299 0 Td[(!k1(ak)]TJ /F1 11.9552 Tf 11.955 0 Td[((2+!k1)b;k;130);( 6k')b;k;110:=1 )]TJ /F1 11.9552 Tf 9.299 0 Td[(2!k1(6a;k20)]TJ /F1 11.9552 Tf 11.955 0 Td[((6)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k0+2!k1)b;k;120)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k0b;k;130):( 5k') 43

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3. Ifw:=!`16=0(e.g.!`1=1forah6;3iedge,or!`1=)]TJ /F1 11.9552 Tf 9.298 0 Td[(1forah3;6iedge)thenfork:=124b;k;110b;k;120b;k;13035=Mw2664b;1;100a;110a;120a;2103775 (3{29)M1:=1 1222422132)]TJ /F1 11.9552 Tf 9.299 0 Td[(34263685)]TJ /F1 11.9552 Tf 9.299 0 Td[(5212693935;M)]TJ /F4 7.9701 Tf 6.586 0 Td[(1:=1 4624)]TJ /F1 11.9552 Tf 9.299 0 Td[(6361422)]TJ /F1 11.9552 Tf 9.299 0 Td[(12497)]TJ /F1 11.9552 Tf 9.299 0 Td[(212)]TJ /F1 11.9552 Tf 9.299 0 Td[(33935:andthenfork=2;:::;`)]TJ /F1 11.9552 Tf 11.955 0 Td[(1,b;k;110:=1 !k0(2ak)]TJ /F4 7.9701 Tf 6.586 0 Td[(1)]TJ /F1 11.9552 Tf 11.955 0 Td[((2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!k0)b;k;100);( 3k")b;k;120:=1 2!k0(6a;k10)]TJ /F1 11.9552 Tf 11.955 0 Td[((6)]TJ /F1 11.9552 Tf 11.955 0 Td[(2!k0+!k1)b;k;110( 4k")+!k1b;k;100):b;k;130:=1 2+!k1(2ak+!k1b;k;120):( 6k") Lemma4. Let^bbetheinnerBernstein-BeziercontrolpointsoutputbyPGS.Ifc=^bthenforb;k;1i0,i2f0;1;2;3gconstructedbyEdgeRecovery,( 3{3 ){( 3{6 )hold. Proof. Theequationlabelsintherangeof3;:::;6ofEdgeRecoveryindicatetheG1constraintsthatareenforcedbysolvingfortheboundarycoecients.Sinceallcoecientsaredeterminedbytheseconstraintsandsincethe^bhavenotbeenperturbed,theoutputofPGShasbeenreconstructed.Thereforealsothemissingequationsoftype(4k')and(5k")musthold. 3.4.3PGSERAlgorithmWedeneanalternativealgorithmPGSERbyreplacingStep0ofalgorithmPGSwiththemoresophisticatedEdgeRecoveryandthenapplyingtheremainingSteps1-5ofPGS.Thisyieldsasurfacewith^b=c,i.e.bispreservedwhenrunningthemodiedalgorithmPGSER. 44

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A B CFigure3-12. SurfaceswiththeircorrespondingBB-netsandhighlightlinesaftereachmajorstepofPGSER.A)InitializedmeshafterStep0.B)EdgeRecoveryappliedtoinput.C)outputofPGSER. Corollary1. ApplyingPGSERwithc:=^breproducesthePGSsurfacewithinnerBeziercoecients^b.Wecannowderivethed.o.f.c`+1oftherepresentationatlevel`+1byapplyingdeCasteljau'salgorithmtotherepresentationatlevel`.Ifthec`+1arenotperturbed,i.e.c`+1=^b`+1thenwerepresentthesurfaceexactly,otherwiseweapplySteps1-5ofPGStoobtainasurfacethatistangentcontinuousexceptpossiblyalongedgesh4;mi,m6=4.Fig. D-2 showsmorecomplexexamplesurfacesgeneratedbyPGSER. 3.5DiscussionSincec=^bchangesunderasecondapplicationofPGS,PGSisnotsuitableforgeneratinganexactrenedrepresentation.However,sinceStep0ofPGSissimple,weuseitfortheinitialconstructionbutuseEdgeRecoveryinplaceofStep0thereafter 45

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forallhigherlevels.Experimentally,weobservedthatsurfacesgeneratedbyPGShaveslightlymoreuniformlydistributedhighlightlinesthanthoseofPGSERprobablyduetotheunderlyingC2splineinterpretation.Toderiveabasisforeachoftherenedspaces,weapplyPGSERwhenonec;klij=1andallothercarezero.ThisyieldsaB-spline-likebasicfunctionanalogoustoabi-cubicB-splinewithdoubleknotsinthetensor-productcasebutwithlargersupport,allalongedges.(Forindividualbasicfunctionsb;klij6=c;klijbutthelinearcombinationofallbasicfunctionswithweightscgeneratesapiecewisepolynomialsurfacewithb=c.)Duetothesubtlebutfundamentalconstraintsprovenin[ 2 ],thealgorithmcannotguaranteesmoothnessforpathswithedgesequenceshn;4iandh4;mi,n6=m.Awell-knownremedyistochooseaquadratic!(t):=2ci(1)]TJ /F3 11.9552 Tf 12.567 0 Td[(t)2,i2fn;mgfortheseedges.However,aquadratic!forbicubicpatchesmeetingwithgeometriccontinuityimpliesthatthesharedboundariesarepiecewisequadraticratherthancubic(orweloosevector-valueddegreesoffreedom).IftheyareadditionallyC2-connectedduetoLemma 2 thenrenementdoesnotgenerateadditionald.o.f.alongthesharedboundaries,becauseaC2piecewisequadraticcurveisasinglequadraticcurve.Moreover,quadraticboundarycurvepiecesresultinshapedefectsnearhigher-ordersaddles.Finally,varyingreparameteriza-tionsmakesrenementmorecomplex.Forallthesereasons,wecurrentlychoosetonotenforcesmoothnessacrosshn;4iedges.ThegoalofEdgeLabelingistoassignoutgoinglabelstoverticesofvalence4and5tominimizethenumberofedgesequenceswithlabelshm;4ih4;niwherem6=n.EdgeLabelingtypicallysucceedsineliminatingsuchsequencesbut,ingeneral,decidingwhetherandhowtooptimizethelabelsoradjustthequadmeshtoguaranteeeliminationisnotpossible(Appendix A ).Wenotethatforrationallinearreparameterizations,therestrictionsonedgevalencelabelsh4;ni,n6=4holdalsoforsurfacesofhigherdegreeandthathigherdegreeresultsinrapidgrowthofd.o.f.whichisoftenundesirable.Forcompleteness,Appendix B 46

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outlinesonesuchconstruction,namelythebi-5versionofthePGSAlgorithm.However,futureresearchisneededtoshowwhethertheexhibitedbettershapeduetotheabsenceofinternalmicropatchtransitionsosetsthedrawbacksofscalabilityandrenementcomplexityforsuchschemes. 47

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CHAPTER4CONSTRUCTINGAWATERTIGHTSPLINEATLASFROMHETEROGENEOUSTRIMMEDSURFACES 4.1MotivationStylists,theprofessionalswhotranslatehigh-enddesignintocomputer-basedmodels,focusontheshapeofsurfacesasexpressedbycurvatureprolesandhighlightlines[ 28 ].Highlightlinesareacommonly-usedsurfaceinterrogationtoolFig. 4-1 approximatingtheparallelarrangementoftubelightsinacarshowroom.Unlessexplicitlyintendedasasurfacefeature,abruptchangesinthehighlightlinedistributionarenotwantedsincedistortedreectionsmaketheproductappearlesswelldesigned.Alsocurvature A B C DFigure4-1. Evaluatingsurfacequalitybyreectionandhighlightlines.CommonlyusedbyA)autobodydesignersandB)dentrepairprofessionals.C)Inaddition,orthogonalplanarcuts(yellow)areusedbystylistsinhigh-enddesignsoftwares(e.g.ICEMSurf[ 26 ]).D)ThegeometricallysmoothG-splinecapof[ 27 ]doesnotdisturbtheuniformdistributionofhighlightlines. 48

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interrogationisappliedtoorthogonalplanarcutsofthesurfacetodetectoscillationsoftheprole.Inthecontextoftheautomotivedesignindustry,splinesurfaceswithaesthetic,non-oscillatinghighlightlinesarereferredtoas`ClassA'surfaces[ 28 29 ].ClassAsurfacessatisfy,dependingontheapplicationareaandcontractualagreement,certainhardgeometricconstraints[ 30 ].Inparticular,theyallow,duetomanufacturingtolerances,amismatchofnormalsalongacurvebetweentwosurfacepiecesofonetenthofadegree.InordertoconstructclassAsurfaces,stylistsalmostuniversallyresorttousingtrimmedpatches.Patchesarepolynomialorrationalsurfacepieces(Section 2.1.3 ).Whentherectangulardomainisfurtherrestrictedbynon-constant,e.g.linearorquadraticcurves,thepatchistrimmed(Fig. 4-2 ).Inhigh-enddesign,primarysurfacesarejux-taposedandtheirsharptransitionsblendedbyintermediatesurfacescalledllets.Theswitchfromprimarytolletsurfaceorbetweensurfacesismodeledbytrimcurvesinthedomain.Thefreejuxtapositionofsurfacepiecesreectsthetraditionalworkowofdesignusingclay[ 31 ]thatisemulatedinleadinghigh-enddesignsoftwares(e.gICEMSurf[ 26 ]).HerepolynomialpiecesinBernstein-Bezierform(BBform)arelaiddowntocaptureprimaryshapeandthencutback(trimmed)toinserttransitions(llets)betweentheprimarysurfaces;additionaltrimmedpatchesarettedtollgaps.Specialdesign A B CFigure4-2. Trimming:Surfacepieces,s:R2!R3,maptheunitrectangleto3-space.Trimcurves,c:R!R2,deneandrestricttheirdomain.A)Untrimmedsurfacepiece.B)Trimcurve.C)Resultingtrimmedsurfacepiece. 49

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toolsframeglobal(non-interpatch)boundariesbystripsofpatchesofpossiblyverydif-ferentscale(Fig. 4-3 right,Fig. 1-1 ).Forthestylist,thisapproachisecientandyieldshighqualityprimarysurfaces;buttheresultinganarchyofparameterizationmakesthepiecewisesurfaceill-suitedfordownstreamdesign,analysisandsimulation.Consequently,currentpracticeistore-approximatethecarefully-designedsurfacebytrianglesorevenvoxels,therebythrowingawaymuchtheoriginalsurfaceinformationandthussacricinganalysisaccuracyandseveringthefeedbackloopinthedevelopmentcycle. Figure4-3. Piecesofunrelatedsizeandparameterizationarejuxtaposed.Theoverlaidgridrepresentsparameterlines,atdierentlevelsofrenement.Theenlargementshowstheorderofmagnitudedierenceinthesizeofthepatchesformingtheheadlightrimcomparedtothetrimmed-ofenderpatchwithitscompletelyseparateparameterization. Academicresearchhas,overthelastquarterofacentury,developedstructuredalter-nativestothisfreejuxtapositionofpolynomialpiecescalledfree-formsurfaces.Thenewtoolscanbeusedtocreateconsistentcomplexesofsmoothlymatchingpolynomialsalsoforirregularlayouts.Butneitherofthetwomainalternatives,hierarchicalrenementbysubdivisionsurfaces[ 32 { 34 ]andgeneralizedgeometricallysmooth(G-spline)constructions[ 7 35 36 ]hasreplacedtrimmedsurfacesinhigh-endmanufacturingdesign.Thetrainingofstylistsandestablishedpracticespartlyexplainthelackofadoption:afterallanimationandgameartistssuccessfullymodelcomplexgeometrybysubdivisionsurfacesandalsotheCADindustryusesubdivisionsurfaceshigh-enddesignsforhigh-qualityrenderingofsore-approximateddesigns.However,therearedeeper,morefundamentalreasonswhythenewapproacheshavenotdisplacedcurrentpracticein,say,automobiledesign.Stylistshaveto 50

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satisfyhighexpectationsonthedistributionandmonotonicityofcurvature.Low-degree,multi-piecesubdivisionandG-splinesurfacesareeitherknownnottodeliverthesamequalityortheyhavenotbeendemonstratedinapracticaldesignow,thelatterbeingamainfocusofthiswork.Asastandardabstractionfromthereal-lifecomplexityofblendingtrimmedsurfaces,G-splineconstructionstypicallyconsideranetworkofquadrilateralfacetsorquadsthatoutlinethenalsurface.SuchaquadmeshmayarisefromonestepofCatmull-Clarkrenement[ 32 ]ofanyinputmesh,orfromreconstructingscanneddatabysophisticatedquad-meshingalgorithms[ 37 { 48 ]orlaidoutbyastylist.Therearetypically`star-like'congurations(bycontrastpolarcongurationsmimicthenaturalgridlaidoutonacone).Focusingonconstructionsthatuseanitenumberoftensor-productsplinepiecesarrangedinsuchalayout,theconstructionsof[ 49 50 ]arehigh-degreerational,whereas[ 51 ]userootsandexponentialfunctions.AmongpolynomialG2constructionsusingasinglepatchperquad,thedegreecanbeashighasbi-18[ 52 ],bi-9[ 53 54 ]withdegreebi-6oeredby[ 55 56 ]iftheexibilitytomodelshapesatextraordinarypointsisrestrictedtoquadratics.Theconstructionsof[ 57 58 ]explorethespaceofdegreebi-7surfacestollann-sidedholeinabi-3splinecomplexbyminimizingahigh-orderfunctional.Recentprogresshasfocusedonoptimizingshaperatherthanjustsatisfyingthealgebraicconstraintsforsmoothnessatpatchtransitions.G2constructionsofdegreebi-5usinga22arrangementofpatchesperquadanddegreebi-6nowdeliverformallyC2surfacesthatshowagoodhighlightlinedistributionforanobstaclecourseofdicultquadmeshes[ 59 ].G1constructionsofevenlowerdegreeshowsimilarlygoodhighlightlinedistributionfortheobstaclecourse(e.g.[ 36 ]).Remarkably,evenaconstructionresultinginformallynon-smoothtransitionscanbeofhighquality[ 60 ].Despitetheapparentmyriadofparametricallyandgeometricallycontinuoussurfaceconstructions,fewworkstacklethegapbetweenhigh-leveldesignandwatertightfree-formsurfacing.ExistingapproachesexploretheconversionoftrimmedNURBSsurfacesto 51

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high-degreeBeziersurfaces[ 61 ],T-splinesurfaces[ 62 ],andsubdivisionsurfaces[ 63 64 ],however,theseconstructionsdonotaddresstheissueofirregularlayoutsandseemtofavorhand-builtisolatedexamplesasinputs.Moreover,shapeguaranteesarenotenforcedinthersttwoanditiswellknownthatshapeproblemsareinherenttothelatter[ 34 ].Inthefollowing,weoerapracticalapproachforconstructingtheglobal,uniform,dierentiable,andgradedatlasesofreal-worldartifactsdesignedbystylistsascollectionsofseparateandheterogeneoustrimmeddesignsurfaces(`patchsoup').Heresplinesurfacescloselyapproximatetheuntrimmedportionsofprimarysurfacepiecesandembedtheminone,bydefaultsmoothly-connected,scaold.Theresultisasplinemeshthatdefaultstoalow-degree(bi3)tensor-productsplinewhereverthelayoutofthequadrilateralsisaregulartensor-productgridandtoabi4splinewheneverthelayoutisirregular(`star-like'). 4.2ConstructingaSplineAtlasfromaCollectionofTrimmedPatches 4.2.1HomogenizingthePatchRepresentationInthepreviousSection 4.1 wearguedthathigh-enddesignprofessionalsdonotcon-strainthemselvesbyconsideringtheconsistencyandgradingofsurfaceparameterizations.Also,attemptstoenticethesestyliststousesubdivisionsurfaces,orG-splines(tensor-productsplinepatchesjoinedwithgeometriccontinuity)havesofarnotbeensuccessful.Aconcretecaseistheindustrialcar-hooddesigninFig. 1-1 andFig. 4-3 .Thehalf-hoodshowsthatorientation,polynomialdegreeandrelativeextentofthepolynomialsurfacepieces(patches)candierdrastically.Thepatchestypicallyonlyjoinuptovisualpreci-sion,i.e.mayevenhavegaps,sincetheiralgebraicintersectioncurvesareapproximatedbyrationalcurvesandhavewidelydieringtruebivariatedegree,e.g.64nextto77.Inaddition,thepatchesaretypicallydeliveredfordownstreamusewithoutconnectivityinformation.Theapparentdisorderallowsthestyliststofocusontheshapeofprimarysurfaces.Thesearesubsequentlytrimmedbacktobejoinedbyllets.Multi-sidedholes 52

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arepatchedbysmalltrimmedsplinepieces.Rimsurfacestripsareautomaticallygen-eratedatamuchnergrade,yieldingalargenumberofsplinepiecesandintersectioncurves.Inordertocreateaglobalatlasfromsuchacollectionofheterogeneoustrimmeddesignsurfaces,aconstructionmust,mostimportantly,re-representtheinputinamorestructuredandhomogeneousfashion. 4.2.1.1Degree-matchingofpatchesAnobviousrststep,therefore,istoensurethatallpatchesareofthesamepoly-nomialbi-degree.Inthiscontext`degree-matching'referstoanappropriatereductionorraisingofthebi-degree,dp,ofapolynomialtensor-productsurfacesothatdnewp=dt,wheredtissometargetdegree.Thechoiceofdtisnotarbitrary,however,anditisde-pendentuponthepreferredbi-degreeoftheG-splineconstructionusedtocreatethenalsmoothsurface.Inparticular,weusebi-degreedt=3aswerelyon[ 1 ].Thefollowingdis-cussionassumesthatthesurfacepiecesarerepresentedintheBernstein-Bezierform.TheBB-formiscommonlyusedinpracticeandothersplinerepresentations(e.g.B-splines,T-splines,etc.)canbeconvertedtoit.Incaseswheredp
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whereAk;k)]TJ /F4 7.9701 Tf 6.586 0 Td[(1(i;j):=8>>>>>><>>>>>>:i=k;ifj=i)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;1)]TJ /F3 11.9552 Tf 11.955 0 Td[(i=k;ifj=i;0;else.Equation( 4{1 )caneasilybeextendedtothebivariate(tensor-product)casebyconsider-ingthecoecientsbbb=(bbbi;j)0idup;0jdvpjbbbi;j2R3,whereduanddvarethepolynomialdegreesofthepatchintheuandvdirectionsrespectively: ~bbb=(Advt;dvp(Adut;dupbbb)T)T:(4{2)Wheneverdp>dt,amoreinvolvedstrategyisneeded.Exactdegreereduction,preservingtheoriginalgeometry,isnotpossibleingeneral[ 66 ].[ 67 ]showedthat,intheunivariatecase,degreereductionintheL2-normequalsbestEuclideanapproximationofBBcoecients.Thus,thedegreereductionoperator,Pdt;dp,formappingtheBBcoecientsb:=(bi)0idpto~b:=(~bi)0idt,wherebi;~bi2R2,accordingto ~b=Pdt;dpb(4{3)canbedeterminedbysimplysolvingtheleastsquaresproblem min~b2Rdp+1kb)]TJ /F3 11.9552 Tf 11.955 0 Td[(Adp;dt~bk(4{4)usingthepseudo-inverseofthedegree-raisingmatrixAdp;dt: A+dp;dt:=(ATdp;dtAdp;dt))]TJ /F4 7.9701 Tf 6.586 0 Td[(1ATdp;dt:(4{5)Ourapproachbuildsupon[ 67 ]andseeksnotonlytoimprovetheapproximationaccuracyofthedegreereduction,butalsotoenableintelligentinsertionsofparameterlineswher-evernecessaryinordertoachievesmoothtransitionsbetweenneighboringpatchesandtoensureconsistentgradationoftheparameterizationofthesought-aftersplineatlas.By 54

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addingBeziersubdivisionSandB-splineknotinsertionKtotheleastsquaresproblem,( 4{4 )becomes min~c2RmkSb)]TJ /F5 11.9552 Tf 11.956 0 Td[(AK~ck(4{6)whichhastheexplicitsolution ~b=K~c;~c=(AK)+Sb:(4{7)Toseethis,rstconsiderthesimplerversion min~b2Rn(dp+1)kSb)]TJ /F5 11.9552 Tf 11.956 0 Td[(A~bk(4{8)withthesolution ~b=A+Sb:(4{9)Letu:=(ui)0in)]TJ /F4 7.9701 Tf 6.587 0 Td[(2jui2(0;1)beanon-decreasing,permissiblynon-uniform,vectorspecifyingparametervaluesatwhichton-arilysubdividetheinputcurveofdegreedppriortodegree-reduction.ThenSdenotesthe(dp+1)n(dp+1)-matrixthatcapturesonestepofthen-aryBeziersubdivision(deCasteljau's)algorithmatu(Section C.1 ).A+representsthepseudo-inverseofthe(dp+1)n(dt+1)ndegree-raisingblockdiagonalmatrixA:= "Adp;dt0...0Adp;dt# :Thisformulationsplitsthettingproblemintonsubproblems,andthereforeprovidesabetterapproximationoftheinputasillustratedinFig. 4-4B .Theresultingnsub-curves,however,donotjoincontinuously(Fig. 4-4C ).Inthebivariatetensor-productcasethiswouldresultincracksinthedegree-reducedsurfaceatsub-patchboundaries,therebyviolatingourgoalofachievingawatertightatlas.One,ratherelegant,solutionistoemployamoregeneralsplinerepresentationthathasbuilt-insmoothness,namelyB-splines,andtwithitscoecients.AsmentionedinSection 2.1.2.1 ,animportantfactisthataBeziercurveofdegreedcanberepresentedasaB-splinesegmentofthesamedegreehavingtheknotvectort:=[tj;tj+1].Here,weusethenotationx=x:=(xi)1imjx1=x2==xmiftheargumentxis 55

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A B CFigure4-4. Thedegreeofadp=5BBcurve(black)isreducedtodt=3(magenta).Theleastsquaresproblemisformulatedasin:A)( 4{4 )[ 67 ];andB)( 4{8 )withu:=[0:5].C)AcloseinspectionofB)revealsthattheresultingcurvepiecesareC)]TJ /F4 7.9701 Tf 6.587 0 Td[(1. repeatedmtimes.Particularly,~cin( 4{7 )representstheunknowncoecientvectorofaCB-splinecurvewiththefollowingknotsequence: t:=[0;u<>;1];u<>:=(ui)0in)]TJ /F4 7.9701 Tf 6.587 0 Td[(2(4{10)Herecanbeusedtocontrolthesmoothnessattheinterfacesoftheresultingsplinesegments(see.g.Fig. 4-5 B,C).If,forexample,onesets=dt)]TJ /F1 11.9552 Tf 12.043 0 Td[(1,thenu<>=uandCsmoothnessisenforcedduringthetting.Anotherwell-knownfact(Section 2.1.2.2 )isthatonecanconvertfromB-splinetoBezierformbyinsertingrepeatedknotsuntilallknotmultiplicitiesareequaltothedegree.Weleveragethistechniqueintheconstructionofthe(dp+1)nmB-splinetoBezierconversionmatrixK,wherem=length(t))]TJ /F3 11.9552 Tf 12.321 0 Td[(dt)]TJ /F1 11.9552 Tf 12.321 0 Td[(1isthelengthofthecoecientvector~c(Section C.2 ).Hence,B-splinecoecients~ccanbeconvertedtoBBcoecients~bsimplyby~b=K~c(Fig. 4-5E ).Notethat,inreality,wemakeuseoftheB-splinecoecients~cwhenconstructingourglobalscaoldastheyaretheinputsexpectedbyourvariousG-splinealgorithms. 56

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A B C D E FFigure4-5. Thedegreeofadp=5BBcurve(black)isreducedtodt=3(magenta).Increasedreducestaccuracy,whereasincreasednumberofuimprovestaccuracy.A)( 4{8 )withu:=[0:5]producesC)]TJ /F4 7.9701 Tf 6.586 0 Td[(1segments.B)( 4{6 )withu:=[0:5]and=1producesC=1segments.C)( 4{6 )withu:=[0:5]and=2producesC=2segments.D)Fittingaccuracycanbeimprovedbyintroducingmoresegmentsasshown:( 4{6 )withu:=[0:250:50:75]and=2.E)B-splinecoecients~c( )areconvertedto~b( )accordingto( 4{7 )(thegraphisthesameasonD)).F)Resultsfromdegreereductionusing( 4{4 )[ 67 ]areshownforcomparison. Theabilitytocontrolthesmoothnessofthedttransitions,via,andtointelligentlyselectwheretheyshouldhappeninthedomain,viau,resultsinamuchimproveddegreereductiontechniquecapableofcloselyfollowingacurveofanyhighdegree(Fig. 4-5D ).Extending( 4{7 )tothebivariate(tensor-product)caseisalmosttrivialandresembleswhatwedescribedearlierin( 4{2 )whendiscussingtheissueofdegree-raising.Forillus-trationconsiderFig. 4-6 wherethetechniqueisappliedtotheuntrimmedheterogeneoussurfacepiecesoftheMiniCoopercarhood(generatedbystylistsinICEMSurf).Thenewdegree-matchingtechniqueproducespatchesofadegreeofourchoosingwithvisually 57

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A B Figure4-6. ComparisonbetweenA)industrialinputcontainingpatchesofdegree3through9andB)patchesofuniformbi-degreedt=3producedbyourdegreereductiontechniqueemployingternarysubdivision(u:=[1 32 3])andenforcingC2(=2)atsub-patchboundaries.Showingfromlefttoright:highlightlinedistributions;Gausscurvaturedistributions;andBBnets.Surfacepiecesarevisuallyidentical. identicalgeometricproperties.TheslightdierencesinGausscurvatureatpatchtransi-tions(Fig. 4-6B ,middle;comparedtoFig. 4-6A ,middle)arecausedbyintentionallynotinterpolatingpatchendpoints{otherwiseourdegreereductionwoulddeviatefromthebestEuclideanapproximation[ 67 ].ThisdoesnotpresentaproblemsinceweconstructourownsmoothpatchtransitionsinSection 4.2.2 .Anaddedimportantbenettoourapproachisthat,inthebivariatecase,itgivesusthefreedomtocreatecustomparame-terizationsofpatchessothatparameterlinesofadjacentsurfacepiecesarecloselyaligned(Section 4.2.1.2 )andthecreationofasingleB-splinescaoldisgreatlyfacilitated. 4.2.1.2EstablishingpatchadjacenciesandgradationAswearguedpreviously,industrialmodels,e.g.createdbyautomotivestylists,representacollectionofheterogeneoustrimmeddesignsurfaces.Makingmattersworse, 58

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A BFigure4-7. Top-leveloctreecontainingtrimcurvesamplesinR3(MiniCoopercarhoodmodel)usedforestablishingpatchadjacencies.A)Bincapacityisdn=10e,wherenisthetotalnumberofsamples;showingallbins.B)Acloseupoftheheadlightregionshowingpatchindicesassociatedwitheachsample. theytypicallydonotcontainanyconnectivityinformation.Developingatechniqueforbest-ttingsuchapatchsoupwithdegree-matched,internallyCsmoothpatchesoftargetbi-degreedt,isarststepintransformingtheinputintoauniformG-splinescaold.WhatfollowsnaturallyistheissueofchoosingtherightpatchpartitioningastoensureanoverallconsistentgradationoftheoutputG-mesh.AsevidentfromtheBBmeshdisplayedinFig. 4-6B (right)theparameterizationsofneighboringdegree-matchedsurfacepiecescanvarygreatlyifuiskeptthesameforallpatches.Thankfully,thisdoesnothavetobethecaseandhencetheproblemisnarroweddowntohowonewoulddeterminetheoptimaluperpatchwithrespecttoitsneighbors.Thisstep,althougheasiertodiscussafterdegree-matching,actuallyprecedesitinimplementation.IdentifyingadjacenciesinanunstructuredcollectionofgeometricprimitivesisaclassicalprobleminCAGD{onethatlacksauniversalsolution.Theapproachesthatdoexistandworkinpracticeareheuristicandoftentailoredtospecicinputsets.Hereweoeronesuchheuristicthatworkswellwiththetypeofindustrialmodelsdescribed 59

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A B CFigure4-8. AchievingaconsistentpatchparameterizationfortheMiniCoopercarhoodmodel;showingB-splinemesh(coecients~c).A)Input.B)Degreereductionemployingindiscriminateternarysubdivision(u:=[1 32 3])forallpatches.C)Usingacustomsubdivisionvectoruperpatchbasedonneighborhoodconsiderationresultsinnicelyalignedparameterlines. earlier.Weleveragethefactthat,despitelackingconnectivityinformation,patchesdocontaintrimmingcurvedata.Trimcurvesresultfromsolvingthesurface-surfaceintersectionproblem,s0(u0;v0)=s1(u1;v1)2R3,andrepresentpolynomialorrationalcurvesc:R!R2thatdeneandrestrictthedomainofasurfaces:R2!R3(Fig. 4-2 ).Therefore,neighboringpatchescanbeidentiedbysamplingtheirtrimcurvesinR3bycomputings(c(t))2R3foradiscretesettofparametersandthenmatchingupthesamples.AnaiveimplementationtakesO(n2)timewherenisthetotalnumberofsamples.Thereforeweemployanoctree-baseddatastructuretoimprovethesearcheciency(Fig. 4-7 ).Theinclusionradiusforidentifyingmatchesissetbasedonthemodelscale.Havingestablishedpatchneighborhoods,wemoveontodeterminingtheoptimalpatchgradation.Thiscantriviallybeachievedbyenablingtheusertosetandmodifythesubdivisionvectoruofeachpatchthroughagraphicalinterface,thusvisuallyaligningparameterlines.Thiswork-owdoesnotsubstantiallydeviatefromwhatisalreadybeingdoneinindustryformodelcreation(Section 4.1 )withthenotableexceptionthatherethegoalisaglobalsmoothsplineatlas.Infact,thisistheapproachweuse(Fig. 4-8 ) 60

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asitcurrentlygivesusgreaterfreedomandcontrolversusthealternative.Thiscanbeautomatedasfollows.Considertwopatches,aprimaryPandaneighborQ.ThenattheithsplitparameteruPiofP,wendtheinterceptwiththetrimcurveandgotoR3togetthepointPi.NowwedeterminePiontheimageofthetrimcurvecQtogettheparameteruQiwhereQshouldbesplit. 4.2.2CreatingaGlobalB-splineScaoldHavingalignedtheparameterizationsofneighboringpatches,wearenowleftwithcreatinginter-patchconnectionstocompletetheconstructionofaglobalsplinemesh.Foreachsurfacepiece,weidentifyinclusionregions,delineatedbyitstrimcurves,viaarasterization-likeapproachinitsdomain.WekeepthecoecientswhoseGrevilleabscissaefallwithintheseregionsdiscardingtherest(Fig. 4-9 ).Thegridonwhichthetrimmedcurvesarerasterizedcanbenon-uniform.Forexample,itstessellationcandependonthepartitioningofthepatchduringthehomogenizationstep(Section 4.2.1 ).TheresultingB-splinemeshconservesasmuchofthestylist-createdprimarysurfacesaspossiblebutleavesholesatthepatchtransitions.DependingonthenalG-splinealgorithm,themethodologyforconnectingthedis-jointB-splinemeshpiecescanvary.Luckilytheconstructionweemploy,[ 1 ],isratherforgiving,beingcapableofhandlingbothextraordinarypointsandT-junctions(Sec-tion 4.2.3 ).Despitethisrelaxationofconnectivityrequirements,thetaskofllingtheholesatpatchtransitionscreatedbytherasterizationremainsnontrivial.AsinSec-tion 4.2.1.2 ,wefavorasystematic,albeitmanual,method,consistentwiththeproof-of-conceptnatureofthiswork.Whileanautomatedapproachseemsfeasible,wedelegateittothefuture.Commonly,thepatchorientationsofprimarysurfacesandtheonesofblendsorlletsareconsistent.Forexample,theblendforasharpfeatureedge(suchasthatofacube)hasparameterlineswell-alignedwiththeonesoftheprimarysurfacesitsmoothsoutand 61

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A B CFigure4-9. Rasterizationoftrimcurvesinthepatchdomainproducesinclusionregionsthatcanbeusedtoidentifycoecientsneededforthenalmeshconstruction.A)Trimcurvesinthedomainoftheoutlinedbrownpatch.B)RasterizationofthetrimcurvesinA).C)ResultinginclusionregionsinR3.Theinsetontherightshowstheinputuntrimmedsurfaces(patchsoup)forcontext. hencetrims.Assumingthatthepatchparameterizationshavebeenappropriatelyhomog-enizedaccordingtoSection 4.2.1 ,connectingtheirindividualB-meshesisstraightforwardanditamountstoasimpleone-to-onematchingofcoecients.Amuchmorecomplicatedscenarioariseswhenstylistscreatearbitrarynewprimaryfeaturesandblendthemintoex-istingsurfaces.Therearenoguaranteesontheorientationofpatches.Itisverylikelythatsuchblendstraceoutafeaturecurvethatisattimesdiagonaltothepreferreddirectionoftheprimarytensor-productsurfaces(Fig. 4-10A ).Therefore,aone-to-onematchingisoutofthequestionandonemustuseextraordinaryverticesandT-junctionsinorderto 62

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successfullyconnectthecontrolnetsofsuchsurfacepieces.Fortunately,wehaveidentiedarelativelystandardwayofhandlingsuchtransitions.Thisinvolvesslightlymodifyingtheconnectivityoftheprimarysurfacemeshesandjudiciouslycreatingafewextraordinaryvertices.WedonotmodifytheoriginalB-coecients,butsimplyreconnectthem.Inthisway,diagonalfeaturescanbehandledwithoutcreatinganexcessiveamountofn-sided(nj2
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A B C DFigure4-10. CompletingB-meshconnectionsatpatchtransitions(e.g.forthecarhoodbulgeinFig. 4-9C ).A)Meshaftercoecienttrimming;alsoshowingthebasicconnectivitymodicationsnecessaryforhandlingfeaturesorienteddiagonallytothepreferreddirectionsofprimarypatches.Edgesmarkedinredandgreenaretoberemovedandaddedrespectively.Notethattheoriginalcoecients(vertices)areuntouched.B)Analwatertightmeshisconstructedviaappropriatevertexmatchingthatresultsinvalence-3verticesforconcaveregionsandvalence-5verticesforconvexones.C)[ 1 ]appliedtotheB-meshyieldsaC1BBsurface:showingregularpatchesingold,extraordinarypointpatchesinred,andT-patchesingray.D)ThedistributionofhighlightlinesfortheregioninC)showsthatthenalsurfaceisofgoodquality. 64

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4.2.3FinalSmoothSurfaceConstructionInthepreviousSection 4.2.2 ,wejudiciouslyintroducedirregularlayouts(star-likepointsandT-junctions)whencreatinginter-patchtransitionswiththeassumptionthatthereexistsasmoothsurfacealgorithmexibleenoughtohandlethem.Extraordinarypointsareubiquitousinmanifoldmeshesandareunavoidableinmeshesforsurfacesofhighertopologicalgenus.Extraordinarypointsappearwhereverthreeormorethanfourquadsmeet,e.g.thethreesidesofacubemeetingatacorner.Joiningsplinesurfacesinthepresenceofsuchirregularitieswhilstensuringgeometrically/parametricallysmoothtransitionsisawellresearchedtopicandcanbeaddressedthroughappropriatereparam-eterizationsoftheinputsplinesurfacepieces.ExamplesincludetheRenablePolycubeG-splineconstructionweintroducedinChapter 3 aswellas[ 27 60 68 69 ]amongothers.Wherestripsofsurfacepatchesareforcedtogether,itisnaturaltoterminatesomeandwheretheyarestretchedout{tospawnadditionalonesinordertokeepthesizeandaspectratioofthepatcheswithinbounds.TheseleadtoT-junctionswheretwonersurfacepiecesmeetonecoarserpiece.T-junctionsallowintroducinggeometryofhigherdetail,ormergingtwoseparately-developedsplinesurfaces.OneapproachtomodelingT-junctionsisviahierarchicalsplines[ 70 { 74 ]whicharebasedontheassumptionthatallsurfacepiecesshareasingleuv-parameterization:foranychoiceofv,theu-knotintervalsmustaddtothesamenumber;andforanychoiceofu,thev-intervalsmustaddtoonexednumber(Rule1of[ 70 ]).Thisrestrictionontheknotsumsisnaturalwhenreningasinglepatchbutwhentheinputisagivenquadrangulationtheknotintervalshavetobeassignedandhencecoordinated.Joiningmanypiecescanthenbecomecumbersomesincethelocalknotintervalshavetogloballyadduptothesamesum.Therefore,whilehierarchicalsplinesarenaturallysuitedforintroducingT-junctionsinquadmeshes,thisisnotthecaseforgeneratingsurfacesfromquadmesheswithT-junctions.Thankfully,arecentlydevelopedconstructionbyKarciauskas,Panozzo,andPeters[ 1 ]{andtheoneweuseforcreatingournalsmoothsurface{supportsbothextraordinary 65

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A B C D E FFigure4-11. FinalC1BBsurfaceproducedbyapplying[ 1 ]toourG-mesh.A)Inputheterogeneouscollectionofpatches(patchsoup),showingBBnets.B)Inputsurfacehighlightlinedistribution.C)Outputsurfaceshowsgoodreections.D)IrregularcapsareredforextraordinarypointsandgrayforT-junctions.E)ResultingBBnet.F)Uniformhighlightlinedistribution. 66

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pointsandT-junctionsthroughvaryingtheparameterization(Fig. E-1 forexamplesur-faces).Thisdoesnotrequiretheglobalcoordinationofknotintervalsmentionedearlier.Herestar-pointandT-junctioncapsconsistofpatchesofdegreebi-4andareframedbybi-3patches(thismotivatedourchoicefortargetbi-degreedt=3inSection 4.2.1.1 ).Theconstruction[ 1 ]yieldsgoodhighlightlinedistributionwherealternatives,suchasCatmull-Clarksubdivision,failtopreservemonotonicity,andhierarchicalsplineparameterizations{toenforceC1.Therefore,obtainingaconsistentandsmoothparameterizationvia[ 1 ]fromournalsplinescaoldenablestheresultingsurfaces(Fig. 4-11 )tobefullyeditableanddierentiablehencemovingclosertodirectlysupportingstablyconvergingsolversforpartialdierentialequations,e.g.forIgAdiscretizationbasedon[ 75 ]. 4.3DiscussionOurapproachforconstructingwatertightsplineatlasesfromindustrialdesignsishighlymodularandintrinsicallysupportsparallelandindependentimprovements.Indeterminingtheoptimalpatchgradation(Section 4.2.1.2 ),forexample,weresorttovisuallymatching/aligningparameterlinesofneighboringsurfacepieces.WeseeahighpotentialforautomatingthisprocesstherebyreducingthenumberofT-facesandensuringtransitionswithimprovedcurvaturedistributions.Althoughourdegree-matchingmethodinSection 4.2.1.1 accommodatesanynon-uniformsubdivisionvectoru,weinfactuseequallyspacedparametervaluesascurrentlytheinitializationoftheG-splineconstruction(Section 4.2.3 )assumesthattheinputG-meshhasuniformknots.AddressingthisaswellasdevisingaprogrammaticsolutionforcreatingB-meshconnectionsatinter-patchtransitions(Section 4.2.2 )isfuturework.Furthermore,asdiscussedearlier,T-junctionsarisenaturallywhenintroducingnewfeaturesintoamesh.Ifsucientlyfarfromafeature,stoppingaparameterline(introducingaT)orcontinuingitdoesnotaectthenalshapeofthesplinesurface.IftheT-faceispartofafeature(Fig. 4-12B ),itinuencesshape.ThisbecomesanotherconsiderationwhenconnectingB-meshesasin 67

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A B C DFigure4-12. SurfaceeectwhenT-facesarepartofameshfeature.A)Inputmesh.B)CorrespondingT-splinesurfaceproducedusingAutodesk'simplementation[ 76 ]inRhino[ 77 ].C)C1BBsurfaceproducedby[ 1 ]alongwithitsD)highlightlinedistributionsaccentuatingthedipsinthesurface. Section 4.2.2 .HerewenoticedthattheG-splineconstructionof[ 1 ]behavessimilarlytoT-splinesagainstwhichitisbenchmarked,withtheexceptionofthepresenceofasmalladditionalartifact(Fig. 4-12C ).Thespreadingoutofhighlightlines(Fig. 4-12D )resultsfromthewaythecentraltotheT-edgeBBcoecientisderivedfromthesurrounding1-ringG-meshneighborhood.Furtherworkneedstobedonehere,inordertoensurethatthesurfacequalityatT-facesnotonlymatchesthebenchmarkbutalsogetsclosetothatofaClass-Asurface. 68

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Thetaskofbridgingthegapbetweenhigh-leveldesignanddownstreamanalysisinordertoenableaneectivefeedbackloopinthedevelopmentcycleofartifactsisadauntingone.Theproposedworkowisarststepinthisdirectionandprovesthatmethodicalapproachesbasedonre-representingtheunstructuredinputinaconsistentfashionandtreatingitwithasuitableG-splinesmoothsurfaceconstructionalgorithmareverypromising.Byshowingthatrealindustrialdesignsrepresentedasheterogeneouscollectionsofunrelatedpatchescanbetransformedintowatertightsplineatlaseswemoveclosertoourlong-termgoalofprovidingdesignersandstylistswiththetoolstodirectlycreatesuchatlaseswhilepreservingtheinterfacestheyhavealreadybeenaccustomedto.Morespecically,stylistswouldbeginthedesignprocessbyusingapredenedgenericG-splinescaold,createdoncepertypeofartifact,thatprovidesaninitialsmoothsurface.Thentheywouldhavethefreedomtoshapeittothetargetdesignspecicationsbyinteractingwithprimarysurfacesasiftheyhadheterogeneouspolynomialdegrees,orientationofparameterlinesandsizeasintheexistingCADenvironments.Underneaththisfamiliarinterface,newsurfaceswouldthenbeconsistentlyre-representedandthetransitionsandmulti-sidedfacetswouldbeautomaticallytakencareofbytheG-splineconstructionwithgradationaddressedbyT-junctions.Theunderlyinghypothesisthattheresultingtransitionsurfacesandmulti-sidedholecapsbetweenprimarysurfacepiecesallowforahighertolerancesanddeviationsfrom,say,aclaymodelaslongasthehighlightlinedistributionissucientlyuniformissupportedbydiscussionwithICEMdesigners. 69

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APPENDIXAEDGELABELINGHEURISTIC A.1Preliminaries Lemma5. Ingeneral,assigningoutgoinglabelstoverticesofvalence4and5inordertoeliminatetheedgesequencesofthetypem)]TJ /F1 11.9552 Tf 11.955 0 Td[(4)]TJ /F3 11.9552 Tf 11.956 0 Td[(n,wherem6=n,isnotpossible. Proof. Consideracompositem)]TJ /F1 11.9552 Tf 12.009 0 Td[(4)]TJ /F3 11.9552 Tf 12.008 0 Td[(nedgewherem6=n2f3;6g.Accordingtoouredgelabelingconvention(Section 3.3.1 ),theedgesincidenttothevalence4vertexreceivelabelshm;nihm;niintheversaldirectionandhm0;4ih4;n0iinthetransversaldirection,therebysatisfying( 3{14 )and( 3{15 )respectively.Nowconsiderthecompositem0)]TJ /F1 11.9552 Tf 12.483 0 Td[(4)]TJ /F3 11.9552 Tf 12.483 0 Td[(n0edgetransversaltothestartingm)]TJ /F1 11.9552 Tf 11.242 0 Td[(4)]TJ /F3 11.9552 Tf 11.242 0 Td[(nedge.Ifm0=n0,thensmoothnessofpatchtransitionsisautomaticallyachievedasshownin[ 22 ].Ifm06=n0,however,G1transitionsneedtobeensuredbyfollowingtheedgelabelingconventiononceagain,thusassigninglabelshm0;n0ihm0;n0iandhm;4ih4;niaccordingly.AselucidatedbyFig. A-1 ,theoutcomeisdualassignmentoflabelsfortheedgesincidenttothe4valentvertex,whichisnotpermissible. m0 h4;m0i hn0;m0i mhm;nihm;nihm;4ih4;nin hn0;4i hn0;m0i n0FigureA-1. Dualassignmentoflabelstoedgesincidenttoa4valentvertexhavingdirectneighborswithvalencesm,n,m0,andn0,wherem6=nandm06=n0. 70

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A.2HeuristicThegoaloftheEdgeLabelingistominimizethenumberofedgesequencesofthetypem)]TJ /F1 11.9552 Tf 11.955 0 Td[(4)]TJ /F3 11.9552 Tf 11.955 0 Td[(nwherem6=n.Itisappliedtotheunrenedquadmesh.Input:A(polycube)quadmeshwithvalences3;4;5;6.Output:Alabelinghp;qiofeachedgeshowingtheapparentvalenceofitsendpointsasin( 3{13 ).Heuristic:Wewilluse,foreachvertexv,anauxiliarysetofnintegers,callededge-vertexmarkers.Thereisonemarkerforeachoutgoinglabelpontheedgelabeledhp;qi.Weproceedasfollows. 1. Initializealledgelabelswiththetruevertexvalenceateitherend.Initializeallmarkersaszero. 2. Foreachedgewithoutgoinglabel=6traceapathuntilavertexofvalencen6=4isencountered.The4-valentverticesalongthepatharetraversedbyalwaysproceedingtotheoppositeedge,i.e.fromedgeeitoei+2withsubscriptsmodulo4.Setthemarkeroftheincomingedgeto1. 3. Forallverticesofvalencep=5,ifpossible,assigntheoutgoinglabelsmarked1with=6asinFig. 3-6 ,left.Foreachedgewithoutgoinglabel6,tracethepathbackuntilalabel6isfound,settingthemarkersofthe4-valentverticesalongthepathto1. 4. Foreachedgewithoutgoinglabel3tracethepathuntilalabel3or6isfound,settingthemarkersofall4-valentverticesalongthepathto1forincomingedgesand)]TJ /F1 11.9552 Tf 9.298 0 Td[(1foroutgoingedges. 5. Foreachvertexvpofvalencep=4suchthatallfourmarkersarenonzero(i.e.twopathscross),collectthelabelsiandi+2ofopposingneighborsiandi+2,i=0;1.Ifi=i+2thenlabeledgeiashi;4iandedgei+2ash4;ii.Otherwiselabeledgeiashi;i+2iandlabeledgei+2ashi+2;ii.(Thisyields,forexample,apathfromavertexwithvalence6toonewithvalence3suchthateverysegmenthasthesamelabel-pairh6;3i.) (a) Ifallfourlabelscorrespondingtovpareeither6or3,thenoverwritethemby4.(ThiswillyieldfourC0edges). (b) Forverticeswith3or1neighbors'labelsset,settheunmatchedneighborlabeltothatofitsopposite:i=i+2andthecorrespondingtwolabelsofvpto4.(Thisyieldsasequencehm;4i;h4;miandhenceaG1construction.) 71

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(c) Forallverticesthathave2or0oftheirneighbors'labelsset,settheremainingneighbors'labelsto6andthecorrespondinglabelsofvpto4.(Thisyieldsasequenceh6;4i;h4;6iandhenceaG1construction.) 6. Foreachvertexvpofvalencep=4suchthatonlytwomarkersarenon-zero(i.e.asinglepath),setthelabelto6wherethemarkis1andto3ifthemarkis)]TJ /F1 11.9552 Tf 9.299 0 Td[(1.(ThisyieldsapathwithG1edgesh3;6i.) 72

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APPENDIXBBI5PGS`=1ALGORITHM B.1GeometricSmoothnessConstraintsThefollowinglemmamotivatesthealgorithmbelowbydeningtheG1continuitywithlinearreparameterizationthatthesurfaceconstructionhastosatisfyforann)]TJ /F3 11.9552 Tf 12.552 0 Td[(nedgeaswellastheconstantreparameterizationfor3)]TJ /F1 11.9552 Tf 11.955 0 Td[(6edges. Lemma6. Letpk:=b;k1andqk:=b;1kbetwopatches(withcoecientspkijandqkij)ofdegreebi-5thatsharetheboundarycurvepk(t;0)=qk(0;t).Forscalars!0and!1,dene;k(!0;!1):=1 5)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(4!0pk20)]TJ /F3 11.9552 Tf 11.955 0 Td[(!1pk00+(10)]TJ /F1 11.9552 Tf 11.955 0 Td[(4!0+!1)pk10;k(!0;!1):=1 10)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(6!0pk30)]TJ /F1 11.9552 Tf 11.955 0 Td[(4!1pk10+(20)]TJ /F1 11.9552 Tf 11.955 0 Td[(6!0+4!1)pk20b;k(!0;!1):=1 10)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(4!0pk40)]TJ /F1 11.9552 Tf 11.955 0 Td[(6!1pk20+(20)]TJ /F1 11.9552 Tf 11.955 0 Td[(4!0+6!1)pk30b;k(!0;!1):=1 5)]TJ /F3 11.9552 Tf 5.479 -9.684 Td[(!0pk50)]TJ /F1 11.9552 Tf 11.955 0 Td[(4!1pk30+(10)]TJ /F3 11.9552 Tf 11.955 0 Td[(!0+4!1)pk40ThenpkandqkjoinG1accordingto)]TJ /F1 11.9552 Tf 5.48 -9.684 Td[((1)]TJ /F3 11.9552 Tf 11.956 0 Td[(t)!0+t!1(@1pk)(t;0)=(@2pk)(t;0)+(@1qk)(0;t)( G1-bi5)ifandonlyifpk01+qk10=!0pk10+(2)]TJ /F3 11.9552 Tf 11.955 0 Td[(!0)pk00; (B{1)pk11+qk11=;k(!0;!1); (B{2)pk21+qk12=;k(!0;!1); (B{3)pk31+qk13=b;k(!0;!1); (B{4)pk41+qk14=b;k(!0;!1); (B{5)pk51+qk15=(2+!1)pk50)]TJ /F3 11.9552 Tf 11.955 0 Td[(!1pk40: (B{6) 73

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Proof. Equation(G1-bi5)isequivalenttomatchingsixBB-coecients.Equations( B{1 )and( B{6 )enforce(G1-bi5)att=0andt=1,respectively.Toverifytheremainingobservethatthesixcoecientsof)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[((1)]TJ /F3 11.9552 Tf 11.955 0 Td[(t)!0+t!1(@1pk)(t;0)are!0(p10)]TJ /F6 11.9552 Tf 11.955 0 Td[(p00);4!0(p20)]TJ /F6 11.9552 Tf 11.955 0 Td[(p10)+!1(p10)]TJ /F6 11.9552 Tf 11.955 0 Td[(p00);6!0(p30)]TJ /F6 11.9552 Tf 11.956 0 Td[(p20)+4!1(p20)]TJ /F6 11.9552 Tf 11.956 0 Td[(p10);4!0(p40)]TJ /F6 11.9552 Tf 11.955 0 Td[(p30)+6!1(p30)]TJ /F6 11.9552 Tf 11.955 0 Td[(p20);!0(p50)]TJ /F6 11.9552 Tf 11.955 0 Td[(p40)+4!1(p40)]TJ /F6 11.9552 Tf 11.955 0 Td[(p30);!1(p50)]TJ /F6 11.9552 Tf 11.955 0 Td[(p40)andthecoecientsof(@2q)(0;t)+(@1p)(t;0)arep01+q10)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p00;5(p11+q11)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p10);10(p21+q12)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p20);10(p31+q13)]TJ /F1 11.9552 Tf 11.956 0 Td[(2p30);5(p41+q14)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p40);p51+q15)]TJ /F1 11.9552 Tf 11.955 0 Td[(2p50: B.2AlgorithmWenowstateindetailthealgorithmthatenforcesLemma 6 .Input:4NB-spline-likecontrolpointsc,4foreachoftheNfacetsofapolycubequadquadmesh.Output:Asurfaceconsistingofbi-quintictensor-productsplinesurfacesbthatjoinG1{exceptalongedgeslabeledh4;niwherethesurfaceisonlyguaranteedtobeC0(C1)]TJ /F7 7.9701 Tf 6.587 0 Td[().Initialization:WeconvertfromB-splinetoBB-formanddegree-raisetwicetoinitializeallbi-5BB-coecientsb;11ij.Thisleavesonlythepointswithlabelb;1100uninitialized,correspondingtothenon-4-valentcorners.Werunthepre-processingEdgeLabelingalgorithmthatpre-labelstheinputquadmeshtominimizethenumberofedgesequenceswithlabelshm;4ih4;niwherem6=n.Overview:ConsidertheG1stripofBB-coecientsb;k1ij;b;k1ij;i=0;1;j=0;1;2;k=1:2`;`=1 74

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involvedintheG1joinbetweentwoquadsbandb,alongacurve(t).Visitingeachpatchb;k1,b;k1,startingfromeachvertexofvalencen2f3;4;5;6gweadjusttheBB-coecientsoftheG1strip.Algorithm: 0. Let`:=2`)]TJ /F4 7.9701 Tf 6.586 0 Td[(1;`=1 1. Forverticesofvalencen=6. (a) Fork=1:`,i.e.uptothemiddleoftheedgeif`>1,set8><>:!k0:=1)]TJ /F4 7.9701 Tf 13.15 5.698 Td[(2(k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1) 2`;!k1:=1)]TJ /F4 7.9701 Tf 13.151 4.707 Td[(2k 2`;forh6;6i,!k0:=1)]TJ /F4 7.9701 Tf 13.15 5.698 Td[(2(k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1) 2`+1;!k1:=1)]TJ /F4 7.9701 Tf 17.261 4.707 Td[(2k 2`+1;forh6;4i,!k0:=1;!k1:=1;forh6;3i (b) Forallnpatchessurroundingthevertex,applythecosineprojectionmatrixPtothetangentcoecientsb;1110 Pb;1110P:=(1+C)=n2Rnn1(i;j):=1;C(i;j):=2cos(2 n(j)]TJ /F3 11.9552 Tf 11.955 0 Td[(i)): (c) Setthevertexofvalencen=6totheaverageb;1100:=1 nn)]TJ /F4 7.9701 Tf 6.587 0 Td[(1X=0b;1110: (d) Adjustthesecondderivativesoftheendsoftheboundarycurves,andthemixedderivativesaccordinglyb;1120 b;1120)]TJ /F1 11.9552 Tf 13.151 8.088 Td[(()]TJ /F1 11.9552 Tf 9.298 0 Td[(1) 4a;a:=n)]TJ /F4 7.9701 Tf 6.586 0 Td[(1X=0()]TJ /F1 11.9552 Tf 9.298 0 Td[(1)R;R:=1 2;1(!10;!11);b;1111:="110000011000001100000110000011100001#+R: 75

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(e) Fork=1:`,i.e.uptothemiddleoftheedgeif`>1,minimallyperturbthetransversalderivativestoenforceG1continuity:b;k111 b;k111+1 2)]TJ /F3 11.9552 Tf 4.948 -9.684 Td[(;k(!k0;!k1))]TJ /F5 11.9552 Tf 11.955 0 Td[(b;k111)]TJ /F5 11.9552 Tf 11.956 0 Td[(b;1k11;b;k121 b;k121+1 2)]TJ /F3 11.9552 Tf 4.948 -9.684 Td[(;k(!k0;!k1))]TJ /F5 11.9552 Tf 11.956 0 Td[(b;k121)]TJ /F5 11.9552 Tf 11.955 0 Td[(b;1k12; 2. Forverticesofvalencen=5. (a) Edgesemanatingfromthen=5vertexareeachlabeledeitherh4;miorh6;midependingonitsapparentvalence.Fork=1:`,i.e.uptothemiddleoftheedgeif`>1,set!k0and!k1accordinglyusingtherulesforeithern=4orn=6 (b) Insertadummycoecientbetweentheedgesmarkedase0ande4inFig. 3-6 andapplythecosineprojectionmatrixPtothenown=6tangentcoecientsb;1110 Pb;1110P:=(1+C)=n2Rnn1(i;j):=1;C(i;j):=2cos(2 n(j)]TJ /F3 11.9552 Tf 11.955 0 Td[(i)): (c) For2f0;1;3;4g,setthecentralcoecientb;1100:=1 4Xb;1110; (d) Adjustb;1111:="1100001100001100001110001#)]TJ /F4 7.9701 Tf 6.586 0 Td[(1R;R:=;1(!10;!11): (e) Proceedasin 1e (above). 3. Forverticesofvalencen=4. (a) Fork=1:`8><>:!k0:=2(k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1) 2`+1;!k1:=2k 2`+1;forh4;3i,!k0:=)]TJ /F4 7.9701 Tf 10.494 5.699 Td[(2(k)]TJ /F4 7.9701 Tf 6.586 0 Td[(1) 2`+1;!k1:=)]TJ /F4 7.9701 Tf 14.604 4.707 Td[(2k 2`+1;forh4;6i,!k0:=0;!k1:=0;forh6;4i. 76

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(b) Sincetheseverticesareordinary,theinitializationstephasalreadyensuredproperplacementofb;1111.Hence,setb;1100:=1 nn)]TJ /F4 7.9701 Tf 6.587 0 Td[(1X=0b;1111; (c) Adjustb;1110inaccordancewith( B{2 )b;1110:=5(b;1111+b;1111)+!1b;1100)]TJ /F1 11.9552 Tf 11.955 0 Td[(4!0b;1120 10)]TJ /F1 11.9552 Tf 11.955 0 Td[(4!0+!1 (d) Proceedasin 1e (above). 4. Forverticesofvalencen=3. (a) Fork=1:`8><>:!k0:=2(k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1) 2`)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;!k1:=2k 2`)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;forh3;3i,!k0:=2(k)]TJ /F4 7.9701 Tf 6.587 0 Td[(1) 2`+1)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;!k1:=2k 2`+1)]TJ /F1 11.9552 Tf 11.955 0 Td[(1;forh3;4i,!k0:=)]TJ /F1 11.9552 Tf 9.298 0 Td[(1;!k1:=)]TJ /F1 11.9552 Tf 9.299 0 Td[(1;forh3;6i. (b) Forn=3,theB-splinetoBB-conversionpullsthetangentcoecientstooclosetogethertotheiraverageb00:=1 3P2=0b;1110.Forbettershape,weoverwritethembyb;k110 b;k110+(b;k110)]TJ /F5 11.9552 Tf 11.955 0 Td[(b00);default:=1=2: (c) Setb;1100:=1 nn)]TJ /F4 7.9701 Tf 6.587 0 Td[(1X=0b;1110; (d) Adjustb;1111:=h110011101i)]TJ /F4 7.9701 Tf 6.587 0 Td[(1R;R:=;1(!10;!11): (e) Proceedasin 1e (above).Thiscompletesthealgorithm.Wenotethatpairsoftensor-productsplinepatchesjoiningacrossa3-6edgeareparametricallyC1connected.Toobtainstillbettershape,onemightoptimizetheB-splinecontrolpointsandhenceBB-coecientsawayfromtheG1strips.However,thisisnotourfocushere. 77

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APPENDIXCALGORITHMSFORGENERATINGMATRICESSANDK C.1SubdivisionMatrixS 1: procedureGetSubdivisionMatrix(dp,u) 2: S Identity(dp+1;dp+1).Createsannnidentitymatrix 3: ifLength(u)1then 4: Stop GetHalfMatrix(dp;u0) 5: Sbot GetHalfMatrix(dp;1)]TJ /F6 11.9552 Tf 11.955 0 Td[(u0) 6: Sbot Rotate90(Sbot;2).Rotatesamatrixby90degreesntimes 7: ifLength(u)>1then 8: u u1:end=1)]TJ /F6 11.9552 Tf 11.955 0 Td[(u0.Setsparametervectorfornextrecursivecall 9: Sbot GetSubdivisionMatrix(dp;u)Sbot 10: endif 11: S VerticalConcat(Stop;Sbot).Verticallyconcatenatestwomatrices 12: endif 13: returnS 14: endprocedure 15: 16: procedureGetHalfMatrix(dp,u) 17: S Zeros(dp+1;dp+1).Createsannnzeromatrix 18: fori 0;dpdo 19: Si;0 (1)]TJ /F3 11.9552 Tf 11.955 0 Td[(u)i 20: endfor 21: fori 1;dpdo 22: forj 1;dpdo 23: Si;j uSi)]TJ /F4 7.9701 Tf 6.586 0 Td[(1;j)]TJ /F4 7.9701 Tf 6.586 0 Td[(1+(1)]TJ /F3 11.9552 Tf 11.955 0 Td[(u)Si)]TJ /F4 7.9701 Tf 6.586 0 Td[(1;j 24: endfor 78

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25: endfor 26: returnS 27: endprocedure C.2B-splinetoBezierConversionMatrixK 1: procedureGetB2BBMatrix(dt,u,) 2: t ConstructKnotVector(dt,u,).Accordingto( 4{10 ) 3: m Length(t))]TJ /F3 11.9552 Tf 11.955 0 Td[(dt)]TJ /F1 11.9552 Tf 11.956 0 Td[(1.NumberofB-splinecoecients~c 4: C Identity(m;m).Coecientmatrix:eachrowisaB-splinecoecientvector 5: K Zeros(Length(u)(dt+1))]TJ /F1 11.9552 Tf 11.956 0 Td[(Length(u);m).InitializeK 6: fori 0;mdo 7: spl MakeSpline(t;Row(i;C)).Constructasplineobject 8: b InsertKnots(spl;u;).Insertutimesintosplandreturnitscoecients 9: Column(i;K) b 10: endfor 11: returnK 12: endprocedure 79

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APPENDIXDEXAMPLESOFPGSANDPGSERSURFACES FigureD-1. SmoothPGSsurfaces(withoutrenement)frompolycubecongurationsalongwiththeircorrespondingBB-controlnetsandhighlightlineplots. 80

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FigureD-2. PGSERsurfacesofmoderate(gold)andhigh(cyan)patchcount. 81

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APPENDIXEEXAMPLESOFSURFACESPRODUCEDBY[ 1 ] A B CFigureE-1. GalleryofG-splinesurfacesproducedby[ 1 ].A)Surface.B)Highlightedirregularregions.C)Meshshowninaddition. 82

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[61] X.LiandF.Chen,\ExactandapproximaterepresentationsoftrimmedsurfaceswithNURBSandBeziersurfaces,"in200911thIEEEInternationalConferenceonComputer-AidedDesignandComputerGraphics,Aug2009,pp.286{291. [62] T.W.Sederberg,G.T.Finnigan,X.Li,H.Lin,andH.Ipson,\WatertighttrimmedNURBS,"ACMTrans.Graph.,vol.27,no.3,pp.79:1{79:8,Aug.2008.[Online].Available: http://doi.acm.org/10.1145/1360612.1360678 [63] N.Litke,A.Levin,andP.Schroder,\Trimmingforsubdivisionsurfaces,"Comput.AidedGeom.Des.,vol.18,no.5,pp.463{481,Jun.2001.[Online].Available: http://dx.doi.org/10.1016/S0167-8396(01)00042-5 [64] J.Shen,J.Kosinka,M.A.Sabin,andN.A.Dodgson,\ConversionoftrimmedNURBSsurfacestoCatmullClarksubdivisionsurfaces,"ComputerAidedGeometricDesign,vol.31,no.78,pp.486{498,2014,recentTrendsinTheoreticalandAppliedGeometry.[Online].Available: http://www.sciencedirect.com/science/article/pii/S0167839614000624 [65] J.PetersandU.Reif,\LeastsquaresapproximationofBeziercoecientsprovidesbestdegreereductionintheL2-norm,"JournalofApproximationTheory,vol.104,no.1,pp.90{97,May2000. [66] J.HoschekandD.Lasser,FundamentalsofComputerAidedGeometricDesign.Natick,MA,USA:A.K.Peters,Ltd.,1993. [67] D.Lutterkort,J.Peters,andU.Reif,\PolynomialdegreereductionintheL2-normequalsbestEuclideanapproximationofBeziercoecients,"Computer-AidedGeometricDesign,vol.16,no.7,pp.607{612,Aug1999. [68] K.KarciauskasandJ.Peters,\Improvedshapeformulti-surfaceblends,"GraphicalModels,vol.8,pp.87{98,22015. [69] ||,\Minimalbi-6G2completionofbicubicsplinesurfaces,"ComputerAidedGeometricDesign,vol.41,pp.10{22,Jan2016. [70] T.W.Sederberg,J.Zheng,A.Bakenov,andA.Nasri,\T-splinesandt-nurccs,"ACMTrans.Graph.,vol.22,no.3,pp.477{484,Jul.2003.[Online].Available: http://doi.acm.org/10.1145/882262.882295 [71] D.Kovacs,J.Bisceglio,andD.Zorin,\Dyadict-meshsubdivision,"ACMTrans.Graph.,vol.34,no.4,pp.143:1{143:12,Jul.2015.[Online].Available: http://doi.acm.org/10.1145/2766972 [72] T.Dokken,T.Lyche,andK.F.Pettersen,\Polynomialsplinesoverlocallyrenedbox-partitions,"ComputerAidedGeometricDesign,vol.30,no.3,pp.331{356,2013.[Online].Available: http://www.sciencedirect.com/science/article/pii/S0167839613000113 88

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BIOGRAPHICALSKETCHMartinSarovwasborninStaraZagora,Bulgaria.Aftergraduatinghighschoolin2006hedecidedtopursuehighereducationintheUS.HereceivedaBachelorofSciencedegreeincomputerscienceandeconomicsin2010fromSt.LawrenceUniversity,NY.Hisresearchfocusesonsmoothparametricsurfaceconstructionsandgeometricmodeling.HereceivedhisPh.D.fromtheUniversityofFloridainthespringof2017. 90