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On the Construction of Smooth Surfaces

Permanent Link: http://ufdc.ufl.edu/UFE0041205/00001

Material Information

Title: On the Construction of Smooth Surfaces
Physical Description: 1 online resource (88 p.)
Language: english
Creator: Fan, Jianhua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: bicubic, continuity, nurbs, quad, smoothness, splines, surfaces
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Smooth, that is once or twice continuously differentiable surfaces play an important role in Computer Aided Design(CAD), digital movie animation and computer game-character design. Non-Uniform Rational Tensor Product B-Spline Patches are used as a standard representation. But because of their inherent restriction to quadrilateral-grids, topologically only a small set of surfaces can be represented by single tensor-products. For arbitrary topology and shape, also n = 4 quadrilateral patches need to be smoothly connected at some points. In particular, converting a quadrilateral input mesh into a C1 smooth surface with one NURBS patch of degree bi-3 per facet is a classical challenge. A part of this dissertation defines and implements an algorithm for this conversion. Another shows that no polynomial construction of lower degree is possible. A third part of this thesis shows that there is an essentially unique change of variables that is rational linear and suitable for smooth connecting always eight patches. A fourth part of this thesis compares four constructions of C2 surfaces.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Jianhua Fan.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Peters, Jorg.
Local: Co-adviser: Sitharam, Meera.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041205:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041205/00001

Material Information

Title: On the Construction of Smooth Surfaces
Physical Description: 1 online resource (88 p.)
Language: english
Creator: Fan, Jianhua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: bicubic, continuity, nurbs, quad, smoothness, splines, surfaces
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Smooth, that is once or twice continuously differentiable surfaces play an important role in Computer Aided Design(CAD), digital movie animation and computer game-character design. Non-Uniform Rational Tensor Product B-Spline Patches are used as a standard representation. But because of their inherent restriction to quadrilateral-grids, topologically only a small set of surfaces can be represented by single tensor-products. For arbitrary topology and shape, also n = 4 quadrilateral patches need to be smoothly connected at some points. In particular, converting a quadrilateral input mesh into a C1 smooth surface with one NURBS patch of degree bi-3 per facet is a classical challenge. A part of this dissertation defines and implements an algorithm for this conversion. Another shows that no polynomial construction of lower degree is possible. A third part of this thesis shows that there is an essentially unique change of variables that is rational linear and suitable for smooth connecting always eight patches. A fourth part of this thesis compares four constructions of C2 surfaces.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Jianhua Fan.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Peters, Jorg.
Local: Co-adviser: Sitharam, Meera.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041205:00001


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ONTHECONSTRUCTIONOFSMOOTHSURFACES By JIANHUAFAN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c r 2010JianhuaFan 2

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Tomyparents 3

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ACKNOWLEDGMENTS Iwanttothankmyadvisor,Dr.J ¨ orgPetersforhisencouragement,patienceandall thesupportthroughtheseyears.AndmycolleaguesAshishMyle s,MinhoKim,Sukitti Punak,TianyunNiandXiaobinWu.Ialsowanttogivethankstomy familyfortheir supportinmanyways. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................10 CHAPTER 1INTRODUCTION ...................................11 1.1Overview ....................................12 1.2SurfacesRepresentation ...........................12 1.3SmoothnessandGeometricContinuity ....................15 1.4LiteratureReview ................................16 2LOWERBOUNDOFBICUBICSPLINESURFACESFROMQUADMESHES .19 2.1Unbiased G 1 Constraints ...........................20 2.2Linear andVertex-localizedConstructions .................26 2.3LowerBoundsforDegreeBi-3 ........................30 2.4DiscussionandConclusion ..........................34 3THEBICUBICSPLINECONSTRUCTIONFORQUADMESHES ........37 3.1TheAlgorithm ..................................40 3.1.1ControlPoints ..............................40 3.1.2InputandOutput ............................40 3.1.3Step1B ezierPatch ..........................41 3.1.4Step2AdjustTangents .........................42 3.1.5Step3DeterminetheBoundary ....................43 3.1.6Step4FirstInteriorLayer .......................44 3.1.7Step5Interior ..............................45 3.1.8DegreesofFreedom ..........................45 3.2SmoothnessVerication ............................46 3.2.1OrdinaryQuadsandOrdinaryEdges .................46 3.2.2ExtraordinaryQuads ..........................47 3.2.3OverallSmoothness ..........................50 3.3Examples ....................................51 3.4DiscussionandExtensions ..........................51 4RATIONALLINEARREPARAMETERIZATIONFORSMOOTHSURFACES ..54 4.1Introduction ...................................54 4.2ConstraintsontheTransitionMapfor G 2 Continuityatan8-valentVertex .57 5

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4.2.1 G 1 Constraints .............................57 4.2.2 G 2 Constraints .............................58 4.3TheProjectiveLinearTransitionMap .....................59 4.4ConstructiveUseoftheTransitionMap ....................61 4.4.1 G 1 Construction .............................63 4.4.2 G 2 Construction .............................66 4.5Conclusion ...................................68 5CURVATURECONTINUOUSSURFACES .....................69 5.1GuidedSurfaces ................................69 5.1.1InitialRenementbyGuidedSubdivision ...............69 5.1.2FiniteCapping .............................72 5.1.3InitialGuide ...............................74 5.2ComparisonofCurvatureContinuousSurfacingAlgorithms ........75 5.2.1ShapeAnalysisSetup .........................77 5.2.2CurvatureComparison .........................78 6CONCLUSION ....................................83 REFERENCES .......................................84 BIOGRAPHICALSKETCH ................................88 6

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LISTOFTABLES Table page 5-1 C 2 constructions ...................................77 7

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LISTOFFIGURES Figure page 1-1Airplanemodels. ...................................11 1-2A)Parametricsurfacepatches.B)Tensor-productpatches ...........13 1-3Univariateuniformcubicspline. ...........................14 1-4Indexingandparametrization. ............................15 1-5Knotdistribution. ...................................17 2-1Indexingandparameterization. ...........................21 2-2Generalizedsplinepatch. ..............................24 2-3Linear .......................................27 2-4Shapedefect. .....................................29 2-5Generalizedsplinepatch. ..............................31 2-6Noshapedefect. ...................................34 3-1Bi-3NURBSconstruction. ..............................37 3-2 G 1 constraints. ....................................38 3-3LayoutandIndexing. .................................41 3-4Smallexampleswithpredictableshape. ......................53 4-1Doubletorus. .....................................55 4-2Compositionof withitself. .............................61 4-3Indexconventionofpolynomialpiece. .......................62 4-4Indexing. .......................................64 4-5 G 1 construction. ...................................65 4-6Highgenussurfaceswithvalence8. ........................67 5-1Ct-maps ......................................70 5-2Sampling. .......................................70 5-3Illustrationoftheapproachtoconstructtheouterlaye rs. .............71 5-4ConstructionofCatmull-Clarkguidedring. .....................71 8

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5-5Structureof and ofdegree(3,3). ........................73 5-6 forvalence5,6,7,8,9. ..............................73 5-7Inputdatafromthree-beamcorner(monkeysaddle). ...............74 5-8Freecoefcients. ...................................74 5-9Fittingtothetensorborderdata. ..........................75 5-101-1correspondence. .................................75 5-11Convexshapeswithvalence 6 and 8 ........................79 5-12Hyperbolicandmulti-saddleshapeswithvalence 6 ...............80 5-13Hyperbolicandmulti-saddleshapeswithvalence 8 ...............81 5-14Curvatureneedlesforconvexshapes. .......................82 5-15 C 1 construction. ...................................82 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ONTHECONSTRUCTIONOFSMOOTHSURFACES By JianhuaFan May2010 Chair:J ¨ orgPeters Major:ComputerEngineering Smooth,thatisonceortwicecontinuouslydifferentiablesu rfacesplayan importantroleinComputerAidedDesign(CAD),digitalmoviea nimationandcomputer game-characterdesign. Non-UniformRationalTensorProductB-SplinePatches areusedasastandardrepresentation.Butbecauseoftheirin herentrestrictionto quadrilateral-grids,topologicallyonlyasmallsetofsur facescanberepresentedby singletensor-products.Forarbitrarytopologyandshape, also n 6 =4 quadrilateral patchesneedtobesmoothlyconnectedatsomepoints.Inpart icular,convertinga quadrilateralinputmeshintoa C 1 smoothsurfacewithoneNURBSpatchofdegreebi3 perfacetisaclassicalchallenge.Apartofthisdissertati ondenesandimplementsan algorithmforthisconversion.Anothershowsthatnopolynom ialconstructionoflower degreeispossible.Athirdpartofthisthesisshowsthatthe reisanessentiallyunique changeofvariablesthatisrationallinearandsuitablefor smoothconnectingalways eightpatches.Afourthpartofthisthesiscomparesfourcon structionsof C 2 surfaces. 10

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CHAPTER1 INTRODUCTION Fromautomobileandplanedesigntodigitalmovieanimation ,videoandcomputer gamecharacterdesign,smoothcurvesandsurfaces(suchasF igure 1-1 C)playa fundamentalroleinthedesignofobjects.Eightypercentoft hemanufacturing(gross nationalproduct)passesthroughCAD,CAM,andCAEsystemsatso mepoint,and asingleanimatedmoviecangrossmorethan 2 billion.Bothmoviesandgamesare movingawayfrompolygonaltosmoothCADqualitysurfaces. A B C Figure1-1.A)Airplanemodelinquadmeshes.B)Zoomedinview.C )The correspondingsmoothsurface. Design( CAD )packagesneedtorepresentthesesurfacesinanefcientfo rm thatiseasytomanipulatealgorithmically,andintuitivef ortheusertomoldintothe desiredshape(Figure 1-1 ).Additionally,suchsurfacerepresentationsshouldbeeas yto evaluateandrenderontothescreen. Thisthesisaddressesthefollowingissuesfundamentaltot hedesignofsurfaces. Anoptimalconstructionforthemostcommonsurfacerepresen tation:bi-cubic splinesoverquadrangulations(Chapter 3 ). Lowerboundsonthecomplexityofsplinesoverquadrangulat ions(Chapter 2 ). Discoveringthereparameterizationofleastcomplexity(m easuredasrational degree)thatallowsformodelingsurfacesofarbitrarygenu s(Chapter 4 ). 11

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Arst,systematictestingandcomparisonofthetopcurvatu recontinuoussurface constructions(Chapter 5 ). 1.1Overview InChapter 2 andChapter 3 ,thefocusisoncreating C 1 surfacesofdegreebi3 from arbitraryquadrilateralmeshes.Earliermethodsarereview edinSection 1.4 .InChapter 2 ,weinvestigatethecomplexity.Westartwithmoregenerals etupoftheproblem, withoutlimitingthedegreeofpatchesorrequiringthatthe ybepolynomialsandderive somegeneralconstraintsintermsoftheboundarycurvesoft hepatches.Weexhibitthe simpleststructureintermsofthenumberofknotsforconstr uctionusingbi3 B-spline patches.InChapter 3 ,weproposesuchanalgorithmwiththeminimalnumberofknot s andprovetheresultingsurfacestobe C 1 Chapter 4 exploreswhetherthereexistsarationallinearreparamete rizationfor constructing C s surfacesofgenus g> 0 .Thecontrolmeshesconsistofquadrilaterals withisolatedverticesofvalence 8 andallotherverticesofvalence 4 .Auniquelinear reparameterizationisproposedandweshowconstructionsf or C 1 surfaceswiththis reparameterization. C 1 surfacesarepopularincomputeranimations,butforCAD/CAM, higherorder continuityisrequired.Forexamplecurvaturecontinuityi scrucialforsmoothreection linesincarmodels. C 2 surfaceconstructionsareusuallycomplicated.InChapter 5 fourofthemostadvanced C 2 constructionsarereviewedandcomparedintermsof curvaturevariationandconvexitypreservation. 1.2SurfacesRepresentation Therearefundamentallytwosurfacerepresentations.Impl icitSurfacesare denedasthezerosetofatrivariatefunction: f x;y;z : f ( x;y;z )=0 2 R g .Implicit representationsconvenientlysupportbooleanoperations ;butvisualizingandrendering thesurfacetypicallyrequiressolvingasetofnon-lineare quations.Wewillnotconsider thisrepresentationinthefollowingbutinsteadfocusonPa rametricSurfacePatches 12

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A B Figure1-2.A)Parametricsurfacepatches.B)Tensor-product patches. denedas f : 2 (R 2 R 3 ;f ( u;v )= x ( u;v ) ;y ( u;v ) ;z ( u;v ) : Here ( u;v ) aretheparametersand ( x;y;z ) arepointsofthesurfacein3-space (Figure 1-2 ). Twoalternativerepresentationsarecommonlyusedforpara metricpatches: theB ezierformandtheB-splineform.Bothuseasetofcontrolpoint s p ij i 2 f 0 ;:::;m g ;j 2f 0 ;:::;n g TheB ezierformofapolynomialpieceofdegree ( m;n ) isdenedas(seeFigure 1-2 B) P ( u;v )= m X i =0 n X j =0 B m;i ( u ) B n;j ( v ) p ij ; where B m;i ( u ) and B n;j ( v ) are i -thand j -thB ezierbasisfunctionsin u and v respectively: B m;i ( u )= m i !( m i )! u i (1 u ) m i ;B n;j ( v )= n j !( n j )! v j (1 v ) n j and ( u;v ) 2 [0 ; 1] [0 ; 1] B eziersurfacesarevisuallyintuitiveandgeometricallyco nvenientbecauseofthe followingproperties: 13

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Afneinvariance:Applyinganafnetransformationtoacontr olmeshresultsinthe sameafnetransformationoftheB ezierpatch. Theconvexhullproperty:AB ezierpatchlieswithintheconvexhullofitscontrol points. u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 p 0 p 1 p 2 p 3 p 4 1 0 1 2 3 controlpolygon spline Figure1-3.Univariateuniformcubicspline(from Myles [ 2008 ]).Controlpoints p :=[1 ; 3 ; 1 ; 2 ; 1] ( red )andknots u :=[ 1 ; 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7] deneacubic splineasthesumofuniformB-splinebases N i; 3 scaledbytheirrespective controlpoints( blue,green,magenta,cyan ). Thesecondrepresentation,theB-splinerepresentation,is forpiecewise polynomials.AB-splinesurfaceisdenedbyasetofcontrolp ointsandtwo knotsequencesoneinthe u andoneforthe v directionrespectively.Thecontrol pointsdenetheshapeofthesplineandtheknotssubdividet hedomain.In particular,inChapters 2 and 3 ,weusebi-3tensorproductB-splinepatcheswith non-uniformlyspacedknots.Thereare n u n v controlpoints,andtwoknotsequences [ u 0 ;u 1 ;:::;u n u +3 ] and [ v 0 ;v 1 ;:::;v n v +3 ] thatdeterminetheextentofthedomain,onefor eachofthe u and v parameters.TheB-splinepatchesaredenedas P ( u;v )= n u X i =0 n v X j =0 N i; 3 ( u ) N j; 3 ( v ) p i;j wherethe N i; 3 arebasisfunctionsasshowninFigure 1-3 BoththeB ezierandtheB-splineformrepresentpiecewisepolynomials and thereforethepatchescannotrepresentcertainshapes.For exampleconicalshapes canonlyberepresentedbyrationalfunctions.TheB-spliner epresentationistherefore extendedtoNURBS( Non-UniformRationalB-Splines )andthereisarationalB ezier 14

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representation.NURBSareusedasstandardrepresentationi nCADindustry.Being basedonaquad-grid,asingleNURBSpatchiscapableofrepres entingonlysurfaces topologicallyequivalenttoplanes,cylindersortori. Tomodelarbitrarytopologyshape,thefollowingtwoapproa cheshavebeenused: SubdivisionSurfaces,whichuseinnitelymanypolynomialor rationalpatches andconstructionswithnitelymanypatches,thefocusofth isthesis. Constructionwithnitelymanypatchesisthefocusofthist hesis. 1.3SmoothnessandGeometricContinuity Smoothnessisimportantincomputergraphicsapplicationsp rimarilyforgoodvisual effect.ButitiscrucialforhighqualityCADmodeling.Forexa mplereectionlinesona carbodyshouldvarysmoothly.Forparametricsurfaceconst ructions(seeFigure 1-2 ), smoothnessisbestexpressedbythenotionofgeometryconti nuity. G k continuity .Apairof k timesdifferentiablesurfacepatches P 1 and P 2 meet G k ,if thereexistsaregular, k timesdifferentiablechangeofvariables : 2 (R 2 2 (R 2 suchthat P 1 meets P 2 withparametriccontinuity C k 1 1 2 2 b k ( u; 0)= b k 1 (0 ;u ) b k 1 b k p Figure1-4. Indexingandparametrization ofadjacentpatchesatavertexofvalence n (if k =1 then b k 1 = b n ),illustratingthe G 1 constraints( 1–2 ). Inparticular,inChapters 2 and 3 wehave n parametrically C 1 patches b k : 2 (R 2 R 3 ;k =1 ;:::;n (1–1) 15

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thatmeetatacentralpoint b k (0 ; 0)= p ,suchthat b k ( u; 0)= b k 1 (0 ;u ) (seeFigure 1-4 ). Andweconsiderunbiased G 1 constraintsbetweenconsecutivepatches: @ 2 b k ( u; 0)+ @ 1 b k 1 (0 ;u )= k ( u ) @ 1 b k ( u; 0) : (1–2) @ ` denotesdifferentiationwithrespecttothe ` thargumentand k : R R isa sufcientlysmooth,univariatescalar-valuedfunction.( If k 0 ,theconstraintsenforce parametric C 1 continuity.) Wenoticethatbeyondcontinuity,thereisnoacceptedmathe maticaldenitionof surfacequality.Oftensurfacesarejudgedbycurvaturevar iationandpreservationofthe featurelines( PetersandKar ciauskas [ 2010 ]). 1.4Reviewofselectedliterature Thereareagoodnumberof G 1 localconstructionsthatusenitelymanypatches. (e.g. GregoryandZhou [ 1994a ]; Peters [ 1994b 2000a 2001 ]; Mylesetal. [ 2008 ]; Nietal. [ 2008 ]).Theydifferinthenumberofpatches,theirdegreeandthe resulting rangeofshapes.Thesealgorithmsprovideefcientcomputa tionwithfairqualityfor applicationslikecomputeranimationwhererenderingandi nteractionspeedisalsovery importantbesidessurfacequality. Inparticularcreating C 1 surfaceswithanitenumberofpatchesofdegreebi-3, i.e.generalizingstandardtensor-productB-splinestosmo othsurfacesfromarbitrary manifoldquadmeshes,isaclassicchallengeofCAGD(seee.g Bezier [ 1977 ]; vanWijk [ 1986 ]; Peters [ 1991b ]).Theassumptionthatasimpleconstructionwithanitenu mber ofpatchesisnotpossiblemotivatestheclassicCatmull-Cl arksubdivision(Figure 1-5 A).(Asanimportantandpopularmethodstomodelarbitrarysha pes,subdivision algorithmsuchas[ CatmullandClark 1978 ; Loop 1987 ]arederivedfromuniform splines.Theycorrespondtoaninnitenumberofnestedring sofsurfacepatches.A detailedtreatmentofsubdivisionsurfacescanbefoundin[ PetersandReif 2008 ].) PCCM[ Peters 2000a ]isaniteconstructionthatapproximatesCatmull-Clarkl imit 16

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surfaceswithsmoothlyconnectedbi-3patches.PCCMrequire suptotwostepsof Catmull-Clarksubdivisiontoseparatenon-4-valentverti ces.Thisprovesthata 4 4 arrangementofpolynomialpatchesperquadsufcesinprinc iple,correspondingto twodoubleinteriorknotsandonesingleknot(Figure 1-5 B),However,PCCMcanhave poorshapeforcertainhigher-ordersaddles(Figure 2-4 Peters [ 2000b 2001 ]).More ABC Figure1-5. Knotdistribution. A)AquadrilateralpiecegeneratedbyCatmull-Clark subdivisionhasinnitelymanysingleknots.B)ApieceofPCCM requires twodoubleandatleastonemoresingleknot.C)Theconstruct ion FanandPeters [ 2008 ]hastwodoubleinteriorknots(whichChapter 2 shows tobetheminimalnumberofknots). recently,anumberofpapersappearedthatarealsopredicat edontheassumptionthat asimpleconstructionwithanitenumberofpatchesisnotpo ssible. Shietal. [ 2004 2006 ]proposeasubdivision-likerenementapproachwithbi-3t ensor-productpatches toobtain C 0 surfaceswhereevermoresingleknotsareinserted.Loopand Schaefer [ LoopandSchaefer 2008a ]proposeabi-3 C 0 surfaceconstructionwithseparate tangentpatchestoconveyanimpressionofsmoothnessasin Vlachosetal. [ 2001 ], whileMyles etal. [ Mylesetal. 2008 ]perturbabi-3basepatchnearnon-4-valent verticestoobtaina C 1 surfaceofdegreebi-5forCADapplications.Hahmannetal. [ Hahmannetal. 2008 ]proposea 2 2 macro-patchperquad;andFanandPeters [ FanandPeters 2008 ]presentanalgorithmthatconstructssmoothlyconnected B ezierpatchesofdegreebi-3whoseinternaltransitionsall owre-interpretationasone tensor-productsplinepatchperquadwithtwointernaldoub leknots(Figure 1-5 C). 17

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Higherqualitysurfaceswith C 2 continuityareneededinapplicationslike mechanicaldesign,carmodeling.Notableconstructionsth atsupportarbitrarydegree ofsmoothnessevenattheextraordinarypointincludefreeformsplines[ Prautzsch 1997 ]andTURBS[ Reif 1998 ],bothofwhichrequiredegreebi(2 k +2) tocreatean everywhereC k surface. MylesandPeters [ 2009 ]presentedasimple-to-implement C 2 subdivisionalgorithmthatgeneratessurfacesofgoodshap eandpiecewise degree-bi3inthepolarsetting.Severalother C 2 schemeshavebeensuggested [ GrimmandHughes 1995 ; Pla-Garciaetal. 2006 ; Hahn 1989a ; GregoryandHahn 1989 ; Ye 1997 ; Reif 1995b ],mostofthemhavethelimitationoftoohighdegreesor at-nessattheextraordinarypoint. Peters [ 2002a ]suggesteda C 2 constructionof leastdegree (3 ; 5) .Thistechniquesuffersfromshapeproblemsduetothelowde gree oftheconstructions. YingandZorin [ 2004 ]createdaneverywhereC 1 surfaceusing exponentialblendingfunctionsbetweenpolynomialpatche s. LoopandSchaefer [ 2008b ] achievedcurvaturecontinuityforquadmeshesusingpatche sofbi-degree 7 withshape optimizationforthefreeparameters.Morerecentworkby KarciauskasandPeters [ 2009 ]introducestheconceptofGuidedSurfacesalsocapableofac hievingarbitrary continuity.For C 2 ,theyemploybi6 splinesurfaceringstoapproximatea C 2 “guide surface”ofgoodquality. 18

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CHAPTER2 LOWERBOUNDOFBICUBICSPLINESURFACESFROMQUADMESHES Eventhougheverynewlyproposedsmoothsurfaceconstructio nseekstobe optimalinsomeaspect,theoveralltheoryofsmoothsurface constructionsoffersfew sharplowerbounds,i.e.proofsthatnopolynomialconstruc tionoflowerdegreeis possibleandthataconstructionofthisleastdegreeexists sothatupperboundand lowerboundmatch.Thischaptergivesthesimplestbicubics tructureofpatchesthat allowaquadmeshtobeconvertedbylocalizedoperationsint oasmoothsurfacewith onesplinepatchperquad. Onewell-appreciatedboundisthedegree-6boundfor C 2 subdivisionsurfaces derivedbyReifandPrautzsch[ Reif 1996 ; Prautzsch 1997 ]andshowntobesharp,for exampleby Reif [ 1998 ]; PrautzschandReif [ 1999 ].Suchsharpboundsallowusto understandthefundamentaldifcultyofthetask,andto guidefutureresearchbyshowingwhereresearchisfutile andwhatassumptionsmustbeside-steppedtoderivesubstan tiallynewresults (seee.g. MylesandPeters [ 2009 ]). Wearemotivatedbyastandardtaskofgeometricdesign:tode termine G 1 -connected tensor-productB-splinepatchesapproximatingaquadrilat eralmeshwhosevertices canhaveanyxedvalence.Whilethischallengecanbemetbyre cursivesubdivision [ CatmullandClark 1978 ],representingthesurfacewithanitesmallnumberofpatc hes denedbythequadanditsneighborsisoftenpreferable,for exampletoparallelize theconstruction(seee.g. LoopandSchaefer [ 2008a ]; Mylesetal. [ 2008 ]).Thisraises thequestion:(Q)whatisthesimpleststructure(indistrib utionandnumberofknots) ofdegreebi-3splinepatchesthatallowaquadmeshtobeconv ertedbylocalized operationsintoasmoothsurfacewithonesplinepatchperqu ad?Surprisingly,thisbasic questionhasnotbeensettledtodate. 19

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Toframethequestion,Section 2.1 takesamoregeneralview.Wedonotconstrain thedomaintobeacollectionofquadrilateralsorthefuncti onstobepolynomialsplines. Also,therelationsinLemmas 1 2 and 3 donotdependonlocalityoftheconstruction butapplytoanycollectionofsufcientlysmoothpatchesco mingtogetherwithalogically symmetric G 1 join: @ 2 b k ( u; 0)+ @ 1 b k 1 (0 ;u )= k ( u ) @ 1 b k ( u; 0) (seeDenition 1 ,page 22 ). AddinglocalityofoperationsasarequirementinSection 2.2 thenrulesouteverywhere (piecewise)linear k ,stillintheverygeneralsetting. InSection 2.3 ,wespecializethesettingtopolynomialtensor-productsp linesof degreebi-3.Forthese,weobtainalowerboundonthenumbera ndmultiplicityof knots.Weprovethatatleasttwointernaldoubleknotsarere quiredperedgetoadmita localconstruction.Thislowerboundistight,becausecons tructionforsmoothsurfaces [ FanandPeters 2008 ]canbere-interpretedasasplineconstructionwithexactl ytwo internaldoubleknots.Together,thelowerandupperboundc onclusivelysettlethe questionQ. LowerboundconstructionisgiveninChapter 3 withexplicitlyinput,output, constructionstagesandformulastoillustratetheB-spline coefcients. 2.1Unbiased G 1 Constraints Weconsider n parametrically C 1 patches b k : 2 (R 2 R 3 ;k =1 ;:::;n (2–1) meetingatacentralpoint b k (0 ; 0)= p suchthat b k ( u; 0)= b k 1 (0 ;u ) (seeFigure 2-1 ).In Section 2.3 ,wewillassumethat 2 istheunitsquare.Fornow,weonlyassumethatthe originisacornerofthedomain 2 andthatexactlytwoboundingedges, e 1 withendpoint (1 ; 0) and e 2 withendpoint (0 ; 1) ,startfrom (0 ; 0) .Thatis,theresultsofthissectionalso apply,say,to m -sidedpatches.Wealsoassumethatthepatchesarenotsingu laratthe origininthesensethat @ 2 b k (0 ; 0) @ 1 b k (0 ; 0) 6 =0 where @ ` denotesdifferentiationwith 20

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respecttothe ` thargument.Thatis,wedonothereconsidersingularpatche ssuchas constructedin[ Peters 1991a ; NeamtuandPuger 1994 ; Reif 1998 ]. 1 1 2 2 b k ( u; 0)= b k 1 (0 ;u ) b k 1 b k p Figure2-1. Indexingandparameterization ofadjacentpatchesatavertexofvalence n (if k =1 then b k 1 = b n ),illustratingthe G 1 constraints( 2–2 ). Tomakethe n patchesforma C 1 surface,wewanttoenforcelogicallysymmetric (unbiased) G 1 constraints.(WewilldiscussthegeneralcaseinSection 2.4 .) Denition1 (Unbiased G 1 constraints) With k : R R asufcientlysmooth, univariatescalar-valuedfunction,theunbiased G 1 constraintsbetweenconsecutive patchesare @ 2 b k ( u; 0)+ @ 1 b k 1 (0 ;u )= k ( u ) @ 1 b k ( u; 0) : (2–2) If k 0 ,theconstraintsenforceparametric C 1 continuity. Weabbreviatethe ` thderivativeof k a k` 2 R ; the ` thderivativeof k evaluatedat 0 (2–3) andthetangent t k := @ 1 b k (0 ; 0) 2 R 3 (2–4) 21

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sothatrelation ( 2–2 ) becomesat (0 ; 0) t k +1 + t k 1 = a k0 t k : ( 2–2 ) u =0 a k0 t k +1 t k t k 1 Thatis,thesuperscriptscountsectorssurrounding (0 ; 0) modulo n whilesubscripts indicatederivatives. WenowmimicthesetupforNURBSpatchesbyassuminghigherdif ferentiability inthevicinityoftheintersectionoftheboundarywithafam ilyoflinesegmentsthat partitionthedomain.Sincewedonotinsistonanorthogonalg ridofknotlines,ora rectangulardomain,wewillcallsuchsmoothfunctionsgene ralizedsplines. Denition2 (Knotlines,edgeknotsandgeneralizedsplines) A C s generalizedspline patchisamap b k : 2 (R 2 R 3 thatis s> 0 timescontinuouslydifferentiable. An edgeknot isoneofanitenumberofpoints ( t j ; 0) ononeoftheboundingedges e 1 ,respectively (0 ;t j ) ontheotherboundingedge e 2 with 0
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Thatmeanswhenwehaverepetitiveknots,thenwehavemultip leidenticalknotslines startingfromthoseknots. For C 1 generalizedsplines,wecanthendifferentiaterelation( 2–2 )along(the respectivedomainedgeof)thecommonboundary b k ( u; 0)= b k 1 (0 ;u ) : ( @ 1 @ 2 b k )( u; 0)+( @ 2 @ 1 b k 1 )(0 ;u ) = k ( u ) @ 2 1 b k ( u; 0)+( k ) 0 ( u ) @ 1 b k ( u; 0) : (2–7) Whenweevaluateat u =0 then at (0 ; 0) ;@ 1 @ 2 b k + @ 2 @ 1 b k 1 = a k0 @ 2 1 b k + a k1 @ 1 b k : (2–8) If n iseventhenthealternatingsumoftheleft-handsidesvanis hes at (0 ; 0) ; n X k =1 ( 1) k @ 1 @ 2 b k + @ 2 @ 1 b k 1 =0 (2–9) andthereforesomusttherighthandside at (0 ; 0) ; 0= n X k =1 ( 1) k a k0 @ 2 1 b k + n X k =1 ( 1) k a k1 @ 1 b k : (2–10) Inparticular,ifthepatchesjoinsmoothlyandthereforeha veauniquenormal n 2 R 3 at p then,with denotingtheEuclideaninnerproduct, if n iseven,at(0,0) 0= n X k =1 ( 1) k a k0 n @ 2 1 b k : (2–11) Thisisthevertex-enclosureconstraint(seee.g.[ Peters 2002b ,p.205],[ Hermannetal. 2009 ]). Webrieyfocusontheimportantgenericcasewhere n =4 patchesmeet. Denition3 (tangentX) If n =4 @ 1 b 1 (0 ; 0)= @ 1 b 3 (0 ; 0) and @ 1 b 2 (0 ; 0)= @ 1 b 4 (0 ; 0) thenthetangents formanX Lemma1 (Xtangent) Ifthetangentsoffour C 1 generalizedsplinesformanX,then a 11 = a 31 and a 21 = a 41 : 23

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Proof. IfthetangentsformanXthen n =4 and a k0 =0 k =1 ; 2 ; 3 ; 4 sothat( 2–10 ) simpliesto at(0,0) ; 0=( a 11 a 31 ) @ 1 b 1 ( a 21 a 41 ) @ 1 b 2 : (2–12) Sincethepatchesareregularatcorners,bothsummandshavet ovanish,implyingthe claim. Wenowconsidertheunbiased G 1 transitionbetweentwo C 1 generalizedspline patches.Wefocusonanedgevertex,theimageofanedgeknot, thatisnotanendpoint oftheboundary.Withoutlossofgenerality,wecanassumetha tatsuchanedgevertex fourpiecesmeetsuchthat b 1 and b 2 belongtoonegeneralizedsplinepatch,and b 3 and b 4 arepiecesoftheedge-adjacentgeneralizedsplinepatch(F igure 2-2 ).For,if anedgeknotdoesnothaveacounterpartintheneighboringge neralizedsplinepatch, i.e.wehavea`T-corner',wecanaddanedgeknotandacorresp ondingknotlineto subdividetheneighboringpatch.Sinceeachgeneralizedspl inepatchisinternally parameterically C 1 ,byDenition 1 2 0 4 : (2–13) b 1 b 2 b 3 b 4 1 2 3 1 2 Figure2-2.Joinacrossan edgeknot ontheboundary(solid)betweentwogeneralized splines.Therstgeneralizedsplinehaspieces b 1 and b 2 24

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Lemma2 ( C 1 generalizedspline,edgevertex) Let (0 ; 0) betheparameterassociated withanedgevertexontheboundarycommontotwo C 1 generalizedsplinesthatare joinedbyunbiased G 1 constraints.Then a 10 = a 30 ; (2–14) at (0 ; 0):0= a 10 ( @ 2 1 b 1 @ 2 1 b 3 )+( a 11 a 31 ) t 1 : (2–15) Proof. Since n =4 a 10 t 1 = t 2 + t 4 = a 30 t 3 andtheparametric C 1 constraintsimply t 1 = t 3 sothat( 2–14 )follows.By( 2–13 ),( 2–10 )specializesto at (0 ; 0) ; 0= a 10 @ 2 1 b 1 + a 30 @ 2 1 b 3 + a 11 @ 1 b 1 + a 31 @ 1 b 3 = a 10 ( @ 2 1 b 1 @ 2 1 b 3 )+( a 11 a 31 ) t 1 asclaimed. So,remarkably,whentwogeneralizedsplinepatchesmeetalo ngacommon boundary,unbiased G 1 constraintsacrossthisboundaryimplytheconstraint( 2–15 ) exclusivelyintermsofderivativesalongtheboundary.Lemma3 ( C 2 generalizedspline,edgevertex) Let (0 ; 0) betheparameterassociated withanedgevertexoftheboundarycommontotwo C 2 generalizedsplinesjoinedby unbiased G 1 constraints.Then,inadditionto ( 2–14 ) ,at (0 ; 0) a 11 = a 31 ; (2–16) 0= a 10 ( @ 3 1 b 1 @ 3 1 b 3 )+4 a 11 @ 2 1 b 1 +( a 12 a 32 ) t 1 : (2–17) Proof. Sincethegeneralizedsplinesare C 2 @ 2 1 b 1 (0 ; 0)= @ 2 1 b 3 (0 ; 0) .Then( 2–15 )is equivalentto( 2–16 ). 25

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Parametric C 2 continuityacrossthespline-internalboundaries(seedas hedlinesin Figure 2-2 )implies for k =2 ; 4 ; at (0 ; 0) ;@ 2 @ 1 @ 2 b k + @ 1 @ 2 @ 1 b k 1 =0 : (2–18) Differentiating( 2–7 )oncemorealongthe(directioncorrespondingtothe)commo n boundaryofthetwogeneralizedsplines,weobtainfor k =1 ; 3 ; at (0 ; 0) ; @ 1 @ 1 @ 2 b k + @ 2 @ 2 @ 1 b k 1 = a k0 @ 3 1 b k +2 a k1 @ 2 1 b k + a k2 @ 1 b k : (2–19) Summingthetwoinstancesof( 2–19 )andsubtractingthetwoinstancesof( 2–18 ) eliminatesthemixedderivativesofthelefthandsideandyi eldsat (0 ; 0) 0= a 10 @ 3 1 b 1 +2 a 11 @ 2 1 b 1 + a 12 @ 1 b 1 (2–20) + a 30 @ 3 1 b 3 +2 a 31 @ 2 1 b 3 + a 32 @ 1 b 3 : Parametric C 2 continuitythenimplies( 2–17 ). 2.2Linear andVertex-localizedConstructions TheTaylorexpansionsuptoordertwoofthepatchesjoininga tapointarestrongly intermeshedbyEquation( 2–8 ).Toavoidsolvinglarge,globalsystems,theexpansionat avertexshouldnotdependontheexpansionsattheneighbori ngvertices. Denition4 (vertex-localizedconstruction) Asurfaceconstructionalgorithmis G 1 vertex-localized if,ateveryvertex(withlocalparameters ( u;v )=(0 ; 0) ),itsetsthe second-orderTaylorexpansion @ i 1 @ j 2 b k 0 i;j;i + j 2 independentoftheexpansions attheneighborverticesandsothattheunbiased G 1 constraints ( 2–2 ) u =0 and ( 2–7 ) hold. Avertex-localizedconstructionenforcesthevertex-encl osureconstraint( 2–11 )at eachvertex.Wewillprexastatementwith `ingeneral' topointoutthatweconsiderall nonsingularchoicesofexpansionssatisfying( 2–11 ). 26

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Notethatavertex-localizedconstructioncanuseapriorik nowninput,forexample thelocalconnectivityandthevalenceoftheneighbors.Nev ertheless,theunbiased G 1 constraintsimplyalocal,unbiasedchoiceofthetangentdi rections,namelysuchthat k (0):=2cos 2 n : (2–21) (Foraproofthatlogicalsymmetryimplies( 2–21 )seee.g.[ Peters 1994a ,Prop3].) Corollary1 (valencesymmetryfor n =4 andlinear ) Let n =4 andlet n k denotethe valenceofthe k thneighborvertex, k =1 ;:::;n .Thenalocal,unbiasedchoiceofthe tangentdirectionsand k lineararecompatiblewithunbiased G 1 constraintsonlywhen thevalencesofoppositeneighborsagree: n k = n k +2 Proof. TheclaimfollowsfromLemma 1 sincebytheunbiasedchoice k (0):=0 and k (1):=2cos 2 n k Corollary 1 isaremarkablystrongrestrictionsinceverticesofvalenc e n =4 are common.Choosinglinear canthereforebeproblematic.Forexample,theconstructio n [ Hahmannetal. 2008 ]canthereforenotsucceedingeneral. regular 0 0 j 1 q 6 =0 1 0 n q 6 =4 n 0 =4 Figure2-3. Propagation oflinear j 0 inLemma 4 Eachscalarfunction k canconsistofpiecesthatcorrespondtotheknotsegmentsof thetwogeneralizedsplinesmeetingalongthecurve.Since,i nthiscontext,weonlydeal withone k atatime,wedropthesuperscript k andpartitionthedomainof := k by 27

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0
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Figure2-4. Shapedefect (starshape)duetoembeddedstraightlinesegmentsata higherordersaddlefrom Peters [ 2000b ]. vector-valuedpolynomialandascalarfactor: c 0 := ` ` `r; (2–22) ` ` ` : R R 3 ; deg ( ` ` ` ) 1 ;r : R R ; deg ( r ) 1 then c isaplanarcurvesegment.Ifthesaddleissymmetricaboutth eplanespannedby thenormalandthetangentthen c isa straightlinesegment Proof. Let n bethenormalat p and,withoutlossofgenerality, r ( u ):=1+ r 1 u forsome r 1 2 R .Then c 0 (0)= ` ` ` (0) c 00 (0)= ` ` ` 0 (0)+ ` ` ` (0) r 1 and c 000 (0)=2 ` ` ` 0 (0) r 1 .Atahigher-order saddlepoint,thenormalcurvatureiszero,andtherefore n c 00 (0)=0 .Thisimplies n ` ` ` 0 (0)=0 and n c 000 (0)=0 establishingplanarity.Ifthesaddleissymmetricthen c 0 (0) and c 00 (0) arecollinearandsois c 000 (0)=2 ` ` ` 0 (0) r 1 Ahigher-ordersaddle,suchasthemonkeysaddleofFigure 2-4 ,shouldhave non-zeroGausscurvatureapartfromthecentralsaddlepoin t.Therefore,wewillin thefollowingdisqualifyconstructionsthatforcestraigh tsegmentsontheboundaryfor non-atgeometry. Tosummarize,weshowedthatvertex-localizedunbiased G 1 constructionswith generalizedsplinesaresubjecttostrongrestrictionsont hereparametrization (Lemma 1 2 and 3 )ortheallowablevalenceofthevertices(Corollary 1 ).Inthenextsection,we applythesegeneralrestrictionstopolynomialsplines. 29

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2.3LowerBoundsforDegreeBi-3 Wenowarguethat,ingeneral,vertex-localizedenforcemen tofunbiased G 1 constraintswithpolynomialtensor-productsplinesofdeg reebi-3(bicubic)ispossible onlyifthesplinepatcheshaveatleasttwointernaldoublek notsperedge. Sincewespecializetopolynomials b k ofdegreebi-3,equalityinthe G 1 constraints impliesthat isarationalfunction, =: r .Infact,wehavealowboundonthedegrees ofthenumerator andthedenominator r Lemma6 ( degreerestricted) Ifthetwobi-3patches b k and b k 1 satisfyanunbiased G 1 constraint ( 2–2 ) theneither k := r isrationalwith (2–23) ( deg ( ) ; deg ( r )) 2f (2 ; 1) ; (2 ; 0) ; (1 ; 1) ; (1 ; 0) ; (0 ; 1) ; (0 ; 0) g and @ 1 b k ( u; 0)= ` ` ` ( u ) r ( u ) ; deg ( ` ` ` ) 2 deg ( r ) (2–24) ortheboundary b k ( u; 0) isforcedtohaveastraightsegment. Proof. Wemayassumethat and r arerelativelycoprime.Sincethelefthandside @ 2 b k ( u; 0)+ @ 1 b k 1 (0 ;u ) ofthe G 1 constraint( 2–2 )ispolynomial, r ( u ) mustbea(scalar) factorof @ 1 b k ( u; 0) 2 R 3 ,the(vector-valued)derivativeoftheboundarycurve.Unl ess b k ( u; 0) isalinesegment, 0 < deg ( @ 1 b k ( u; 0)) 2 .Consequentlydeg ( r ) 2 andsince deg ( r )=2 impliesthat @ 1 b k ( u; 0)= v r foraconstantvector v 2 R 3 ,deg ( r ) 1 must holdtoavoidthat b k ( u; 0) isastraightsegment.Sincedeg ( @ 2 b k ( u; 0)+ @ 1 b k 1 (0 ;u )) 3 alsodeg ( @ 1 b k ( u; 0) ) 3 andthereforedeg ( ) 2 Afterscalingnumeratoranddenominator,wemayassumethat r ( u ):=1+ r 1 u .A non-linear thenforcesaparticularboundarycurve. Corollary2 (Non-linear restrictsboundarycurves) If ( deg ( ) ; deg ( r )) 2 f (2 ; 1) ; (2 ; 0) ; (1 ; 1) ; (0 ; 1) g thenthecorrespondingdegree3boundarycurvesegment isoftheform ( 2–22 ) 30

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Proof. Thederivativeofthecurvesegmenteitherhasalinearfacto r r oritislinear becausedeg ( )=2 Lemma 5 andCorollary 2 togetherimplythatingeneral,atendpoints, mustbe linearorconstantifwerequiremoreexibilitythanforced straightlinesegments. Corollary3 (Non-linear atahigher-ordersaddle) If b k ( u; 0) emanatesfroma symmetrichigher-ordersaddlepointandisofdegreeatmost threethen k inthe unbiased G 1 constraints ( 2–2 ) mustbelinearorconstantfor b k ( u; 0) nottobeastraight segment. b 1 b 2 b 3 b 4 1 2 3 1 2 Figure2-5.(Figure 2-2 repeated)Joinacrossan edgeknot ontheboundary(solid) betweentwosplines.Therstsplinehaspolynomialpieces b 1 and b 2 Thenextlemmashowsthatatedgeknots,neighboringpieceso f constrainone anothermorethanjustby( 2–15 )and( 2–17 ). Lemma7 ( non-linearatsingleknot) LetthesegmentsbearrangedasinFigure 2-5 theedgeknot single andtheleftboundarysegment( b 3 ( u; 0) sharedbythetwobi-3 splines)xedbutgeneral(inthesensethatthecontrolpoin tscannotbeassumedtobe inaparticularrelation).Then 1 canonlybenon-linearif a 30 =0 ;a 31 =0 ; and a 32 = a 12 6 =0 : (2–25) Inparticular, 3 mustalsobenon-linear. Proof. If := 1 isnon-linearthenLemma 6 implies ( deg ( ) ; deg ( r )) 2f (2 ; 1) ; (2 ; 0) ; (1 ; 1) ; (0 ; 1) g 31

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andtherefore @ 1 b 1 ( u; 0):= ` ` ` ( u ) r ( u ) ,alinearvector-valuedpolynomialtimesthescalar (possiblyconstant)factor r ( u ):=1+ r 1 u .By( 2–14 )and( 2–16 )andthe C 2 constraints fortheboundarycurve,constraint( 2–17 )becomes at (0 ; 0) ; 0= a 30 ( @ 3 1 b 3 2 r 1 ` ` ` 0 (0) | {z } =: v )+4 a 31 @ 2 1 b 3 +( a 32 a 12 ) t 3 : (2–26) By C 1 continuity ` ` ` (0) r (0)= ` ` ` (0)= t 3 andhencethe C 2 constraint @ 2 1 b 3 = ` ` ` (0) r 1 + ` ` ` 0 (0)= t 3 r 1 + ` ` ` 0 (0) implies ` ` ` 0 (0)= t 3 r 1 + @ 2 1 b 3 (0 ; 0) : (2–27) Therefore,at (0 ; 0) v = @ 3 1 b 3 2 r 1 ( t 3 r 1 + @ 2 1 b 3 ) .Since,ingeneral, @ 3 1 b 3 (0 ; 0) @ 2 1 b 3 (0 ; 0) and t 3 arelinearlyindependent,thescalar r 1 cannotforce v =0 (recallthat b 3 isxed), andsince v @ 2 1 b 3 (0 ; 0) and t 3 arelinearlyindependent,wemusthave a 30 =0 and a 31 =0 and a 12 = a 32 inorderfor( 2–26 )tohold. If 3 islinearthen a 32 =0 andsince 00 (0)= r 00 (0)= 00 (0) when (0)= 0 (0)=0 (notethat r (0)=1 andhence (0)= 0 (0)=0 ),wehave 1 0 contradictingthe assumptionthat 1 isnon-linear. Wenowhaveallthepiecesinplacetoprovethemaintheoremof smoothsurface constructionwithbi-3splines.Theorem1 (twodoubleedgeknotsneeded) Ingeneral,usingsplinesofdegreebi-3for avertex-localizedunbiased G 1 constructionwithoutforcedlinearboundarysegments requiresthesplinestohaveatleasttwointernaldoublekno ts. Proof. Ingeneral,iftheboundarycurvehasonlyasingle1-foldkno t(hencetwo C 2 -connectedsegments)therearenotenoughdegreesoffreedo mtoenforce C 2 continuityofthepiecewisecurve.Iftherearetwo1-foldkn ots(three C 2 -connected segments), C 2 continuityuniquelydeterminesallboundarycoefcients. Ifthereis one2-foldknot(two C 1 -connectedsegments), C 1 continuityuniquelydeterminesall boundarycoefcients.However,intheselasttwocases,( 2–17 )isunresolvedatthe 32

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(two,respectivelyone)edgeknots f i g andtherefore,ingeneral,thesebasecasesallow forconstructinga C 2 boundarycurvebutnotforenforcing( 2–2 ). Insertingoneadditionaledgeknotthatis1-foldcreateson eadditionalboundary curvesegment j ofdegree3constrainedbyfourvector-valuedconstraints: the parametric C 0 C 1 and C 2 constraintsplus( 2–17 )or,equivalently,onefreesplinecontrol pointsubjectto( 2–17 ).If j islinear,itstwocoefcientsaredeterminedvia( 2–14 ) and( 2–16 )bythoseoftheneighborsegment,andthereforethefree(B-s pline)control pointmustbeusedtoresolve( 2–17 ).Thatis,if j islinear,wedonotgaindegreesof freedomthatwouldenableenforcing( 2–17 )attheedgeknots f i g ofthebasecase. ByCorollary 3 ,thestartingsegment's 0 canbeassumedtobelinear.Let j be non-linearwhile l l =0 ;:::;j 1 j 1 ,arelinear.Bythereasoningoftheprevious paragraphall b l ( u; 0) l =0 ;:::;j 1 aredeterminedsothatLemma 7 applies:thatis, j canonlybenon-linearifthereisatleastbyonedoubleknotb etweensomesegment b l 1 ( u; 0) and b l ( u; 0) Thesymmetricargumentattheotherendimpliestheclaim. TheproofofTheorem 1 revealsslightlymorethanitsclaim:theinteriorsegment with j non-linearmustbeseparatedbydoubleknotsfromeitherend segment.The simplestsuchconstructionisthenbasedonthreesegmentsw iththemiddlesegment bracketedbytwodoubleknots,andsuchthat 0 and 2 arelinearand 1 quadratic(see Figure 1-5 C). Corollary4 (lowerboundissharp) Theconstructionin FanandPeters [ 2008 ]usesthe fewestknotswhencreatingasmoothsurfacethathasonebi-3 splineassociatedwith eachquadofageneralquadmeshandnoforcedlinearsegments Proof. Bycoveringeachquadwitha 3 3 arrangementofparametrically C 1 -connected bi-3patchesinBernstein-B ezier-form,theconstructionin FanandPeters [ 2008 ] usesexactlytwoedgeknots,both2-fold.Byitschoiceofquad ratic 1 justforthe G 1 33

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constraintsacrossthemiddlesegmentandlinear 0 and 2 fortheendsegments,it neednothavetheshapeproblemcharacterizedbyLemma 5 Figure2-6. NoShapedefect (noforcedstraightlinesegments)inahigherordersaddle (cf.Figure 2-4 ). 2.4DiscussionandConclusion Remarkably,theresultsinSection 2.1 donotdependonthedegreeoreventhe polynomialnatureofsplines,butassumeonlysufcientlys moothfunctionsthatare piecewisewithsmoothtransitionsbetweenthepieces.Inpa rticular,theresultsapplyto niterenementbysubdivisionwhichcreatesparametrical lysmoothtransitionswithin eachgeneralizedspline.Theextensiontogeneralizedspli nesmappingto R d d> 3 is straightforward. Forbi-3splinesthesegeneralconstraintsimplyalowerbou ndonthenumberand distributionofknots.Theconstructionin FanandPeters [ 2008 ]showsthelowerbound tobetight. Theresultsextendtoconstructionsbasedon G 1 transitionsoftheform k ( u ) @ 2 b k ( u; 0)+ r k ( u ) @ 1 b k 1 (0 ;u )= k ( u ) @ 1 b k ( u; 0) forwhichthereisasufciently richsetofinputdatathatimply = r .Forexample,if ( k ; k ;r k ) reectthelocal geometricdistributionoftheinputdata,anylocallysymme tricinputyields = r andthe resultsofthepaperapply. 34

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Theboundsprovideachecklistforconstructions.Theorem 1 impliesforexample thatthereisasubtleerrorintheproofofthenon-trivialco nstruction[ Hahmannetal. 2008 ]whichusesonedoubleedgeknotonly:theconstructionfall sfoulofCorollary 1 Sucha 2 2 splitconstructioncanonlysucceedinspecialcases.Choos inggeneric inputdata,suchastwocubesjoinedatonefacetoadouble-cu beinitialcontrolnet, showsaproblematthesplittingpoints, 1 4 ,respectively 3 4 downthedoubleedges,where n 1 = n 2 = n 3 =4 but n 4 =3 .Asasecondexample,Lemma 4 preventsavertex-localized solutionwithall j linear.Whenthislemmaisspecializedbyxingthedegreetob e3, byincreasingthepatchcontinuityto C 2 andbychoosing 0 j := q j q 00 + j q 0 q thenit yieldsaproofoftheclaim[ Shietal. 2004 ,Thm3.1].(Inlightof( 2–17 ),wemightadjust thetitlesof Shietal. [ 2004 ]and Shietal. [ 2006 ]sincewecannotobtain G 1 surfacesby addingsingleknots.) Whenwerestrictconnectivity,i.e.droptheassumptionmade attheoutsetthat theconstructionappliestogeneralinputandusesonetenso r-productsplineperquad, thenconstructionswithfeweredgeknotsarepossible.Forr estrictedconnectivity,itis wellknownthatifallvalencesareoddortangentsformanX,th envertex-enclosure doesnotimposeconstraintsandsimpleB ezierconstructionsarepossible(e.g. vanWijk [ 1986 ]; Peters [ 1991b ]; GregoryandZhou [ 1994b ]).If n 0 = n 1 alwaysholds,say whensmoothingacube,thenwecanchooselinear 1 and 3 with a 11 = a 31 and a 10 =0 toenforce( 2–15 ).Thatis,aconstructionwithonedoubleedgeknotis possible.Suchaconstruction,coveringaquadby 2 2 bi-3patches,isproposed in Hahmannetal. [ 2008 ].Asimilarbutdual,spline-likeconstructionappearsin ZhaoandTeh [ 1995 ].Globalconstructions,singularparameterizations[ Peters 1991a ; NeamtuandPuger 1994 ; Reif 1998 ],controlofthevalence,forexamplebysplitting patches[ Prautzschetal. 2002 ,9.11], Peters [ 1991b 1995b ],canallowforsimpler constructions.Reif's G 1 construction[ Reif 1995a ]usesmultiplepatchesperquad, notfollowingthetensor-productsplineparadigmbutwiths omebiasedtransitions. 35

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Remarkably,thepatchesareofdegreebi-2.Butthisimpliesu ndesirableboundary curvesbyLemma 5 Ifweallowhigherdegree,thengeneralconstructionsofsmo othsurfaceswithone patchperquadareshownpossiblefordegreebi-5,forexampl e Mylesetal. [ 2008 ].For degreebi-4,asingleknot(a2x2-split)mustbeintroduced( seee.g. Peters [ 1995a ]). Thecaseofseveral G 1 -connectedpatchesperquadstillawaitsfullinvestigatio n,as doesthecaseofrationalbi-3patchesandthegeneralizatio noftheproblemtounbiased G k transitionsfor k> 1 36

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CHAPTER3 THEBICUBICSPLINECONSTRUCTIONFORQUADMESHES Whenassociatingonebi-cubictensor-productsplinepatchw itheachfacetofa quadrilateralmesh,themainchallengecomesfromconverti ngextraordinaryquads, i.e.quadsthathaveoneormoreverticesofvalence n 6 =4 (redquadsinFigure 3-1 D). Regularquadshaveonlyverticesofvalence4andthereforea llowinterpretationofthe surrounding 4 4 gridofverticesascontrolpointsofbi-cubicB-splinesurfa ce(gold quadsinFigure 3-1 D). A B C D E Figure3-1.Bi-3 NURBSconstruction. A)QuadmeshwithB)enlargedsectionandC) surfacegeneratedbythealgorithm.D)Redquadsareextraor dinarywith possiblyseveralnon-4-valentvertices.Regulargoldquad scorrespondto bi-3polynomialpatchesthatjoin C 2 withtheirneighbors,alsoacross T-junctions.E)Aminimalnumberofbi-3polynomialpieces(r andom coloring). ThischapterpresentsexplicitB-spline(NURBS)controlpoint formulasfor aconstructionwithonedegreebi-3NURBSpatchperquad, atmosttwointeriordoubleknots,and acompletesmoothnessanalysis. 37

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EachNURBSpatchhas,pertensor-direction,eithernointerio rknot(andrepresentsa pieceofastandard C 2 splinecomplex)ortwointeriordoubleknots.Themainalgor ithm hasbeenpublishedin FanandPeters [ 2008 ],asmallimprovementmadehereistohave C 2 continuityacross n =4 regularvertices.Itisexplicitlyshownintheformulasint he followingSectionsandalsodemonstratedintheresult(seeF igure 3-4 G,H). Section 3.1 givesexplicitlyinput,output,constructionstagesandfo rmulaswith illustrationsoftheB-splinecoefcients.Section 3.2 formallyveriesthattheconstruction satisesthe G 1 continuityconstraints,i.e.resultsforgenericinputina natleast C 1 surfaceeverywhereand C 2 transitionsacrossedgeswith4-valentvertices.Section 3.3 examinesthequalityoftheresultingsurfacesbyexamplesa ndSection 3.4 points togeneralizationssuchasusingmorethantheminimalnumbe rofknotsinthespline patches. 1 1 2 2 b k ( u; 0)= b k 1 (0 ;u ) b k 1 b k A @ 2 b k @ 1 b k @ 1 b k 1 b k ( u; 0) B Figure3-2. G 1 constraints: A) n patchjoin.B)Coplanarpartialderivativesalonga boundarycurve. Consider n patchesthatareinternally(parametrically) C 1 b k :[0 :: 1] 2 (R 2 R 3 ;k =1 ;:::;n; (3–1) andthatmeetatacentralpoint b k (0 ; 0) andsuchthat b k ( u; 0)= b k 1 (0 ;u ) asdisplayed inFigure 3-2 .Foreverypairofpatchessharingacurve,weenforcelogica llysymmetric 38

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unbiased G 1 constraints @ 2 b k ( u; 0)+ @ 1 b k 1 (0 ;u )= k ( u ) @ 1 b k ( u; 0) ; (3–2) where @ ` meansdifferentiationwithrespecttothe ` thargumentand k :[0 :: 1] R isaunivariatelinearorquadraticscalarfunction.If k =0 ,theconstraintsenforce parametric C 1 continuity. Todelineateeachbi-3splinepatch,weplace 4 -foldknotsattheendsoftheknot sequence.Tocopewithextraordinaryquadswithoneormorev erticesofvalence n 6 =4 PetersandFan [ 2009 ]provesthatatleasttwodoubleinternalknotsareneeded.Sc aling tomultiplesof3yieldsaknotsequence :=( 0 ; 1 ;::: 11 )=(0 ; 0 ; 0 ; 0 ; 3 ; 3 ; 6 ; 6 ; 9 ; 9 ; 9 ; 9) thatcannicelybeusedtoassociatecontrolpointswiththei rGrevilleabscissae r (cf. Figure 3-3 C).Thatisforthe k thpatchsurroundingthecentralpoint, b k ( u;v ):= 7 X i =0 7 X j =0 b kr i ;r j N i ( u ) N j ( v ) ;r :=( r 0 ;:::r 7 )=(0 ; 1 ; 2 ; 4 ; 5 ; 7 ; 8 ; 9) (3–3) where N i isthedegree3B-splinewithknotsequence i ;::: i +4 .Below,wewillalso refertothebi-3(=bi-cubic)Bernstein-B ezierform(short:BB-form)ofapolynomial 3 X i =0 3 X j =0 c r i ;r j f i ( u ) f j ( v ) ;f ` ( t ):= 3 ` (1 t ) 3 ` t ` : ToconvertauniformsplinewithB-splinecoefcients p ij totheBB-formweneedonly specify 36 c 00 =16 p 11 +4( p 21 + p 12 + p 01 + p 10 )+( p 22 + p 02 + p 00 + p 20 ) 18 c 10 =8 p 11 +2( p 10 + p 12 )+4 p 21 + p 22 + p 20 (3–4) 9 c 11 =4 p 11 +2( p 21 + p 12 )+ p 22 sincetheremaining13formulasareobtainedbycombinatori alsymmetry. 39

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3.1TheAlgorithm Wefactorthealgorithmintovelocalizedstagesthatareex plainedindetailin Sections 3.1.3 – 3.1.7 .Thecoefcientsofastagecanbedeterminedinparallel,fo rall quads,respectivelyallvertices.Algorithm Foreachquad,we 1.computeabi-3tensorproductpatch Q inBernstein-B ezierform; Ifthequadisregular,set b = Q andSTOP, elseinserttworepeatedknotsin u and v direction. 2.adjusttangentstoenforce( 3–2 ); 3.completetheboundarytoenforce( 3–2 ); 4.modifytherstinteriorlayerofcontrolpointstoenforc e( 3–2 ); 5.computetheremaininginteriorcontrolpointsbyaveragi ng. 3.1.1ControlPoints Belowwegivetheformulasforthecontrolpoints b kr i r j .Thepresentationis intentionallytersetofocusoneasyimplementation.(Howe ver,wementionthechoiceof k toexplainthederivationandpreparetheproofofsmoothnes sinSection 3.2 .) Intheguresillustratingtheformulas,older,alreadycom putedcontrolpointsare presentedasgraydisks;thenewcontrolpoints,computedby theformulas,areshown asblackdisks.Onlyafewcontrolpointsneedtobeshownsinc etheremainingonesare obtainedsymmetrically,i.e.bycombinatorialsymmetry.3.1.2InputandOutput[Input: p ] (Figure 3-3 A)Avertex p 0 ofvalence n := n 0 surroundedbyvertices p 1 ; p 2 ;:::; p 2 n forming n quads.Thevalenceofthedirectneighbors p k 2f p 1 ; p 2 ;:::; p n g of p 0 is denoted n k [Output: b ] 40

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1 1 1 2 3 4 n n n p 0 p 1 p n +1 p 2 p 2 n 90 99 09 04 07 00 p Q b Q 00 Q 30 ABC Figure3-3. LayoutandIndexing. A)Localinputmesh p .B) Q patchinitialization.C) NURBScontrolpointindicesofapatch b (Figure 3-3 B,C)OneNURBSpatchofdegreebi-3perquadjoiningitsneighb orsto formasmoothsurfacesuchthatadjacentpatchessatisfyunb iased G 1 constraintsand join C 2 ,alsoacrossT-junctions,acrossedgewherebothend-point shavevalence4. Ifthequadisordinary,theoutputisabi-3patch Q (Figure 3-3 B)inBB-form,i.e.a NURBSpatchwithknots :=(0 ; 0 ; 0 ; 0 ; 9 ; 9 ; 9 ; 9) ineachdirection. Otherwise,theoutputsplinepatch b (Figure 3-3 C)hastheknotsequence := (0 ; 0 ; 0 ; 0 ; 3 ; 3 ; 6 ; 6 ; 9 ; 9 ; 9 ; 9) .Section 3.4 discusseswhattodowhenmorethanthis minimalnumberofknotsarewanted.3.1.3Step1B ezierPatch Weuseformula( 3–4 )toderiveallpoints,exceptthatwegeneralizetheformula of Q k00 tothewell-knownlimitpointofCatmull-Clarksubdivision : 41

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Q p 6 Q 00 Q 10 Q 11 Q k00 := P nl =1 n p 0 +4 p l + p n + l n ( n +5) ; (3–5) Q k10 := 8 p 0 +4 p k +2 p k +1 +2 p k 1 + p n + k + p n + k 1 18 (3–6) Q k11 := 1 9 (4 p 0 +2( p k + p k +1 )+ p n + k ) : (3–7) Symmetricconstructionoftheotherthreecornersofthequad yieldscoefcients Q kij that weinterpretasthecontrolpointsofNurbspatchwithknotss equence(0,0,0,0,9,9,9,9). Ifthequadisregular,thenwestopandoutputthepatch b := Q .Wenotethatifwe applytheremainingstepsofthealgorithm,wecanchoose Q k00 tobesomeotherpoint thatthesurfaceshouldinterpolate,e.g.theoriginalmesh point p 0 .Thisisillustratedin Figure 3-4 D. Ifthequadisnotregular,inserttwodoubleknots(rstinth e u theninthe v direction)toobtain b 00 b 10 b 40 Q 10 Q 20 b k00 := Q k00 ; b k10 := 2 3 Q k00 + 1 3 Q k10 ; b k20 := 4 9 Q k00 + 4 9 Q k10 + 1 9 Q k20 ; b k40 := 4 27 Q k00 + 12 27 Q k10 + 1 3 Q k20 + 2 27 Q k30 : 3.1.4Step2AdjustTangents[if n 0 6 =4 ]thecoefcientsareadjustedtofallintothetangentplane spannedby e 1 and e 2 : 42

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p b k10 b k11 Q k11 b k10 := b k00 + e 1 c kn + e 2 s kn 3 ; e i := n 3(2+ n ) n X l =1 r i p l + i p n + l c kn :=cos 2 k n ;s kn :=sin 2 k n ;c n := c 1n ;! n :=16 n 4 ; n := 1 16 ( c n +5+ p ( c n +9)( c n +1)) ; n := n 0 : 53 if n =3 1 4 n if n> 3 r 1 := n c ln ; 1 := c ln + c l +1 n ;r 2 := n s ln ; 2 := s ln + s l +1 n ; b k11 := 6( b k10 + b k01 ) 4 b k00 + Q k11 9 and b k11 isplacedaccordingtothe 1:1:1 splitofknotinsertion. 3.1.5Step3DeterminetheBoundary3a.[if n 0 =4= n k ] skipSteps3and4. 3b.[if n 0 6 =4 6 = n k ] Wechoose k i ( u ):= ki (1 u )+ ki +1 u i =0 ; 1 ; 2 ,in( 3–2 )and k0 :=2cos( 2 n 0 ) ; k3 := 2cos( 2 n k ) ; k1 := 2 k0 + k3 3 ; k2 := k0 +2 k3 3 (3–8) b 00 b 20 b 40 andtherefore b k20 := b k10 + 3( b k11 + b k 1 11 2 b k10 ) k1 ( b k10 b k00 ) 2 k0 ; b k40 := 4 3 b k20 1 3 b k80 + 2 3 b k70 2 3 b k10 ; b k50 ; b k70 arecomputedsymmetrically. 3c.[if n 0 6 =4= n k ] By PetersandFan [ 2009 ],notall k i canbelinear.Wechoose k 0 ( u ):= k0 (1 u )+ k1 u; k 1 ( u ):= k1 (1 u ) 2 + k2 (1 u ) u + k3 u 2 ; k 2 ( u )= k3 (3–9) andset k0 :=2cos( 2 n 0 ) ; k1 := k0 2 ; k2 :=0 ; k3 :=0 43

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b 00 b 20 b 40 b 40 b 50 Then b k20 := b k10 + 3( b k11 + b k 1 11 2 b k10 ) k1 ( b k10 b k00 ) 2 k0 ; b k40 := 41 25 b k20 + 4 25 b k70 4 5 b k10 ; b k50 := 36 25 b k20 + 9 25 b k70 4 5 b k10 ;: Notethat b k70 remainsunchangedfromknotinsertionguaranteeing C 2 continuity acrossregular n =4 vertices.Thisisdifferentfrom FanandPeters [ 2008 ],wherewe modied b k70 afterwardssuchthattheresultingsurfaceisonly C 1 acrossregularvertices. 3.1.6Step4FirstInteriorLayer4a.[if n 0 6 =4 or n k 6 =4 ] computepreliminarycoefcients ~ b kij : ~ b 21 ~ b 41 ~ b k21 := 4 9 b k01 + 4 3 b k11 + 1 3 b k81 2 9 b k91 ; ~ b k41 := 20 27 b k01 + 4 3 b k11 + b k81 16 27 b k91 ; ~ b k51 ; ~ b k71 arecomputedsymmetrically.Then 4b.[if n 0 6 =4 6 = n k ] h 1 := b k20 + 0 b k40 b k20 2 +2 1 ( b k20 b k10 ) 6 ;h 2 := b k40 + 2 1 ( b k50 b k40 )+ 2 b k40 b k20 2 6 ; b 21 b 41 b k21 := h 1 + 1 2 ( ~ b k21 ~ b k 1 12 ) ; b k41 := h 2 + 1 2 ( ~ b k41 ~ b k 1 14 ) ; b k51 ; b k71 arecomputedsymmetrically. 4c.[if n 0 6 =4= n k ] h 1 := b k20 + k0 b k40 b k20 2 + k0 ( b k20 b k10 ) 6 ;h 2 := b k40 + k0 b k70 b k50 2 12 ; 44

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b 21 b 41 b k21 := h 1 + 1 2 ( ~ b k21 ~ b k 1 12 ) ; b k41 := h 2 + 1 2 ( ~ b k41 ~ b k 1 14 ) ; b k51 := b k50 + 1 2 ( ~ b k51 ~ b k 1 15 ) ;: Note that b k71 remainsunchangedfromknotinsertionguaranteeing C 2 continuityacross regular n =4 vertices.Twolayersofcontrolpointsalongeachboundarya renowxed. Thishasbeenimprovedfrom FanandPeters [ 2008 ],wherewemodied b k71 suchthat theresultingsurfaceisonly C 1 acrossregularvertices. 3.1.7Step5Interior Interiorcontrolpoints b k44 arecomputedsuchthateachofthecurves b ( u;t ) and b ( t;v ) with 0 t 1 canbeasinglecubiccurvesubdividedintothreepiecesandt he remainingcoefcientsasaleastdeviationthereof. b 44 b 20 b 40 b k44 := 20 27 b k04 + 4 3 b k14 + b k84 16 27 b k94 2 + 20 27 b k40 + 4 3 b k41 + b k48 16 27 b k49 2 : b 22 b 42 b 20 b 40 if n 0 6 =4 or n 1 6 =4 b k42 := 1 2 b k41 + b k44 1 2 b k45 ; if n 0 6 =4 b k22 := 1 2 b k12 + b k42 1 2 b k52 2 + 1 2 b k21 + b k24 1 2 b k25 2 : WehavenowspeciedallNURBScoefcients.3.1.8DegreesofFreedom Ourapproach,althoughminimalinthenumberofknots,offer ssomedegreesof freedom.Theseweresetheuristicallyasfollows.Wecancho osethepositionpoint Q k00 tobesomeotherpointthatthesurfaceshouldinterpolate,f orexampletheoriginalmesh pointasshowninFigure 3-4 D.AlsothetangentplanecanbesetinStep2bychoosing 45

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e 1 and e 2 differently.Ourchoiceofpositionandtangentplanemimic sCatmull-Clark subdivision.Thecoefcients b 11 arefreetochoose.Onealternativewaytosetthemis tomakethemdependontheboundarycurveasinPCCM[ Peters 2000a ].Weopted insteadforsimplicity.Accordingto PetersandFan [ 2009 ],thereisnoadditionalfreedom inthechoiceoftheboundaryinStep3.InStep4,thecoefcient pairs ( b k21 ; b k 1 12 ) and ( b k41 ; b k 1 14 ) eachofferadegreeoffreedom.Wesetthemsoastominimallyp erturb theirpositionsafterknotinsertionjustasmuchasisneces sarytoaccommodatethe G 1 constraints.Havingsettheboundarycurvesandtherstder ivativesacrossthe boundaries,theremainingB-splinecoefcientsareallfree tochooseifweonlyrequire (parametric) C 1 continuity.Whileitisnotalwayspossibletoenforcethatal lremaining secondderivativesmatchacrossinternalboundaries,wech osetheinteriorcoefcients tomakethetransitionsassmoothaspossible. 3.2SmoothnessVerication Thepolynomialequalities @ 2 b k ( u; 0)+ @ 1 b k 1 (0 ;u )= k ( u ) @ 1 b k ( u; 0) oftheunbiased G 1 constraints( 3–2 )holdforthe k thcurveexactlywhenallsplinecoefcientsareequal. Wewillshowthisbyexpandingtheleftside,calledLeftbelo w,andtherightside,called Right.Wewillsuperscript = signswiththeequationorassignmentofthealgorithmthat impliestheequality. Wenowshowthisequalitystartingatthevertices. 3.2.1OrdinaryQuadsandOrdinaryEdges SinceStep1convertsB-splinecontrolpoints p toBBcontrolpoints Q according to( 3–4 )(( 3–5 )agreeswiththeformulain( 3–4 )for n =4 ),ordinaryquadsjoin C 2 Also,ifoneorbothquadsarenotordinary,butbothendpoints oftheedgehave valence n =4 ,thenthesplinecoefcientsuptosecondorderaredetermin edbythe conversionfollowedbyknotinsertion.Therefore,thesurf aceis C 2 acrossT-junctions. Wesummarizethisobservationasfollows. 46

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Lemma8. Ifbothendpointsofaboundarycurvehavevalence 4 thentwopatches meetingalongthecurvejoin C 2 3.2.2ExtraordinaryQuads Nextweconsiderthegeneralcase.Toverifythe G 1 constraintsalongtheshared NURBSpatchcurves,itisconvenienttodene b k3 j := b k2 j + b k4 j 2 ; b k6 j := b k5 j + b k7 j 2 ;j =0 ;::: 9 : Thisallowsustointerpretthe b kij ascoefcientsofnine( 3 3 ) C 1 -connectedpatches inBB-form.Thecoefcientsofthepolynomials @ 2 b k ( u; 0) @ 1 b k 1 (0 ;u ) ,and @ 1 b k ( u; 0) in ( 3–2 )aredifferencesoftheseBBcontrolpoints: v k i := b ki 1 b ki 0 ; w k i := b k 1 1 i b k 1 0 i ; u ki := b ki +1 ; 0 b ki 0 : (3–10) Thedifferencesneedonlyhaveasinglesubscriptsince( 3–2 )hasonlyonevariable, u The G 1 constraint( 3–2 )isthenequivalenttothe 8 equations v k 0 + w k 0 = k0 u k0 (3–11) 3( v k + w k )=2 ki u k + ki +1 u k 1 ; ( i )=1 ; 4 ; 8 (3–12) 3( v k + w k )= ki u k +2 ki +1 u k 1 ; ( i )=2 ; 5 ; 7 (3–13) v k 9 + w k 9 = k3 u k8 (3–14) ExpansionatverticesofExtraordinaryquads .Werstfocusagainon smoothnessatthevertices.Lemma9. The G 1 constraintholdsat Q 00 47

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Proof. Weverifythe G 1 constraint( 3–11 )at u =0 byseparatelyexpandingtheleftside LeftanditsrightsideRight.Let n = n 0 3 ,thevalenceat Q 00 Left : v k 0 + w k 0 ( 3–10 ) = b k +1 10 b 00 + b k 1 10 b 00 step2 : = e 1 ( c k +1 n + c k 1 n )+ e 2 ( s k +1 n + s k 1 n ) 3 : Right : k0 u k0 ( 3–10 ) = k0 ( b k10 b 00 ) step2 : = k0 ( e 1 c kn + e 2 s kn ) ( 3–8 ) ; ( 3–9 ) = 2 c 1n c kn e 1 +2 c 1n s kn e 2 3 : EqualityofLeftandRightfollowsbystandardtrigonometric expansionforallvalences including n 0 =4 Smoothnessacrosspatchboundaries .Foreachedge,wenowanalyzethethree remainingvalencecombinations.ByLemma 8 ,if n 0 =4= n k then( 3–2 )holdswith k 0 Lemma10. If n 0 6 =4 6 = n k then ( 3–2 ) holds. Proof. ByLemma 9 andsymmetryoftheconstraintstructureatthetwoendpoint s,we needonlyverify( 3–12 )for i =0 and i =1 and( 3–13 )for i =0 For i =0 ( i )=1 ,( 3–12 )hasalefthandside,Left,andarighthandside,Right, thatagree: Left :3( v k 1 + w k 1 ) ( 3–10 ) =3( b k11 b k10 + b k 1 11 b k 1 01 ) : Right :2 k0 u k1 + k1 u k0 ( 3–10 ) =2 k0 ( b k20 b k10 )+ k1 ( b k10 b 00 ) 3b : =2 k0 ( b k10 + 3( b k11 + b k 1 11 2 b k10 ) k1 ( b k10 b 00 ) 2 k0 b k10 )+ k1 ( b k10 b 00 ) =3( b k11 b k10 + b k 1 11 b k 1 01 ) : 48

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For i =1 ; ( i )=4 ,( 3–12 )holdssince Left :3( v k 4 + w k 4 ) ( 3–10 ) =3( b k41 b k40 + b k 1 14 b k 1 04 ) 4b : =3( h 2 + 1 2 ( ~ b k41 ~ b k 1 14 ) b k40 + h 2 + 1 2 ( ~ b k14 ~ b k 1 41 ) b k 1 04 ) =6( h 2 b k40 ) 4b : =6( b k40 + 2 1 ( b k50 b k40 )+ 2 b k40 b k20 2 6 b k40 ) =2 1 ( b k50 b k40 )+ 2 b k40 b k20 2 : Right :2 k1 u k4 + k2 u k3 ( 3–10 ) =2 k1 ( b k50 b k40 )+ k2 ( b k40 b k30 ) : Thetwoexpressionsareequalbecause b k30 = b k40 + b k20 2 Weverify( 3–13 )for i =0 ; ( i )=2 Left :3( v k 2 + w k 2 ) ( 3–10 ) =3( b k21 b k20 + b k 1 12 b k 1 02 ) 4b : =3( h 1 + 1 2 ( ~ b k21 ~ b k 1 12 ) b k20 + h 1 + 1 2 ( ~ b k 1 12 ~ b k21 ) b k 1 02 ) =6( h 1 b k20 ) 4b : =6( b k20 + 0 b k40 b k20 2 +2 1 ( b k20 b k10 ) 6 b k20 ) = 0 b k40 b k20 2 +2 1 ( b k20 b k10 )= 0 ( b k30 b k20 )+2 1 ( b k20 b k10 ) : Right : k0 u k2 +2 k1 u k1 ( 3–10 ) = 0 ( b k30 b k20 )+2 1 ( b k20 b k10 ) : Thiscompletestheproof. Lemma11. If n 0 6 =4= n k then ( 3–2 ) holds. Proof. b k30 3 b k40 +3 b k50 b k60 = b k20 + b k40 2 3 b k40 +3 b k50 b k50 + b k70 2 = b k20 5( 41 25 b k20 + 4 25 b k70 4 5 b k10 )+5( 36 25 b k20 + 9 25 b k70 4 5 b k10 ) b k70 2 =0 : (3–15) Fortheleftmostsegment, i =0 ,( 3–11 )holdsbyLemma 9 ;and( 3–12 )and( 3–13 )have alreadybeenveriedinLemma 10 .Fortherightmostsegment, i =2 ,Lemma 8 applies. 49

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Weneedonlyverify( 3–2 )forthemiddlesegment, i =1 .Bytheformulasin 3c : ,the thirdderivativeofthemiddleboundarycurvesegmentis( 3–15 ).Thatis,thissegment isofdegree 2 andcanequivalentlybedenedbycoefcients b [2]j 0 ofthepolynomialin quadraticBBform: b k [2] 30 := b k30 ;b k [2] 40 := 3 b k40 b k30 2 = 3 b k50 b k60 2 ;b k [2] 50 := b k60 : (3–16) By( 3–9 ), k 1 ( u )= k1 (1 u ) 2 isaquadraticfunctionand k2 = k3 =0 ( k0 and k1 aregiven near( 3–9 )).For i =1 ,( 3–12 )and( 3–13 )specializeto 9( v k + w k )=2 ki u k [2] ; ( i )=4 (3–17) 3( v k + w k )=0 ; ( i )=5 ; (3–18) where u k [2] = b k [2] +1 ; 0 b k [2] 0 Weverify( 3–17 ), Left :9( v k 4 + w k 4 ) 4c : =18( h 2 b k40 )=18( k0 b k70 b k50 2 12 )= 3 k0 ( b k70 b k50 ) 4 ; Right :2 k1 u k [2] 4 = k0 ( b k [2] 50 b k [2] 40 ) ( 3–16 ) = 3 k0 ( b k60 b k50 ) 2 = 3 k0 ( b k70 b k50 ) 4 and( 3–18 ), Left :3( v k 5 + w k 5 ) 4c : =3( b k50 + 1 2 ( ~ b k51 ~ b k 1 15 )+ b k50 + 1 2 ( ~ b k 1 15 ~ b k51 ) 2 b k50 ) =0 : Thiscompletestheproof. 3.2.3OverallSmoothness SincetheNURBSpatchesaredegreebi-3andhaveatmostdoublei nternalknots, theyareinternallyparametrically C 1 .TogetherwithLemma 10 andLemma 11 this impliessmoothness.Inparticular,byLemma 8 ,ordinarypatchesjoinextraordinary patches C 2 50

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Theorem2. Asurfacegeneratedbythealgorithmis C 1 and C 2 acrossregularedges wherebothendpointshavevalencesof4. 3.3Examples Totesttheshapeofthesurfaceconstruction,weusesmallex ampleswith predictableshape.Inparticular,Figure 3-4 (J–L)showsthattheconstructioncan handlehigher-ordersaddlepointswithouttheshapedecie nciesofclassicalPCCM. Figure 3-4 B,G,KshowtheGausscurvaturedistributionandFigure 3-4 C,I,Lthe highlightlines.Figure 3-4 Nillustratesthedistributionofpolynomialpiecesforthe inputmeshFigure 3-4 M.TheT-meshpresentsnoproblemsincethesurfacesare C 1 andthesplinepatchesareevaluatedevenly.Weautomatical lyevaluatenerinthe patchescorrespondingtoordinaryquads.Figure 3-4 G,Hcomparesthecurvatureofa lozenge-shapegeneratedbythealgorithmwiththatof FanandPeters [ 2008 ].Themain differencesareinthetransitionsacrossregularedges. 3.4DiscussionandExtensions Forsomecombinationsofvertexvalences,thenumberofknot scouldbelowered stilltojustoneinternalknot.Forexample,when n 0 = n 1 onopposingedgesthen oneinternalknotsufcestoconstructasmoothlyconnected patch.However,this wouldcreatemanyspecialrulesandsuchaminimizationofkn otsislikelynotthekey argumentforusingthepresentalgorithmasthenextparagra phexplains. Inapplicationssuchassurfacereconstruction,manymorek notsandtherefore polynomialpiecesperquadmaybeneededtotgivendata.Byin sertingknotsin thebi-3NURBSpatchofthispaper,wecanrenetherepresenta tionofthepatch.All interiorcoefcientsarethenfreelyadjustabletottheda ta.Forexample,byaugmenting 2 to (0 ; 0 ; 0 ; 0 ; 1 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 8 ; 9 ; 9 ; 9 ; 9) ,allcoefcientswithGrevilleabscissae i;j 1
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freedomalongtheboundarycurves.Thisisageneralandnota specicrestrictionof thepresentconstruction. Themainimprovementofthepresentedalgorithmover Peters [ 2000a ]isthennot thelowernumberofpolynomialpieces,buttheimprovedshap eatthecornersofthe patch.Thekeytothisisthatthenewalgorithmusesadiffere ntreparameterizationat thecorners:both k 0 and k 2 in( 3–8 )or( 3–9 )arelinear,allowingfordegree3boundary curvesemanatingfromthevertex.Bycomparison Peters [ 2000a ]usesquadratic k 0 and k 2 toquicklyswitchfrom G 1 constraintsto C 2 continuityacrosstheboundaries.At higherordersaddlepoints,thiscanleadtoatspots,where asthenewalgorithmdoes notsufferfromthispotentialshapedeciency. 52

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A B C D E F G H FanandPeters [ 2008 ] I J K L M N O P Figure3-4. Smallexampleswithpredictableshape. A),E),J)Quadnetandsurface. B),G),K)Gaussiancurvatureshading,H)Constructedaccordi ngto FanandPeters [ 2008 ],C),I),L)Highlightlines.D)Interpolationofthequad net(seeSection 3.1.3 ).F),N),P) Q and b patchesforregularandirregular quads.M)ObtainedfromJ)byonestepofCatmull-Clarksubdi vision,O)by twosteps. 53

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CHAPTER4 RATIONALLINEARREPARAMETERIZATIONFORSMOOTHSURFACES 4.1Introduction TheEulerformula 1 (Euler-Poincar echaracteristic)allowsforclosedpolyhedron oftopologicalgenus g> 0 consistingofanarrangementofquadrilateralsthatisa rectangulargridexceptfor =2 g 2 isolatedverticesofvalence8. 2 Figure 4-1 shows anexampleofgenus g =2 ,withtwoisolatedverticesofvalence8(onthefrontandbac k ofthewaist-lineofagure-8doubletorus)andFigure 4-6 showsexamplesofhigher genus. UsingRicciow,Guandco-workershavegeneratedre-approx imationsof triangulationsbyquadrilateralmesheswiththecorrectnu mberof8-valentvertices (see Guetal. [ 2008 ]foranillustrationofthepipeline 3 ).Moreover,thesequad-meshes areangle-preservingimagesofsubsetsoftheEuclideanplan ewhoseboundarieshave beensuitablyidentied.Wemaythereforeviewthemasconfo rmalparametrizationof triangulateddatain R 3 Tomodelsmooth, s timesdifferentiable,conformallyparametrizedsurfaces ,we canassociateatensor-productsplinewitheachquadrilate ral.However,the8-valent pointsinterrupttherectangulargridconnectivityrequir edbytensor-productsplinesand represent'topologicalobstructions'wheresmoothsurfac esare'difculttoconstruct, unstable,anderror-prone'accordingto Guetal. [ 2008 ].ByEuler'scountwecannot 1 v e + f = where v;e;f arerespectivelythenumbersofvertices,edgesandfaces inagivenpolyhedronand =2 2 g istheEulercharacteristic 2 Toseethis,observethatwecanexclusivelyassociatewithe ach4-valentvertex fourhalfedgesandfourquartersofattachedquadssothatit snetcontributionto theEulercountis 1 4 = 2+4 = 4=0 .Foreach8-valentvertexthecontributionis 1 8 = 2+8 = 4= 1 3 Guetal. [ 2008 ]alsoproposesdecreasingthenumberofsingularities.Thi sisnot importantinthepresentpaper. 54

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A B Figure4-1.Figure-8doubletorus.A)Fourregulararrangeme ntsofquadrilaterals(one percolor)with 2 g 2=2 cornerpointsofvalence8.B)Corresponding smoothsplinesurface. hopetoeliminatethe8-valentvertices.And,byaresultofMi lnor[ Milnor 1958 ],we cannothopetousejustlinearchangesoftheparameterstotr ansitionbetweenthe tensor-productsplinepiecesofaregularlyparametrizeds moothclosedsurfacesof genus g> 1 in R 3 ,i.e.haveanafneatlas.(Forgenus g =1 ,theconstructionofan afneatlasistrivial,sincethetorusisjustatensor-prod uctgridwithopposingedges identied.) Thenextsimplestalternativetoanafneatlas,compatible withconformal parameterizations,isafractionallinearatlas.Indiffer entialgeometry,linearfractional transformationsareusuallyassociatedwiththeclassofMo ebiustransformations: ( z ):= az + b cz + d : C C where a;b;c;d 2 C and ad bc =1 .Moebiustransformations arebijectiveconformalorientation-preservingmaps.The ysufcetobuildacomplex fractionallinearatlaswhenprovingtheUniformizationTh eorem[ Abikoff 1981 ](which yieldsaclassicationofsurfacesaccordingtothesignofc onstantcurvature). 55

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Bycontrast,thefocusofthepresentpaperisonrealrational linearmaps ofthe type ( u;v ):= 264 [1] [2] 375 := 264 a 1 + b 1 u + c 1 v d 1 + e 1 u + f 1 v a 2 + b 2 u + c 2 v d 2 + e 2 u + f 2 v 375 ; (4–1) where a i ;b i ;:::;f i arerealscalars.Thereasonforthischoicebecomesapparen twhen using ,respectively asreparameterization r : R 2 R 2 sothattwosurfacepieces p and q haveidenticalderivativesuptoorder s alonganedgeparametrizedby t : @ i p ( t; 0)= @ i ( q r )( t; 0) ;i =0 ;:::;s: (4–2) Naively(withoutoverlap)using ,thebasicrequirementforadjustingtheorientation ofcounterclockwisearrangedpatches, ( t; 0)=(0 ;t ) ,xes ( u;v )= v + u p 1 andhenceaccommodatesonly4patchesjoining,not8.Thisis becausethegroupof Moebiustransformations,theprojective,speciallinearg roupoverthecomplexnumbers, PSL (2 ; C ) ,endows withonly6realdegreesoffreedom.Bycontrast, ischosen fromthe8-dimensionalgroup SL (3 ; R ) which,aswewillshow,canaccommodate constructionswith8-valentvertices.Nevertheless, isinsomesensethesimplestmap: expressedasarealrationalmap, isofdegree2over2,while isofdegree1over1. InSection 4.2 ,wecollecttheconstraintsthat( 4–2 )imposeson .InSection 4.3 ,theseconstraintsarethenappliedtosetthefreeparamete rsof .Somewhat surprisingly,wendthatthereisanessentiallyuniquemap thatcan(potentially) serveasreparameterization,givenaquadrilateralcontro lnetwithisolatedvertices ofvalence8andvalence4everywhereelse.InSection 4.4 ,weshowthat indeed enablesconnectingtensor-productsplines,oneperquad:w eoutlinealgorithmsforthe constructionof C 1 and C 2 surfacessatisfyingthe G 1 ,respectively G 2 constraints( 4–2 ). Inkeepingwithourfocus,wefollowtheconstructiveapproa chtosurfacesrather thantheanalyticapproachviachartswhichassumestheexis tenceofasurface:our splinepatchesandreparameterizationsaredenedonnite ,closedsets,namelythe 56

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unitsquare,denoted 2 R 2 .Andwejoinabuttingpatchesratherthanoverlapping charts.Onecan,ofcourse,obtainanatlasfromthepatch-ba sedconstruction.Fora constructionviacharts,seee.g. YingandZorin [ 2004 ],forapolynomialconstruction withgeneralconnectivity,seee.g.. LoopandSchaefer [ 2008b ].Theintroductionof Kar ciauskasandPeters [ 2009 ]givesanoverviewofrecent C 2 constructionswith generalconnectivity.Alltheseapproachesusenon-linearr eparameterizations. 4.2ConstraintsontheTransitionMapfor G 2 Continuityatan8-valentVertex Wearelookingtodeterminearationallinearreparameteriz ation : 2 (R 2 R 2 (4–3) where 2 istheunitsquare.Thereparameterizationistoallowtwote nsor-product patches p : 2 (R 2 R d and q : 2 (R 2 R d tojoinsmoothlyacrossthecommon boundary q ( t; 0)= p (0 ;t ) : (4–4) Equation( 4–4 )implies ( t; 0)=(0 ;t ) (4–5) andtherefore @ k 1 [1] ( t; 0)=0 ;@ 1 1 [2] ( t; 0)=1 and @ k 1 [2] ( t; 0)=0 for k> 1 : (4–6) 4.2.1 G 1 Constraints Thereparameterizationtobedeterminedisusedinthe G 1 constraints ( @ 2 q )( t; 0)=( @ 2 ( p ))( t; 0) =( @ 1 p ) @ 2 [1] ( t; 0)+( @ 2 p ) @ 2 [2] ( t; 0) : (4–7) 57

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Wewantthe G 1 constraintstobeunbiased,i.e.structurallysymmetricas weexchange p with q .Therefore ( @ 2 [1] )( t; 0)= 1 : (4–8) Also,sinceexactly8tensor-productpatchesalwaysjoinate achcommonvertexand 2cos 2 8 = p 2 [2] mustsatisfy @ 2 [2] (0 ; 0)= p 2 : (4–9) Thisyieldstheunbiased G 1 constraints ( @ 2 q )( t; 0)+( @ 1 p )( t; 0)=( @ 2 p ) @ 2 [2] ( t; 0) ; ( 4–7 ') and,at ( u;v )=(0 ; 0) ;@ 2 q + @ 1 p = p 2 @ 2 p : (4–10) Ifwedifferentiate( 4–7 ')alongtheboundary,i.e.differentiatewithrespectto t ,thenby ( 4–6 ) ( @ 1 @ 2 q )( t; 0)+( @ 2 @ 1 p )(0 ;t ) =( @ 2 2 p )(0 ;t )( @ 2 [2] )( t; 0)+( @ 2 p )(0 ;t )( @ 1 @ 2 [2] )( t; 0) (4–11) and,at ( u;v )=(0 ; 0) ;@ 1 @ 2 q + @ 2 @ 1 p = p 2 @ 2 2 p + @ 2 p @ 1 @ 2 [2] : (4–12) 4.2.2 G 2 Constraints Thereparameterizationtobedetermineddenesthe G 2 constraints ( @ 2 2 q )( t; 0)=( @ 2 2 ( p ))( t; 0) (4–13) =( @ 2 1 p )(0 ;t ) 2( @ 1 @ 2 p )(0 ;t ) @ 2 [2] ( t; 0)+( @ 2 2 p )(0 ;t )( @ 2 [2] ) 2 ( t; 0) +( @ 1 p )(0 ;t )( @ 2 2 [1] )( t; 0)+( @ 2 p )(0 ;t )( @ 2 2 [2] )( t; 0) : 58

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Inparticular,at ( u;v )=(0 ; 0) : @ 2 2 q = @ 2 1 p 2 @ 1 @ 2 p p 2+2 @ 2 2 p + @ 1 p @ 2 2 [1] + @ 2 p @ 2 2 [2] : (4–14) Substituting( 4–12 )and( 4–10 ),wesymmetrizethe G 2 constraintat ( u;v )=(0 ; 0) @ 2 2 q @ 2 @ 1 q p 2 ( 4–12 ) = @ 2 1 p @ 1 @ 2 p p 2+ @ 1 p @ 2 2 [1] + @ 2 p ( p 2 @ 1 @ 2 [2] + @ 2 2 [2] ) ; @ 2 2 q @ 2 @ 1 q p 2+ 1 2 @ 2 q @ 2 2 [1] ( 4–10 ) = @ 2 1 p @ 1 @ 2 p p 2+ 1 2 @ 1 p @ 2 2 [1] + @ 2 p ( p 2 @ 1 @ 2 [2] + @ 2 2 [2] + p 2 2 @ 2 2 [1] ) : (4–15) Consideringthedirectionalignedwith t := @ 2 p (0 ; 0)= @ 1 q (0 ; 0) ,foranunbiased constructionat ( u;v )=(0 ; 0) weseethat t ( @ 2 2 q @ 2 @ 1 q p 2+ 1 2 @ 2 q @ 2 2 [1] )= t ( @ 2 1 p @ 1 @ 2 p p 2+ 1 2 @ 1 p @ 2 2 [1] ) : Sinceweruleoutsingularconstructions, t t 6 =0 andtherefore, at u = v =0 p 2 @ 1 @ 2 [2] @ 2 2 [2] = p 2 2 @ 2 2 [1] (4–16) musthold. 4.3TheProjectiveLinearTransitionMap Wenotethat,althoughweaimatconstructingpiecewisetens or-productspline surfaces,thepreviousSectiondidnotmakeanyassumptionon thesurfacepieces p and q orthemap otherthanthattheybesufcientlysmooth. Remarkably,inthisgeneralsetting,theprojectivelinear reparameterizationis essentiallyunique.Theorem3. Thetransitionmap : R 2 R 2 of ( 4–1 ) forthe G 2 constructionofa C 2 surfaceisuniqueuptothevalueof := @ 2 [2] (1 ; 0) : ( u;v ):= 1 1+ v ( p 2 ) 264 v u + p 2 v 375 : ( 4–1 ”') 59

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Proof. By( 4–5 ), ( t; 0)=(0 ;t ) ,wehave [1] : a 1 + b 1 t d 1 + e 1 t =0= ) a 1 = b 1 =0 (4–17) [2] : a 2 + b 2 t d 2 + e 2 t = t = ) a 2 = e 2 =0 ;b 2 = d 2 : (4–18) Therefore simpliesto ( u;v ):= 264 c 1 v d 1 + e 1 u + f 1 v d 2 u + c 2 v d 2 + f 2 v 375 : ( 4–1 ') Next( 4–8 )implies 1= @ 2 [1] ( t; 0)= c 1 d 1 + e 1 t = ) e 1 =0 ;d 1 = c 1 : (4–19) Ontheotherhand, @ 2 [2] ( t; 0)= c 2 d 2 t f 2 d 2 and( 4–9 )implies @ 2 [2] (0 ; 0)= c 2 d 2 = p 2 : (4–20) Therefore ( u;v ):= 264 d 1 v d 1 + f 1 v u + p 2 v 1+ vf 2 =d 2 375 : ( 4–1 ”) Ifwedene := @ 2 [2] (1 ; 0)= p 2 f 2 d 2 then [2] ( u;v )= u + p 2 v 1+ v ( p 2 ) andwecanabbreviate ( t ):= @ 2 [2] ( t; 0)= p 2(1 t )+ t; (4–21) @ 2 2 [2] ( t; 0)= 2( p 2 ) ( t ) : (4–22) Next,weobservethat @ 2 2 [1] ( t; 0)= 2 f 1 c 1 = @ 2 2 [1] (0 ; 0) .Therefore( 4–16 )implies @ 2 2 [1] = 2 f 1 c 1 = 2 p 2 p 2 @ 1 @ 2 [2] @ 2 2 [2] =2( p 2)+4( p 2 ) (4–23) =2( p 2 ) : Alltogether,thisyields(1”'). 60

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Figure4-2.Compositionof withitself:The brown square 2 intheupperrightquadrant ismapped,inorder,rstto ( 2 ) ( blue ,upperleftquadrant),thenin counterclockwiseorderto 2 ( 2 ) ( mediumblue ),etc.toarrivebackat 8 ( 2 )= 2 Indeed,onechecksbysymbolicsubstitution,thatourratio nallinearmapsatises thecompositionconstraint[ Hahn 1989b ]uptoanyorder.Thisallowsittoserveas reparameterizationforconstructing C s manifoldsforany s Lemma12. The8-foldcomposition ::: = id. SeeFigure 4-2 foranillustration. 4.4ConstructiveUseoftheTransitionMap Topracticallyverifythatourchoiceof isnotonlynecessarybutalsosufcientfor thegenerationofsmoothsurfaces,wesketchtwosuchconstr uctions,for s =1 ; 2 Thelayoutofthequadrilateralsaregroupedintorectangul argridswhosecorner verticesare8-valentandwhoseinteriorpartitionissotha talwaysfourquadrilaterals meetataninteriorvertex.Forexample,Figure 4-1 A,showsfoursuchrectangulargrids formingagure-8.(Duetothelowgenus,eachuseseach8-val entvertextwice.)Each checkerboardwillcorrespondtoatensor-productsplinepa tchwithuniformknotsinthe interior.Alongtheboundary,thesplinepatcheshavetheful lmultiplicityofknotssothat 61

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therelevantcoefcientsareinBernstein-B ezierform(BB-form;see( 4–24 )below).We willchoosethepolynomialdegreeofthesplinestobe 2 s +1 Inthislayout,weconnectthe8-valentverticesbyachainof m 1 verticesof valence4correspondingtotheknotsofthesplinepieces.Th atis,eachsplinepiece willhavefour8-valentverticesascornersandbesubdivide dintoarectangulargrid accordingtotheuniformchoiceofknotlines(seeFigure 4-5 A).Wealwaysassume m> 1 sincethisisanecessaryandsufcientconstraintalreadyf orthe G 1 construction ofdegreebi-3[ PetersandFan 2009 ].Correspondingly,eachcurveemanatingfrom an8-valentpointhasafunction associatedwithit,wepartitionthegraphofthe uniformlyinto m pieces.Thatis,weset = k := p 2 m 2 m b k;;00 b k;;10 b k;;20 b k;;30 b k;;40 b k;;02 b k;;03 b k;;01 b k;;22 b k;;13 b k;;31 b k;;04 ::: :::::: ::: Figure4-3.Cornerofthepolynomialpiecewithindex ; intherectangulargridofthe k thspline, k =1 ;:::; 8 .Wefocuson ( ; )=(0 ; 0) ,thecornerBB-patchesof thespline,wherethe8patchesmeetandpropagatethe G s constraints alongboundaries ; = ; 0 .ThesubscriptsidentifytheBB-coefcients:for example, b k; 00 00 correspondstothepointsharedbythesplineswhichtogethe r with b k; 00 10 and b k; 00 01 denesthetangentplane. Wedenotethepolynomialpiecewithindex ; intherectangulargridofthe k th splineby b k;; .InBernsteinB ezierform(BB)patch, k =1 :::n 0 b k;; ( u;v ):= d X i =0 d X j =0 b k;;ij d i u i (1 u ) d i d j v i (1 v ) d j : (4–24) Here d isthedegree.Wewillchoose d =2 s +1 .Asusualinmulti-variateconstructions, theavailabledegreesoffreedomdonotexactlymatchthenum berofconstraints.Our 62

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strategyforsettingthedefaultlocationoffreevariables byinterpretingtheverticesof theinputquadmeshassplinecoefcientswithsingleknotsa ndtocomputealeast perturbationofthecorrespondingBB-coefcientsalongthes plineboundariestoensure a G s transitionbetweenthesplinepieces. Bothforthe G 1 andthe G 2 constructions,weaimonlyatoutliningthemainpoints oftheconstruction.Theformulasforthe G 1 constructionsaresufcientlycomplete forimplementationbyanon-specialist.The G 2 constructionisonlyexplainedatthe specialistlevelsoasnottodistractfromthemainpointoft hepaper. 4.4.1 G 1 Construction Wechoose d =3 .Theunbiased G 1 constraintsis ( @ 2 q )( t; 0)+( @ 1 p )(0 ;t )= ( t )( @ 2 p )(0 ;t ) : (4–25) Abbreviating k := p 2 m 2 m k; j := v k; j + w k; j andsetting u k;j := b k; 0 j +10 b k; 0 j 0 ;v k; j := b k; 0 j 1 b k; 0 j 0 ;w k; j := b k 1 ; 0 1 j b k; 0 j 0 ; relation( 4–25 ),splitamong m segmentsalongtheboundarycurve,yieldsfor = 0 ;:::;m 1 : k; 0 = k u k;0 (4–26) 3 k; 1 =2 k u k;1 + k +1 u k;0 (4–27) 3 k; 2 = k u k;2 +2 k +1 u k;1 (4–28) k; 3 = k +1 u k;2 : (4–29) Nominally,theseare 4 m equationsbut,byenforcingcontinuitybetweenthepieces, ( 4–29 ) = i =( 4–26 ) = i +1 ,i.e.constraint( 4–29 )whensubstituting = i isidenticalto constraint( 4–26 )for = i +1 sothatthereareonly 3 m constraintstocheck. 8-valentvertexneighborhood 63

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p 0 p 1 p 2 p 3 p 4 p 15 p 16 x 00 x k10 x k11 AB Figure4-4.A)Extraordinaryvertex p 0 with 8 directneighbors p 2 k 1 k =1 ::: 8 .B)Limit point x 00 ,tangentpoints x k10 and`twist'coefcients x k11 WeinitializetheBB-coefcients x ij := b k; 00 ij for i + j< 3 byapproximatingthelimitof theCatmull-Clarksurface.Supposeavertex p 0 ofvalence 8 (Figure 4-4 )issurrounded by8patchesinBB-form(toformthecornerpiecesofaspline).W eset x 00 tothelimit of p 0 underCatmull-Clarksubdivision(redcircleinFigure 4-4 B)andplacethe x k10 (blue circleinFigure 4-4 B)ontheCatmull-Clarktangentplanefor k =1 ::: 8 x 00 = x k00 := 1 104 8 X j =1 8 p 0 +4 p 2 l 1 + p 2 l ; (4–30) x k10 := x k00 + e 1 c k + e 2 s k ; (4–31) e i := 4 3 ( 2) 8 X j =1 i p 2 j 1 + i p 2 j ; (4–32) x k11 := 1 9 (4 p 0 +2( p 2 k +1 + p 2 k +3 )+ p 2 k +2 ) ; (4–33) wherethescalarweightsaredenedas c k :=cos k 4 ;s k :=sin k 4 ; := r +5+ p ( r +9)( r +1) ; r := 1 p 2 ;! := 4 ; 1 := !c j 1 ; 1 := c j 1 + c j ; 2 := !s j 1 ; 2 := s j 1 + s j : 64

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A B Figure4-5. G 1 construction.A)TheBB-control-netwiththenumberofsegment sis chosenas m =2 ; 8 ; 6 ; 8 ; 2 ; 8 ; 6 ; 8 .B) C 0 -connectedpatches(notethe discontinuitiesinthehighlightlines)inBB-formafterinit ialization.(Thered surfacestemsfromtheunperturbedquadmeshinterpretedas controlnetof atensor-productsplineofdegreebi-3.) Figure 4-5 showstheBB-patchesafterinitialization. Propagationofthe G 1 correction alongsplineboundaries.Let b k; 00 20 and b k; 00 30 bethe BB-coefcientsderivedfrominterpretingtheverticesofthe quad-meshasbi-3spline coefcients.Weadjust b k; 00 20 and b k; 00 30 toenforce( 4–27 )fortherstBB-pieceofthespline (andsymmetricallyattheother8-valentvertex),setting b k; 00 20 := b k; 00 10 + u k; 0 1 ;u k; 0 1 := 3( v k; 0 1 + w k; 0 1 ) k1 u k; 0 0 2 k0 ; (4–34) b k; 00 21 := b k; 00 21 + 0 ; 0 := b k; 00 20 b k; 00 20 (4–35) b k; 00 30 := b k; 00 20 + b k; 10 10 2 ; 1 := b k; 00 30 b k; 00 30 ; (4–36) b k; 00 31 := ~ b k; 00 31 + 1 (4–37) 65

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andpropagatetheadjustmentbymodifying v k; j v k; j + k; j = 2 ;w k; j w k; j + k; j = 2 ; (4–38) for j =1 ; 2 ; 3 and =1 ;:::;m 1 tosatisfy( 4–26 ) ;k ,( 4–27 ) ;k ,( 4–28 ) ;k 4.4.2 G 2 Construction Thesplinesofour G 2 constructionareofdegreebi-5.Asinthe G 1 case,we inheritthebasicshapefromthequadmeshbyinterpretingth emeshverticesasspline controlpoints.Wecanthereforefocusontherstthreeboun darylayersinBB-form,i.e. coefcientswhereatleastoneindexislessorequalto2. Weneedtoenforce ( @ 2 2 q )( t; 0)=( @ 2 1 p )(0 ;t ) 2 ( t )( @ 1 @ 2 p )(0 ;t )+ 2 ( t )( @ 2 2 p )(0 ;t ) ( 4–13 ') 2( p 2 ) ( t )( @ 2 p )(0 ;t )+2( p 2 )( @ 1 p )(0 ;t ) : 8-valentvertexneighborhood Asinthe G 1 case,weneedtorstdeterminethetotaldegree 2 s -jetatthevertices, i.e.theBB-control-pointswithindexsummingtoatmost4.The searethepoints emphasizedinFigure 4-3 .Thepointsaretreatedinthreegroupsindicatedbycolors. All greenpointswithindexsummingtoatmost2,ofalleightpatc heshavenomorethan 6degreesoffreedomtotal,sincethatdenesthelocal C 2 expansion.Asareasonable heuristic,weborrowtheexpansionsofthe G 1 construction,degree-raisedto5.Foreach patch,wetracea C 2 expansionaroundthe8-valentvertexandthenaveragethe C 2 expansions.Thebluelayerissolvedforthe16coefcientsw ithsubscript 12 and 21 (for xedboundarycoefcientswithindex 30 whichagreewiththosewithindex 03 ofthe previouspatch).Forthis,wesubstitutethe G 1 constraintsinvolving b k 1 ; 00 12 and b k; 00 21 into the G 2 constraintsinthesamecoefcientsandeliminatethecoef cientswithsubscript 21 .Thisyieldsa 8 8 systemofconstraintsintermsofthe8-vector x 12 whose k thentry 66

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A B C D E F Figure4-6.A),B),C)Genus3surfacewithfourverticesofvale nce8.D),E),F)Genus5 surfacewitheightverticesofvalence8. is x k12 := b k; 00 12 Mx 12 = r ; M kj := 8>>>>>><>>>>>>: p 2 2 ; j k j j =1 2 ;k = j 0 ;else: ; (4–39) r k :=(2 p 2 ) x k11 +2 x k30 +(2 p 2 4) x k20 2 x k 1 11 + p 2 x k 1 30 Thematrix M isinvertibleandwegetauniquesolutionforthecoefcient swith subscripts 12 and,bybacksubstitutionintothe G 1 constraint,forthosewithsubscript 21 Finally,withthecoefcientswithsubscript 22 takenfromthesplinesurface,wecansolve the G 1 constraintsinvolving b k 1 ; 00 13 and b k; 00 31 uniquelyinthosevariables. 67

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Propagationofthe G 2 correction alongsplineboundaries.Asinthe G 1 case,we propagateaperturbationoftheinitializationalongthebo undarytoobtaina G 2 transition betweenthesplinepatches,leavingthesecondlayerofcoef cients b k; 0 j 2 unchanged. 4.5Conclusion Theorem 3 showedtheuniquenessofthelowest-degree,namelyrationa l linear(projectivelylinear)bi-variatereparameterizat ionforconstructingatleast C 2 -smoothsurfaceswithunbiasedgeometriccontinuityconst raintsoverquad-meshes whoseverticesareofvalence8.Moreover,Lemma 12 showedthattherational linearreparameterizationiscompatiblewith C s constructionsforany s .Itisthe simplestsuchmap.Conversely,twoconcreteconstructions inSection 4.4 illustrate thisreparameterizationcangeneratesmoothmanifolds.We expectthatsurface parameterizationswithsimplesttransitionfunctionswil lnduseascomputational domains. 68

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CHAPTER5 CURVATURECONTINUOUSSURFACES ThischapterdescribesmyC++implementationofGuidedSurfa ces,inparticular guidedcappingandguideinitializationandcomparesittoi tsthreeclosestcompetitors. Whilethealgorithmgoesbackto KarciauskasandPeters [ 2009 ],theexistingMaple implementationwasbroughttoC++tobeabletocompareittot hreeotheralgorithmsfor constructing C 2 surfaces. 5.1GuidedSurfaces GuidedSurfacesallowshapecontrolovertheholeregionbypr escribingan intermediatelocal”guidesurface”.Theresultingsurface notonlyhassmoothness andlowdegreebutalsoapproximatesthegivenshapeofthegu ide.Inthefollowing Sections,werstreviewtheconceptofGuidedSurfaces.Thena possibleinitial renement,guidedcappingandnallyanalgorithmofconstr uctingguidefrom tensor-borderdataareexplained.5.1.1InitialRenementbyGuidedSubdivision Theinitialrenementusesone(ormoreifdesired)ringsofg uidedsubdivision ( KarciauskasandPeters [ 2009 ]).Suchasurfaceringisconstructedasthecomposition ofaguide g withaparametrization whichhasthestructureofanannulus.Wethen applyaHermitesampleoperator H toreconstructthesurface x whichhastheshapeof theguideandisrepresentedaslowdegreeoftensorproductp atches. x m := H ( g m ) : g isapiecewisepolynomial R 2 R 3 TheconcentricParametrization (seeFigure 5-1 )isdenedasthecharacteristic mapofCatmull-Clarksubdivision.Itmaps n copiesofL-shapedpatchestoaring in R 2 .Theringand scaledcopyofit ( S= 2):= ( S ) joinsmoothly( isthe sub-dominanteigenvalueoftheCatmull-Clarksubdivision ).Theunionofthese scaledringsparametrizetheneighborhoodaroundthedomai noriginoftheguide. So connectsthedomainofthenalsurfacewiththedomainofthe guide. 69

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u v 1 1 n A u v 1 1 n B Figure5-1.(From Kar ciauskasandPeters [ 2007 ])A)showsct-maps .B)showsif domainisscaledby 1 2 ,therangeisscaledby .Botharefor 2 -sprocket (Catmull-Clarksubdivision)layout. ABC Figure5-2.(From Kar ciauskasandPeters [ 2007 ])A)Combiningfour (2 ; 2) -jetsinB ezier formintoasegment x mj ofapolynomialpatchofdegree (5 ; 5) .B)Averaging (3 ; 3) -jetstodeneapatchofdegree (6 ; 6) .C)Averaging (3 ; 2) -jetstodene apatchofdegree (6 ; 5) HermiteSampling H GuidedSurfacesapproximatetheshapeofguidebysamplingth epositions andderivativesatthelocationsdenedbytheparametrizat ion .Thesampling operator h (notyet H )fortensor-productpatchesdeterminesatensor-product patch h ( f ) inB ezierform,matchingthederivativesofagivenmap f denedon 2 atthefourcorners,atleastuptosecondorder.Theoperator h 66 generatesapatchofdegree (6 ; 6) .Foreachcorner,itsamples thepartialderivatives,the (3 ; 3) -jetof f f@ s f@ 2 s f@ 3 s f @ t f@ s @ t f@ 2 s @ t f@ 3 s @ t f @ 2 t f@ s @ 2 t f@ 2 s @ 2 t f@ 3 s @ 2 t f @ 3 t f@ s @ 3 t f@ 2 s @ 3 t f@ 3 s @ 3 t f representsthembyits 3 3 expansioninB ezierform.Togetherthesefour groupsofsixteencoefcientsdenethe 49 coefcientsofpatchofdegree (6 ; 6) (Figure 5-2 B). 70

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6 6 6 6 6 6 Subdivide Degree Raise Extend Figure5-3.Illustrationoftheapproachtoconstructtheou terlayersfromboundaryor previouslayer.Therststepdegreeraiseisonlynecessary forextending fromboundarypatcheswithdegreelowerthanbi6. A B C Figure5-4.(From Kar ciauskasandPeters [ 2007 ])ConstructionofCatmull-Clarkguided ring:A) h 6 ; 6 denesfourlayersofcontrolpointsalongtheinwardcorner edgesofpatchesofdegree (6 ; 6) .B)Thethreeoutermostlayersofadegree (6 ; 6) patch,obtainedbysamplingabicubicextensionoftheprevi ouslayer aresubdivided(asshowninFigure 5-3 tomatchthegranularityofthepoints generatedby h 6 ; 6 (C). Withtheabovecomponentsdened,GuidedSurfacesareconstru ctedasillustratedin Figure 5-4 ,theinnerfourlayersareconstructedbyHermitesamplingt he 3 -jetsatthe cornersof g m .TheouterthreelayersareconstructedasillustratedinFi gure 5-3 EachoftheL-shapedpatchesisdegreeraisedto 6 6 patch,thenthesepatchesare extended C 2 andsubdividedtoformtheoutermostthreelayersofthenext ring. ImplementationNotes .Wegeneratedtheexplicitformulatocompute H ( g m ) usingthesymboliccomputationpackageMaple.Insteadofco mputingthefull composition,wetookadvantageofchainruletoonlycompute thenecessaryjetsofthe guideand inordertogetpositionandderivativestoconstruct x .Theexplicitformulas 71

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areexportedbyMapleasCcodeandonlyneedtobecomputedonc e.Thentheguided ringisgeneratedasablackbox.Usersonlyneedtoprovideth eguide(seeSection 5.1.3 ),boundaryinformation. 5.1.2FiniteCapping Guidedringsjoin C 2 .In KarciauskasandPeters [ 2009 ],afamilyofschemesare developedtocapamulti-sidedholewithanitenumberoften sor-productpatches.Here continuityistradedforthenumberofpatches.Themaincomp onentofthealgorithmis toconstructaparametrizationwhichcoverstheneighborho odofthedomainoriginofthe guide.Below,adegree (4 ; 4) G 1 constructionisexplained. Algorithms .Theconstructionissimilartoguidedsubdivision.Wecomp ose aguidewithparametrization, and ,thenHermitesampletoconstructthenal surface.Thedifferenceisinsteadofscalingtheparametri zations repeatedlyto covertheneighborhoodofthedomainoriginasinguidedsubd ivision,hereanite parametrization and isconstructedtocovertheneighborhoodofthedomainorigi n andmeetcontinuouslywith AsinFigure 5-5 meets and in C 2 ,oncewehave dened, isxedby continuityconstraints.Wechoose ofdegree (3 ; 3) and (i) islinearalongthesectorpartitioningray,i.e. i 0 :=(1 i= 3) 0 +( i= 3) v 0 i =0 ::: 3 ;thepoints 0 i aresymmetrictothepoints i 0 (ii)therstderivativeof acrossthesectorpartitioningrayisofdegree2,i.e. i 1 i =0 ::: 3 ,deneaquadraticindegree-raisedform.Thesegments i 1 ; i 0 i =2 ; 3 mustbeperpendiculartotheedge 0 ; v 0 and,bysymmetry, 11 mustbeonthebisectrix between ( s; 0) and (0 ;t ) Thisleavestwodegreesoffreedom(oneforeachredcircledp oint). Wechoosethethreepieces 1 ; 2 ; 3 of ofdegree (3 ; 3) andsothat(see Figure 5-5 ) (iii) and join C 2 and 72

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00 102030 01 11 31 Figure5-5.(From KarciauskasandPeters [ 2009 ])Structureof and ofdegree(3,3). Grayunderlayindicatesprolongationfrom (iv) and are C 1 connectedeverywhere(thisxes 30 and 31 ). Wepindownthelastfreeparameter,thepointonthebisectri xindicatedby inFigure 5-5 ,andminimize F 3 ( 1 )+ F 3 ( 2 )+ F 3 ( 3 )+ F 3 ( ) .For n> 4 ,a goodapproximationtotheexplicitsolutionistoplacethe at( : 068815616cos( 2 n )+ : 7098539220 )timesthedistanceofthepointmarked 2 totheorigin.Theresulting tessellationsareshowninFigure 5-6 Figure5-6.Theparametrization thatcoverstheneighborhoodofthedomainoriginof theguideforvalence5,6,7,8,9 ImplementationNotes .Implementationofguidedcappingalgorithmisverysimila r totheimplementationofguidedsubdivision.Oncewecomput etheparametrization and ,thesamemapleroutinecanbeusedtocomputethejetsneeded toconstructthe nitecap. 73

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5.1.3InitialGuide Intheaboveconstructions,weassumethereisaguidewhichc haracterizes theshapewewanttohaveinsomeregionandiseitherintensor productformorin triangularB ezierform.Aguidecanbeprovidedbydesignersexplicitlyo ritcanbe generatedtomatchexistingboundarydata(seeFigure 5-7 ). A B C Figure5-7.(From KarciauskasandPeters [ 2009 ])Inputdatafromthree-beam corner(monkeysaddle).A)Facetedinput.B)Tensor-borderof degree 3 .C) Degreeraisedto 5 A B Figure5-8.(From KarciauskasandPeters [ 2009 ])Freecoefcients(markedasblack lledcircles)of C 2 pcw-mappingswhen d =5 (A)or d =4 (B). Generatingapiecewisepolynomialguidebasedonthetensor borderdata(see Figure 5-7 ).Thetensorborderisdenedtomimicthebehavioroftensor -product patchesalongtheboundary. KarciauskasandPeters [ 2009 ]describeitasthereverse proceduretoguidedsubdivision.Aguideisrstinitialize dby C 2 continuityconstraints 74

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A B C Figure5-9.(From KarciauskasandPeters [ 2009 ])ttingtothetensorborderdata.A) Red,greenandbluepointsaresamplelocations.B)Sampleddat ais convertedinto 3 3 blocksofbi5patches.C)Tensorborderdata. A B C Figure5-10.(From KarciauskasandPeters [ 2009 ])1-1correspondence–”sampled 3 3 corner”(B)and”circledcorner”(A).(C)Directdeterminatio noffree controlpointsofpcw-quintic. betweenpieces.Theremainingcoefcients(markedasblack lledcirclesinFigure 5-8 ) arecomputedtobesttthetensorborderasillustratedinFi gure 5-9 Fortotaldegree5guide,aftertherststepofinitializati onbasedoncontinuity constraintsinternally,therestoffreepointscanbesolve dfromthefollowingexpression: min g j free n X j =1 k H ( x j ; il ) b j ; il k 22 : (5–1) 5.2ComparisonofCurvatureContinuousSurfacingAlgorithm s Thedifcultyinconstructingsmoothsurfacesliesinthene ighborhoodof extraordinaryvertices,ofvalence n 6 =4 forquadrilaterals.Herewecompare 75

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GuidedSurfaceswiththreeotherapproachestobuildingsmoo thsurfacearoundthe extraordinaryvertices. FreeformSplines .Freeformsplines[ Prautzsch 1997 ]arearegularlyparametrized G 2 constructionwithbisexticpatchesaroundextraordinaryv erticesandbicubic otherwise.Themethodcanbeextendedto G k withtensorproductpatchesofdegree 2 k +2 .Thisapproachrequirestheextraordinaryverticesarewel lseparated,sogenerally twostepsofCatmull-Clarkareneeded. G 2 Bi-7Surfaces LoopandSchaefer [ 2008b ]builddegreebi7 tensorproduct patchesaroundextraordinaryverticeswith G 2 continuity.Thesurfaceisdenedasa constrainedminimizationproblemoveranenergyfunctionw hichreducestozerofor bicubicpatches.AsforallschemesinthisSection,basisfunc tionscanbeprecomputed independentofthesurface. ANon-polynomial C 1 Construction YingandZorin [ 2004 ]proposeda manifold-basedconstructionforarbitrarysmoothnesssur faces.Theresultingsurfaceis notpolynomial. Table 5-1 comparesthefourconstructionswithrespecttothenumbero fpatches andthedegreeofthepatches.Thesetwowilldecidethecompl exityofevaluationof thesurfaceandthecomplexityofthefurtherprocessingina designpipeline.Sinceall fouralgorithmsarecomplicated,itisgoodtonotethatthey generatelinearcombination ofprecomputablebasisfunctions.Thatiswecanmultiplyea chbasisfunctionwithits controlpointintheinputmeshandtakethesummationtogett hesurface.(Thebasis functionsprovidedby C 1 schemedonotsumupto1,soanadditionalaveragebythe summationofthebasisisneeded.) Fromtheabovetable,Freeformsplines,Loop'sbi-7schemea nd C 1 allrequireup to 2 stepsofCatmull-Clarksubdivision,whichgenerate 16 n patches.Soeventhough GuidedSurfaceshaveapparentlymorepolynomialpiecesatth eextraordinaryquads, thetotalnumberofpatchesitgeneratesisstillsmallercom paredtotheotherthree. 76

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Besidesthecomplexityofthealgorithms,wealsowanttocomp arethequalityofthe surfacestheygenerate.Theyallguaranteeatleastsecondo rdercontinuity,(andcan inprinciplegivehighercontinuity).Weconsiderasurface ashighqualityifitdoesnot createcurvaturefeaturesnotimpliedbytheinputdata,tha tmeansitshouldminimize curvaturevariationandpreservefeaturelines( PetersandKar ciauskas [ 2010 ]).This cannotbemeasuredwellbyasinglenumberorcapturedbyasim pletheorem.Rather wedisplaydenselysampledGausscurvatureforchallenging inputcongurations. 5.2.1ShapeAnalysisSetup Ourtestmeshesfeatureisolatedextraordinarypoints,sur roundedbytwo layersofregularmesh,sonoCatmull-Clarksubdivisionnee dstobeapplied.This preventstheshapeproblemthatallschemeswithCatmull-Cl arkstepsareproneto ( Karciauskasetal. [ 2004 ]:ForCC,thelimitshapegenericallyconvergetoahyperbol ic shapealsoforaconvexinputmesh.) Figure 5-11 comparestherstthreealgorithmsonconvexinputmesheswi thcentral valence 6 and 8 .Foreachmodel,werstdrawthehighlightlines,thatissho wnasthe rstrowineachblock.ThenwecomputetheGausscurvaturean dscaleitintothe samerange(butadifferentonefordifferentcongurations toobtainbestresolution)and mapcolorstothecurvature.Thesurfaceisrenderedbytheco lorcorrespondingtoits curvatureateachsamplepoint.Theresolutionissetto 64 64 perpatch. Toeasilyseewhetherthetheconvexitysuggestedbythegeom etryoftheinput meshispreserved,Figure 5-14 comparesthecurvatureneedlesofthesurfacesforeach scheme.Thatis,thesurfaceisrenderedwithnormalneedles inplaceofsurfacepoints. Table5-1. C 2 constructions.NotethattwoCCstepscreate 16 n patches. Algorithms CCsteps&Poly# ofpatchesdegree Freeformsplines[ Prautzsch 1997 ] 2 & n (6 ; 6) Loop's G 2 bi-7[ LoopandSchaefer 2008b ] 2 & n (7 ; 7) GuidedSurfaces[ KarciauskasandPeters 2009 ] 0 & 4 n (6 ; 6) C 1 [ YingandZorin 2004 ] 2 & ? 1 77

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ThelengthofeachneedleisproportionaltotheGausscurvat ure.Eachneedleisalso coloredaccordingtotheGausscurvature.5.2.2CurvatureComparison InFigure 5-14 ,forthevalence 6 ellipticintherstrow,thesurfacesgeneratedbyall threeschemeshavepositiveGausscurvature,butforvalenc e 8 shape,Loop&Schaefer (L&S)andFreeformdonot.Thisisvisiblebytheblueleft-poin tingspikesthatshowthe dramaticswitchtonegativeGaussiancurvatureintheareac losetotheextraordinary vertex. ForYingandZorin's C 1 scheme,wealsondthatitdoesnotgenerateaconvex surface.SincethedisplaypackageBezierViewdoesnotworkwit hthenon-polynomial patches,weclipthe C 1 surfacewithaplane.(Figure 5-15 B,C) Figure 5-12 andFigure 5-13 havehyperbolicandmulti-saddleshapeswithvalence 6 and 8 respectively.BothL&SandGuidedschemegivesimilarcurvatu revariation,the Freeformschemegivesmoredramaticcurvaturevariation.T heundesirableoscillations arevisibleonthesurfacewithregularlighting. Weconcludethatthethreecomparisonschemesfailtoprovid eaconvexshape whenadesignermightexpectthisduetotheinputmeshbecaus etheinputmeshcan betriangulatedintoaconvexmesh.WhileGuidedSurfacingpas sesthistest,thereisno proofthatitwillalwaysgenerateconvexsurfaces. 78

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Input L&S(7,7) Freeform(6,6) Guided(6,6) Figure5-11.Convexshapeswithvalence 6 and 8 79

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Input Loop's(7,7) Freeform(6,6) Guided(6,6) Figure5-12.Hyperbolicandmulti-saddleshapeswithvalen ce 6 80

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Input Loop's(7,7) Freeform(6,6) Guided(6,6) Figure5-13.Hyperbolicandmulti-saddleshapeswithvalen ce 8 81

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Input Loop's(7,7) Freeform(6,6) Guided(6,6) Figure5-14.Curvatureneedlesforconvexshapes. A B C Figure5-15. C 1 schemeonellipticshape.A)Surfaceafterclippingapplied.B) Surface withsamplemeshafterclipping.C)Zoomedinviewof(B). 82

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CHAPTER6 CONCLUSION Inconclusion,thisthesisexplainedanoptimalconstructi onforthemostcommon surfacerepresentation,bi-cubicsplinesoverquadrangul ations.Itprovedlower boundsonthecomplexityofsplinesoverquadrangulationsa nddiscoveredthe reparameterizationofleastcomplexity(rationaldegree) thatallowsformodeling surfacesofarbitrarygenus.Finallyitpresentedarst,sy stematictestingand comparisonofthenewestcurvature-continuoussurfacecon structions. 83

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BIOGRAPHICALSKETCH JianhuaFanwasborninChina.ShereceivedherBachelorofScien cedegreein computersciencein1999fromNortheasternUniversity,Shen yang,China.Hermajor researchareaisincomputergraphics. 88