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Curvature-Continuous Bicubic Subdivision Surfaces for Polar Configurations

Permanent Link: http://ufdc.ufl.edu/UFE0022832/00001

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Title: Curvature-Continuous Bicubic Subdivision Surfaces for Polar Configurations
Physical Description: 1 online resource (68 p.)
Language: english
Creator: Myles, Ashish
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bicubic, c2, configuration, connectivity, continuous, curvature, mesh, non, polar, quad, stationary, subdivision, surface
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: Subdivision surfaces are popular in animation as a way of smoothing coarse control meshes. On the other hand, the Computer-Aided Design (CAD) industry typically prefers the simplicity and predictability of NURBS when constructing high-quality surfaces for the manufacture of cars and planes. Since a single NURBS patch is capable only of modeling the topologies of planes, cylinders, and torii, it is complex to use a NURBS atlas to construct a surface of arbitrary topology that is curvature-continuous everywhere. While popular subdivision algorithms of low parametric degree, like Catmull-Clark and Loop subdivision, are not inherently restricted in topology, they su?er from shape artifacts at so-called 'extraordinary vertices'. This makes them unattractive for CAD. Subdivision theory requires a (bi)degree of at least 6 in order for stationary subdivision to be non-trivially curvature-continuous and mitigate some of these shape artifacts. We circumvent this restriction by designing a curvature-continuous non-stationary bicubic subdivision algorithm which has the implementational simplicity of stationary algorithms. We hope techniques such as ours make subdivision surfaces more attractive for high-quality constructions in CAD.
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 Ashish Myles.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Peters, Jorg.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-06-30

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022832:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022832/00001

Material Information

Title: Curvature-Continuous Bicubic Subdivision Surfaces for Polar Configurations
Physical Description: 1 online resource (68 p.)
Language: english
Creator: Myles, Ashish
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bicubic, c2, configuration, connectivity, continuous, curvature, mesh, non, polar, quad, stationary, subdivision, surface
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: Subdivision surfaces are popular in animation as a way of smoothing coarse control meshes. On the other hand, the Computer-Aided Design (CAD) industry typically prefers the simplicity and predictability of NURBS when constructing high-quality surfaces for the manufacture of cars and planes. Since a single NURBS patch is capable only of modeling the topologies of planes, cylinders, and torii, it is complex to use a NURBS atlas to construct a surface of arbitrary topology that is curvature-continuous everywhere. While popular subdivision algorithms of low parametric degree, like Catmull-Clark and Loop subdivision, are not inherently restricted in topology, they su?er from shape artifacts at so-called 'extraordinary vertices'. This makes them unattractive for CAD. Subdivision theory requires a (bi)degree of at least 6 in order for stationary subdivision to be non-trivially curvature-continuous and mitigate some of these shape artifacts. We circumvent this restriction by designing a curvature-continuous non-stationary bicubic subdivision algorithm which has the implementational simplicity of stationary algorithms. We hope techniques such as ours make subdivision surfaces more attractive for high-quality constructions in CAD.
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 Ashish Myles.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Peters, Jorg.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022832:00001


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CURVATURE-CONTINUOUSBICUBICSUBDIVISIONSURFACES FORPOLARCONFIGURATIONS By ASHISHMYLES ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2008 1

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c r 2008AshishMyles 2

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Tothebettermentofeducationeverywhere 3

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ACKNOWLEDGMENTS Thanksgotomysupervisorycommitteeandeveryoneinvolved inSurfLab,especially myadvisorJorgPeters,forgivingmesomuchintuitionandp ositiveexperienceduringmy 8yearsworkingingraphicsandgeometry.Iadditionallytha nkXiaobinWuforhelpingme developmyearlyintuitiononsubdivisionsurfacesduringo urinternshipatATI,andmy roommateIl(Memming)Parkforourmathbrainstorm,whichim provedmyintuitionof thetheoreticalstructureunderlyingthiswork. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES .....................................7 LISTOFFIGURES ....................................8 LISTOFSYMBOLS ....................................9 ABSTRACT ........................................11 CHAPTER 1INTRODUCTION ..................................12 2GENERALIZATIONSOFUNIFORMBICUBICSPLINES ............18 2.1UniformB-SplineRepresentation .......................18 2.1.1Univariate ................................18 2.1.2Tensor-productbivariate ........................20 2.2Catmull-ClarkandBi-3PolarSubdivision ..................22 2.2.1Generalizationsofquadgridmeshes ..................22 2.2.2Subdivisionasrenementoperations .................22 2.2.3Subdivisionaspiecewisepolynomials .................25 2.2.4Behavioraroundandattheextraordinarypoints ...........26 3RADIALTAYLORSUBDIVISION(RTS) .....................27 3.1NotationandLabeling .............................27 3.2RadialTaylorSubdivision(RTS)Denition .................30 3.3Analysis .....................................32 3.3.1SpectralanalysisofRTS ........................32 3.3.2ReformulatingRTSineigenspace ...................39 3.3.3Eigenspaceexpansionandcurvaturecontinuity ............41 3.4ApproximationviaMeshRenement .....................44 4 C 2 POLARSUBDIVISION( C 2 PS) .........................47 4.1Semi-StationarySubdivision ..........................47 4.2Analysis .....................................48 4.2.1Preservationofeigencoecients ....................49 4.2.2Reformulationof C 2 PSintermsoftheeigencoecients .......50 4.2.3Convergenceoftheeigensplines ....................51 4.2.4Proofofcurvaturecontinuity ......................52 5RESULTSANDDISCUSSION ...........................54 5

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6CONCLUSION ....................................58 APPENDIX: C 2 PSINTERMSOFTHEEIGENCOEFFICIENTS ..........60 REFERENCES .......................................64 BIOGRAPHICALSKETCH ................................68 6

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LISTOFTABLES Table page 1-1Variousmeshrenementalgorithms .........................13 3-1SpectralbehaviorofRTS ...............................35 7

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LISTOFFIGURES Figure page 1-1ANURBSsurfaceinCAD ..............................12 1-2Polarcongurationexamples ............................16 2-1Univariateuniformcubicspline ...........................19 2-2Generalizationsofmeshconnectivity ........................21 2-3CommutativityofregularB-splinesubdivision ...................21 2-4Catmull-Clarkconvertsn-gonstoquads ......................22 2-5Catmull-Clarkvs.polarsubdivision .........................23 2-6Catmull-Clarkcausesripplesonpolarcongurations ...............24 2-7SplineringsofCatmull-Clarkandbi-3polarsubdivisio n .............25 3-1Polarconguration ..................................27 3-2RadialTaylorsubdivisionsplinering ........................28 3-3RadialTaylorsubdivision(RTS)rules .......................31 3-4Eigenvectorsandeigensplines ............................36 3-5CombiningCatmull-ClarkandRTS .........................45 4-1 C 2 polarsubdivisionrules ..............................48 5-1ComparisonofRTS,RTS 1 ,and C 2 PS .......................55 5-2RTS,RTS 1 ,and C 2 PSonhigh-valentpolarcongurations ............55 5-3 C 2 PSshapegallery ..................................56 5-4ConvertingCatmull-Clarkextraordinaryverticestopo larcongurations .....57 8

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LISTOFSYMBOLS P nh P S equivalentto P n 1 h =0 forpositiveintegers n and P 2 S forsets S respectively A block-circulantsubdivisionmatrix ^ A block-diagonalizationofsubdivisionmatrix A computedviaFourier transform ^ A k k th Fourierblockof ^ A k indexoftheFourierblock ^ A k whichcontributestheeigenvalue ` k c c j : n cos(2 ),cos 2 j n e k eigenspline ^ e k Fourierradialeigenspline F Fourierblockmatrixwith6 6blocks. G operatorthat,whenappliedtoapolarconguration q ,returnsthe setofallitscircularGrevilleabscissae G q = j n j 2Zn G m operatorthatconvertsapolarconguration q m totheuniform periodiccubicsplinering G m q m :[2 m ; 4 m ] R1 !R(seeFigure 3-2 ,pp. 28 ) G m q m ( r; ):= 5 X i =1 n X j q mij N ( m ) i ( r ) N ( n m ) j ( ) ; := 1 2 ^ G m operatorthatconvertsavector u 2R6 totheuniformcubicspline ^ G m u :[2 m ; 4 m ] !R^ G m u ( r ):= 5 X i =1 u i N ( m ) i ( r ) : ` k eigenvalueof A withthe k th largestmodulus,countingmultiplicity 9

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diagonalmatrixoftheeigenvaluesof A (respectively, ^ A ) L operatorthatconvertsapolarcongurationtoitslimitsur facein polarparameterization: L ( q ):= x n m valenceofthepolarvertexofthepolarconguration q m N ( m ) i ( r )the i th uniformcubicB-splinebasiswithknots 1 2 m [ 1 ; 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6] N ( n ) j ( )the j th uniformperiodiccubicB-splinebasiswithknots 1 n Z n op k ( ) c k if k n= 2 s k otherwise p k eigencoecientcorrespondingtoeigenspline e k q m an m -timesrenedpolarcongurationR,R1 thesetofrealsandthesetofrealsmodulo1,respectively s s j : n sin(2 ),sin 2 j n V R (resp. V C ) matrixwhosecolumnsarethereal(respectively,complex)r ight eigenvectorsof A ^ V matrixwhosecolumnsaretherighteigenvectorsof ^ A v k ^ v k righteigenvectorof A and A k ,respectively,correspondingto ` k w k ^ w k lefteigenvectorsof A and A k ,respectively,correspondingto ` k x polarlimitsurfaceinpolarparameterization x polarlimitsurfaceinCartesianparametrizationZ,Zn thesetofintegersandthesetofintegersmodulo n ,respectively Z n thestrictlyincreasingsequenceofintegersinZn 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy CURVATURE-CONTINUOUSBICUBICSUBDIVISIONSURFACES FORPOLARCONFIGURATIONS By AshishMyles December2008 Chair:JorgPetersMajor:ComputerEngineering Subdivisionsurfacesarepopularinanimationasawayofsmo othingcoarsecontrol meshes.Ontheotherhand,theComputer-AidedDesign(CAD)i ndustrytypicallyprefers thesimplicityandpredictabilityofNURBSwhenconstructi nghigh-qualitysurfaces forthemanufactureofcarsandplanes.SinceasingleNURBSp atchiscapableonlyof modelingthetopologiesofplanes,cylinders,andtorii,it iscomplextouseaNURBS atlastoconstructasurfaceofarbitrarytopologythatiscu rvature-continuouseverywhere. Whilepopularsubdivisionalgorithmsoflowparametricdeg ree,likeCatmull-Clark andLoopsubdivision,arenotinherentlyrestrictedintopo logy,theysuerfromshape artifactsatso-called\extraordinaryvertices".Thismak esthemunattractiveforCAD. Subdivisiontheoryrequiresa(bi)degreeofatleast6inord erforstationarysubdivision tobenon-triviallycurvature-continuousandmitigatesom eoftheseshapeartifacts. Wecircumventthisrestrictionbydesigningacurvature-co ntinuousnon-stationary bicubicsubdivisionalgorithmwhichhastheimplementatio nalsimplicityofstationary algorithms.Wehopetechniquessuchasoursmakesubdivisio nsurfacesmoreattractivefor high-qualityconstructionsinCAD. 11

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CHAPTER1 INTRODUCTION Fromautomobileandplanedesigntodigitalmovieanimation tovideoandcomputer gamecharacterdesign,smoothcurvesandsurfacesplayafun damentalroleinthedesign ofobjects.StandardComputer-AidedDesign( CAD )packagesneedtorepresentthese surfacesinanecientformthatiseasytomanipulatealgori thmically,andintuitivefor theusertomoldintothedesiredshape.Additionally,suchs urfacerepresentationsshould beeasytovisualizeandrenderontothescreen. SmoothsurfacerepresentationsinCADpackagescanbelarge lyclassiedintotwo categories:implicitandparametric.Implicitsurfacesar edenedintermsofzero-sets.For example, x 2 + y 2 + z 2 1=0istheimplicitrepresentationoftheunitsphere.While this representationisusefultocreatebasicshapeandtoapplyb ooleanoperations,visualizing andrenderingthesurfacetypicallyrequiressolvingaseto fnon-linearequations. Thealternativeistouseparametricrepresentations.Inco ntrasttoitsimplicitform,a unitspherecanberepresentedusingthreeequationsinterm softwoparameters s and t as follows. x ( s;t )=cos( s )cos( t ) ;y ( s;t )=cos( s )sin( t ) ;z ( s;t )=sin( s ) As s isvariedfrom0to and t isvariedfrom0to2 ,thepointsonthesurfaceofthe Figure1-1.ANURBSsurfaceinatypicalCADpackagedetermin edbyacontrolnet consistingofallquadsandinternalverticesofvalence4. 12

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spherearegenerated.Astandardformofparametricreprese ntationiscalled Non-Uniform RationalB-Splines ,orNURBSforshort.Inthiscase,the x -, y -,and z -coordinatesare representedseparatelybypiecewiserationals.Thesurfac eisdenedbyA)a control mesh ,orcontrolnet,whichforNURBSisaquadmeshwitheachinter nalvertexhaving valence4,asshowninFigure 1-1 ;B)two knotsequences thatdeterminetheextentand eectofthedomain,oneforeachofthe s and t parameters;and,C) weights associated witheachvertexinthecontrolmesh.Eventhoughbooleanope rationsonNURBSare notstraightforward,NURBSarepopularduetotheirintuiti vemanipulabilityandease ofrendering.However,beingbasedonquad-gridcontrolstr uctures,NURBSarecapable ofrepresentingonlytopologicalplanes,cylinders,ortor ii.WhilethetheoryofNURBS willnotbediscussed,aspecialcaseofitisimportantinthi sstudy:the uniformB-spline withuniformlyspacedknotsandallweights1.SurfacesinBsplineformcanbeconverted directlytoclosedform,whichisusefulforanalysis.Alter natively,thesurfacecanbe denedviaaniterativemeshrenementalgorithm,whichise asiertogeneralize.Chapter 2 discussesuniformbi-degree-3splinesingreaterdetail. Table1-1.Variousmeshrenementalgorithms(notcomprehe nsive).Quad/triangleis only C 1 overcertainedgesandisolatedpoints.ExceptforTURBS,al l producedsurfacesaregenericallyonly C 1 atisolatedpoints.Thelastcolumn indicateswhetherornotthealgorithminterpolatesitscon trolpoints. YearAlgorithmSmoothDegreeBasisInterp. 1978Catmull-Clark[ CatmullandClark 1978 ] C 2 bi-3 2 no 1978Doo-Sabin[ DooandSabin 1978 ] C 2 bi-2 2 no 1987Loop[ Loop 1987 ] C 2 4 4 no 1990Butterry[ Dynetal. 1990 ] C 1 N/A 4 yes 1996Kobbelt[ Kobbelt 1996 ] C 1 N/A 2 yes 1997Simplest[ PetersandReif 1997 ] C 1 2 2 no 1998TURBS[ Reif 1998 ] C k bi-(2 k +2) 2 no 2000 p 3[ Kobbelt 2000 ] C 2 N/A 4 no 20014{8[ VelhoandZorin 2001 ] C 4 6 2 no 2001Circlepreserving[ Morinetal. 2001 ] C 2 3&trig. 2 no 2002Ternarytriangle[ Loop 2002b ] C 4 4 4 no 2003Quad/triangle[ StamandLoop 2003 ] C 2 bi-3,4 4 2 no 20044{3[ PetersandShiue 2004 ] C 2 4 4 2 no 13

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ToaddresstheinherentlimitationsofNURBS,subdivisions urfaceswereintroduced simultaneouslyby CatmullandClark [ 1978 ]and DooandSabin [ 1978 ].Thetwo subdivisionsurfacealgorithmsaregeneralizationsofB-s plineiterativemeshrenement rules,supportingarbitraryconnectivityandmanifoldtop ology.Theserulesspecifywhere pointsareadded;howthepositionsofthesepointsarecompu ted;andhowthemeshis reconnected.Afteraninniteiterativeapplicationofthe sesubdivisionrules,themesh convergestoalimitsurface.Somesubdivisionalgorithmsw erecreatedspecicallyfor triangularmeshes,whereasotherswerecreatedforquadmes hes.Somewerecreated specicallytointerpolatetheverticesofthecontrolnet. Someweredesignedfortangent continuity( C 1 ),andothersforcurvaturecontinuity( C 2 ).Table 1-1 summarizesseveral well-knownsubdivisionalgorithms,andisbynomeanscompl ete.Thesurfacequalityof various C 2 algorithmslistedisdecientatcertainisolatedpoints,c alled extraordinary points ,wheretheyareonly C 1 .Whatisconsideredtobeanextraordinarypointdepends onthedetailsofeachalgorithm.Section 2.2 ,forinstance,willdenetheextraordinary pointforCatmull-Clarksurfacesanddescribethesurfaceb ehaviorinitsneighborhood. Theliteratureontheanalysistechniquesisenumeratedatt heendofSection 2.2.4 Varioussurfaceconstructionalgorithmswereinventedora daptedforapplicabilityor quality.Forexample,quad/trianglesubdivisionmentione dinTable 1-1 isacombination ofCatmull-ClarkandLoopsubdivisionsappliedtothequada ndtriangularportions ofthemeshseparately.Newrulesweredevelopedfortheboun darybetweenthequad andtrianglemeshes,andthebehaviorofthesurfacealongth oseedgesisonly C 1 SinceCatmull-Clarkbyitselfwasdesignedforquadsandhas undesirableshapeon trianglemeshes,itscombinationwithLoop'salgorithmimp rovesoverallsurfacequality andtheapplicabilityofthesubdivisionalgorithm.Addres singsurfacesofrevolution, Morinetal. [ 2001 ]designedasubdivisionalgorithmcapableofreproducingc ircles,which polynomialalgorithmscannotdo.Thistechniquereproduce scubicpolynomials,circles, andhyperbolicfunctionsdependingonatensionparameter. Bytensoringthealgorithm 14

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onaquadmesh,theyobtainedasurfacethatis C 2 exceptatextraordinarypoints,where itis C 1 .Whilemostsubdivisionalgorithmsapproximatelyquadrup lethenumberofpoints inthemeshaftereveryrenement,somearespecicallydesi gnedtoreneslowly:simplest subdivision[ PetersandReif 1997 ]and4{8subdivision[ VelhoandZorin 2001 ]quadruple everytwoiterations; p 3subdivision[ Kobbelt 2000 ]increases9-foldeverytwoiterations. Slowingtherenementgivesgreatercontroloverthesizeof therenedmesh.Thisis usefulforrenderingnomorethanisnecessary. Catmull-ClarkandLoopsubdivision,themostwell-knownsu bdivisionalgorithms forquadandtrianglemeshes,respectively,areknowntohav eunboundedcurvaturein thevicinityoftheextraordinarypoint.Manyattemptshave beenmadetoimproveupon them. Sabin [ 1991 ]re-tunedCatmull-Clarksothatityieldedsurfaceswithbo unded curvature. Augsdorferetal. [ 2006 ]wentastepfurthertominimizeGaussiancurvature variationwithinthespaceofboundedcurvaturealgorithms .Variousmodicationshave beenmadetoLoopsubdivisiontosupportcurvaturecontinui ty,albeitwithalocalrat spotwithzerocurvature[ PrautzschandUmlauf 1998 2000 ];boundedcurvaturewiththe surfacelyingwithintheconvexhullofthecontrolpoints[ Loop 2002a b ];and,curvature control[ GinkelandUmlauf 2006 ]. Umlauf [ 2005 ]summarizedmanyofthesere-tuning techniques. Notableconstructionsthatsupportarbitrarydegreeofsmo othnessevenatthe extraordinarypointincludefree-formsplines[ Prautzsch 1997 ]andTURBS[ Reif 1998 ],bothofwhichrequiredegreebi-(2 k +2)tocreateaneverywhereC k surface. YingandZorin [ 2004 ]createdaneverywhereC 1 surfaceusingexponentialblending functionsbetweenpolynomialpatches.Morerecentworkby KarciauskasandPeters [ 2007b 2008 ]introducedtheconceptofguidedsubdivisionalsocapable ofachieving arbitrarycontinuity.For C 2 ,theyemployaninnitesequenceofbi-degree-6splinesurf ace ringstoapproximatea C 2 \guidesurface"ofgoodquality.In[ KarciauskasandPeters 2007d ],theyemploysequencesofbicubicsplineringscontaining anexponentially-increasing 15

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numberofpolynomialstoreproducetheguidesurface'sseco ndorderbehavioratthe extraordinarypointinspiteofthelowdegreeoftheoverall construction.Ourconstruction implicitlyalsousesbicubicsplinesurfaceringsofexpone ntially-increasingnumberof polynomialstoachievecurvaturecontinuity;however,thi sincreasecomesaboutnaturally inouralgorithm. Avarietyofotherapproacheshavebeenusedtoimproveshape nearextraordinary points. Peters [ 2000 ]approximatedCatmull-Clarksurfaceswithanitenumbero fbicubic patchesthatjointangent-continuously.Asanalternative Peters [ 2002 ]suggesteda C 2 constructionofdegree(3 ; 5).Boththesetechniquesstillsuerfromshapeproblemsdu e tothelowdegreeoftheconstructions. LoopandSchaefer [ 2008 ]achievedcurvature continuityforquadmeshesusingpatchesofbi-degree7with shapeoptimizationforthe freeparameters. KarciauskasandPeters [ 2007c ]usedtheconceptsofguidedsubdivisionto constructa C 2 surfacewithanitenumberofbi-degree-6patches.[ Levin 2006 ]perturbed Catmull-Clarksurfacesusingpolynomial-square-rootble ndingfunctionsbetweenlocal polynomialpatches.Inthesamevein, Zorin [ 2006 ]perturbedLoopsubdivisionsurfacesto be C 2 usingablendingfunctionthatwasitselfasubdivisionsurf ace. AB Figure1-2.PolarcongurationonA)ngertipsandB)thetop ofthemushroom. Manyofthesurfaceconstructionalgorithmsmentionedabov earecomplexor suerinshapenearhigh-valencevertices. KarciauskasandPeters [ 2007 ]recognized onecommonly-occurringcongurationofhighvalenceinqua d-dominantmeshes: the polarconguration ,whichisthefocusofthisstudy.Thepolarcongurationcon sists 16

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ofahigh-valencecentralvertex{the polarvertex {inthemiddleofatrianglefan surroundedbyaquadgridneighborhood.Thiscongurationo ccursnaturallyatthe endsofelongatedobjectsliketipsofngers,andinthelati tude/longitudeconnectivity ofthesphere(Figure 1-2 ),anditisstructurallyfarsimplerthantheneighborhoodo f Catmull-Clarkextraordinarypoints,asweshowinSection 2 .Catmull-Clarkonpolar congurationsresultsinmacroscopicoscillationsinthep olarneighborhood.Treatingpolar asaspecialcasegivesgoodresults,evenwhenthecentralva lenceisveryhigh(Figure 2-6 ). Karciauskasetal. [ 2006 ]adaptedguidedsubdivisiontopolarcongurationtocreat e C 2 polarjetsubdivision,whichemploysacontrolnetstructur etomakesplinesurface ringsofdegree(6 ; 5). KarciauskasandPeters [ 2007a ]introducedverysimplebicubic C 1 subdivisionalgorithmwithboundedcurvature,whichwassu bsequentlyadaptedby Mylesetal. [ 2008 ]tobecompatiblewithCatmull-Clarksubdivision. Mylesetal. [ 2008 ] alsooereda C 1 bicubicNURBSpatchconstructionwithboundedcurvatureto coverthe neighborhoodofthepolarconguration. Thereisnoacceptedmathematicaldenitionofsurfacequal ity.Forsimulation,itis oftenusefultohavewell-denedcurvatures.Additionally ,theintroductionofcurvature continuitytendstoimprovevisualqualityofthemodeledsu rface.Subdivisiontheory [ PetersandReif 2008 ]statesthatCatmull-Clarksubdivisioncannotbere-tuned tobe non-trivially C 2 attheextraordinarypointwithdegreelessthanbi-6.Inthi sstudy, wesidesteptheassumptionsunderlyingthistheoremtotake advantageofthenatural subdivisionstructureofpolarcongurationstocreatea C 2 algorithmthathasdegreeonly bi-3.Wealsoshowthatoursimplesubdivisionalgorithmyie ldssurfaceswithhighvisual qualityandgoodcurvaturedistribution. 17

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CHAPTER2 GENERALIZATIONSOFUNIFORMBICUBICSPLINES 2.1UniformB-SplineRepresentation Weintroducenotationanddenitionstosimplifythediscus sion. Forintegers n P nh isanalternativenotationfor P n 1 h =0 .Forsets S P S isan alternativenotationfor P 2 S Zisthesetofintegers,andZn istheintegersmodulo n .Risthesetofreals.R1 isRmodulo1. Z n isthestrictlyincreasingsequenceofintegersinZn An anecombination isalinearcombinationwheretheweightsaddto1.A convex combination isananecombinationwheretheweightsarepositive. 2.1.1Univariate AdetailedtreatmentoftheB-splineformcanbefoundin[ Prautzschetal. 2002 ]. ApiecewisepolynomialinB-splineformisdenedbyasequen ceof controlpoints that denestheshape,andauniformly-spacedknotsequencethat denesthedomain.The piecewiselinearinterpolantforagivenorderingofcontro lpointsisknownasthe control polygon (seeFigure 2-1 ).Aunivariatecubic(i.e.degree3)uniformspline f :R!Rwith n controlpoints b :=[ b 0 ;b 1 ;:::;b n 1 ]requires n +4uniformly-spacedknots t :=[ t 0 ;t 1 ;:::;t n +3 ]andisdenedby f ( t ):= n X i b i N i ( t ) ; wherethe n cubicB-splinebases N i ( t )are N i ( t ):= 1 6 8>>>>>>>>>><>>>>>>>>>>: u i ( t ) 3 if t 2 [ t i ;t i +1 ] 3 u i +1 ( t ) 3 +3 u i +1 ( t ) 2 +3 u i +1 ( t )+1if t 2 [ t i +1 ;t i +2 ] 3 u i +2 ( t ) 3 6 u i +2 ( t ) 2 +4if t 2 [ t i +2 ;t i +3 ] (1 u i +3 ( t )) 3 if t 2 [ t i +3 ;t i +4 ] 0 otherwise u i ( t ):= t t i t i +1 t i : (2{1) WhilethesplineistechnicallydenedonallofR,itisrestrictedforpracticalpurposes 18

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t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 b 0 b 1 b 2 b 3 b 4 1 0 1 2 3 controlpolygon spline t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 b 0 b 1 b 2 b 3 b 4 1 0 1 2 3 A B Figure2-1.Univariateuniformcubicspline.A)Acubicspli ne f ( t )withcontrolpoints b =[1 ; 3 ; 1 ; 2 ; 1]( red )andknots t =[ 1 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8]isthesumof uniformB-splinebasesscaledbytheirrespectivecontrolp oints( blue,green, magenta,cyan ).B)Anequivalentdenitionusingiterativecontrolpolyg on renement. to t 2 [ t 3 ;t n ],asinFigure 2-1 ,whereatleastfournon-zerobasesoverlap.Thebasis functionsarenon-negativeandsumtooneinthisinterval,i mplyingthateachpointonthe splineisaconvexcombinationofthecontrolpoints.Thisyi eldstwoimportantgeometric propertiesofB-splines. Aneinvariance: Applyingananetransformationtothecontrolpolyhedron appliesittothetransformationsplineaswell. Convexhullproperty: AparametriccurveinB-splineformalwaysliesinthe convexhullofitscontrolpoints. UniformcubicB-splinesalsohavebuilt-insecond-orderco ntinuitysothatadjacent polynomialpiecesjoin C 2 The t -coordinateassociatedwitheachcontrolpoint b i iscalledthe Grevilleabscissa t i andisdened,ingeneral,via t i := 1 d P dj t i + j +1 ,where d isthedegreeofthespline.For uniformcubics,thissimpliesto t i = t i +2 .Itwillbeusefullatertoindexcontrolpointsby theirGrevilleabscissaewhentheknotsequenceischosenso that t i = i n .Tothisend,we denetheoperator G G b := f t i g i 2Zn ; (2{2) 19

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andthebracketedfractionalindexingnotation. b [ t i ] = b [ i n ] := b i ;N [ t i ] ( t )= N [ i n ] ( t ):= N i ( t ) : (2{3) Usingthisnotation,oursplineisequivalentlydenedas f ( t ):= G b X b [ ] N [ ] ( t ) : Onecansimilarlydeneperiodicuniformcubics f :R1 !R,requiringtheknot sequencetoliewithinR1 .Sincetheknotsequencecyclesaround,weneedonlyspecify n knots{e.g. t = 1 n Z n {andassumebyconventionthattherstcontrolpointhasGre ville abscissa0. TheB-splineformcanalternativelybedenedviaacontrolp olygonrenement procedureasillustratedinFigure 2-1 B.Theonce-subdividedcontrolpoints b 1 := [ b 11 ;b 12 ;:::;b 12 n 3 ]arecomputedfromtheoriginalcontrolpoints b 0 :=[ b 00 ;b 01 ;:::;b 0n 1 ]via thefollowingequations. b 12 i = 1 8 b 0i 1 + 6 8 b 0i + 1 8 b 0i +1 ;b 12 i +1 = 1 2 b 0i + 1 2 b 0i +1 (2{4) Applyingthisrenementproceduread-innitumconvergest othesplinecurve. 2.1.2Tensor-productbivariate TheB-splinebasescanbeeasilygeneralizedtosurfacesbyt ensoringtheunivariate bases,sothatthe bi-3 (i.e.bicubic,bi-degree-3,ordegree(3,3)),surface f ( s;t )isdened as f ( s;t ):= n s X i n t X j b ij N s i ( s ) N t j ( t ) ; wherethesplineisdenedbythe n s n t controlmesh b ofcontrolpoints,andtwoknot sequences s :=[ s 0 ;s 1 ;:::;s n +3 ]and t :=[ t 0 ;t 1 ;:::;t n +3 ]whichdenetheB-splinebases N s i ( s )and N t j ( t ),respectively.TheGrevilleabscissaofacontrolpoint b ij isapair( s i ;t j ) 20

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insteadofasinglereal.Figure 1-1 illustratessuchauniformbi-3splineinatypicalCAD environment. AB Figure2-2.Generalizationsofmeshconnectivity.A)Aquad -onlygeneralizationtomesh connectivityallowsvertexvalencesotherthan4.B)Polarm eshconnectivity arrivesnaturallywhenmanycontrollinesalongthesameten sordirectionmeet atasingularity. A C B Figure2-3.CommutativityofregularB-splinesubdivision .Bi-3splinesubdivisionA)in onedirectionfollowedbyB)theother,orC)simultaneousre nementasin Catmull-Clark. Thesurfacecanalsobedenedusingameshrenementprocedu re.Thecontrolmesh maybesubdividedineithertensoreddirectionindependent ly,toyieldthesamesurface inclosedform.Figure 2-3 illustratesthisprocedureonaparametricspline,whereth e meshis( A )renedstrictlyinonedirectiontwicefollowedby( B )theotheronetwice. Alternatively,onecantensorthesubdivisionmaskssothat onemay( C )directlysubdivide inbothdirectionstwice,convertingeachoriginalquadint ofourrenedonesafterevery tensoredsubdivision. 21

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2.2Catmull-ClarkandBi-3PolarSubdivision 2.2.1Generalizationsofquadgridmeshes Quad-gridcontrolmeshesarelimitedtorepresentingtheto pologyofplanes,cylinders, andtorii.Generalizingthemeshconnectivitytoallowarbi trary-valenceverticesand polarcongurations(Figure 2-2 )admitsmeshesencodingarbitrarysurfacetopology.The B-splinequad-gridconnectivityiscalledthe ordinarycase .The extraordinarycase consists ofaquadneighborhoodwithan extraordinaryvertex ofvalence 6 =4,anditsneighboring quadsarecalled extraordinaryquads .The polarconguration ,asdenedpreviously, consistsofacentral polarvertex ofarbitraryvalence 3,surroundedbyatrianglefan withinringsofregularquads(Figure 2-2 B). Whiletheutilityofarbitraryvalencesmaybeobvious,anap preciationforpolar congurationsrequiresmoreobservation.Figure 1-2 alreadydemonstratestheirutilityon certainmeshes,butwewilljustifytreatingpolarcongura tionsspeciallywhenexamining subdivisionsurfacesinthefollowingsection.2.2.2Subdivisionasrenementoperations TheCatmull-Clarksubdivisionalgorithm CatmullandClark [ 1978 ]takesanarbitrary inputmeshandsubdividesittoproduceadensermeshonwhich thealgorithmcanagain beapplied.Thelimitsurfacecorrespondingtothissequenc eofevermorerenedmeshesis calledtheCatmull-Clark(limit)surface.Forsimplicityo fdiscussion,wewillassumeclosed meshes{i.e.thosewithnoboundaries. Figure2-4.OnestepofCatmull-Clarkrenementconvertsam eshtoallquads. AnapplicationoftheCatmull-Clarksubdivisionprocedure convertseveryfaceinto multiplequadsby:1.splittingeveryedgeintotwobytheintroductionofanewv ertex(edgevertex), 22

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2.introducinganewvertexatthemiddleoftheface(facever tex),and 3.connectingthefaceverticestotheirneighboringedgeve rtices. SeeFigure 2-4 fortheendresultonthreedierentpolygons.Subdivisions plitsafaceof size n into n quadsandcreatesan n -valentvertexatthecenter.Sincethemeshfacetsare four-sidedafteronerenement,allsubsequentsubdivisio nsquadruplethesizeofthemesh. Thenewpointsareanecombinationsoftheirneighborsandt heoldverticesaremodied usinganecombinationsoftheiroldneighborsaswell.Catm ull-Clarksubdivision, demonstratedinFigure 2-5 A,canhencebethoughtof(andwasoriginallyderived)asa generalizationofFigure 2-3 C,wherebothdirectionsofthemesharesubdividedinasingl e step. A B radial circular3 Figure2-5.A)Catmull-Clarksubdivisionsplitseveryquad directlyintofour,using specialrulesinthevicinityofextraordinaryvertices,li kethe3-valentoneson thecube.B)Bi-3polarsubdivisionrenesstrictlyinthera dialdirectionthe desirednumberoftimes(thricehere),andnallyinthecirc ulardirectionto achievethesamegranularityasCatmull-Clark. Theexactrenementweightsarenotrelevantinthisdiscuss ion,andareomitted. However,itisworthnotingthattheseweightsdependonlyon thelocalconnectivityofthe controlmesh.Theweightsaresaidtobe stationary .Additionally,theconnectivityisalso stationary ,inthatthelocalconnectivityaroundtheextraordinaryve rtexhasthesame 23

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structureasbefore{avalence n vertexsurroundedbyquadswithallothervalencesbeing 4.Subdivisionschemeswithstationaryweightsandconnect ivityarethemselvessaidtobe stationary Bi-3polarsubdivision[ KarciauskasandPeters 2007a ; Mylesetal. 2008 ]canbe thoughtofasgeneralizingFigure 2-3 (A-B),wheresubdivisionshappenstrictlyinone directionfollowedbytheotherone.Werefertothedirectio nalongthecontrollines emanatingradiallyfromthepolarvertexasthe radialdirection ,andtheperiodicdirection asthe circulardirection .Thelimitsurfaceisdenedinthiscasebyapplyingsubdivi sion intheradialdirectionadinnitum,followedbysubdividin ginthecirculardirection. However,forthepurposesofapproximatingthelimitsurfac ewiththemesh,wesubdivide onlyanitenumberoftimesintheradialdirectionbeforewe switchtothecircular directionasillustratedinFigure 2-5 B.Bi-3polarsubdivisionisstationary,andits subdivisionweightsareirrelevantforthisdiscussion.We willinsteaddetailandanalyzea slightlymorecomplexversionofthisalgorithminSection 3 ABC Figure2-6.EvenwhenthepolarcongurationA)isconvex,th eCatmull-Clark subdivisionsurfaceB)hasunseemlyripples,whilebi-3pol arsubdivisionC) yieldspredictablesurfaces. AsFigure 2-6 demonstrates,applyingCatmull-Clarktoaconvexpolarcon guration resultsinmacroscopicripplesthatareclearlynotspecie dbythecontrolnets.However, treatingpolarcongurationsspeciallyyieldspredictabl esurfaceswithgoodbehavior despiteahighvalencepolarvertex.Thisjustiesconsider ingthepolarcongurationasa separatecase. 24

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2.2.3Subdivisionaspiecewisepolynomials Whileuniformsplinescanbewrittenoutinclosedform,Catm ull-Clarkandbi-3 polarsubdivisionsurfacesarenotasstraightforward.AsF igure 2-7 demonstrates,at anyrenementlevel,thelimitsurfacenearextraordinaryv erticesandpolarvertices isnotdirectlyavailable.However,subdividingoncerevea lsaregularringofquads, whichundergouniformbi-3subdivisioninsubsequentrene ments.Thesurfacedened bythisringcanhencebewrittenoutinclosedform.Therefor e,nearextraordinary verticesandpolarvertices,thesurfaceconsistsofan innite sequenceof splinesurface rings approachingalimitpoint.IntheCatmull-Clarkcase,wecal lthislimitpointan extraordinarypoint ,whileinthepolarcase,itiscalleda pole A ? ? ? ? ? ? ? ? ? ? ... ? B ... ? ? ? ? ? ? ? ? ? ? ? ? ? Figure2-7.TheinnitesequenceofsplineringsofA)Catmul l-ClarkandB)bi-3polar subdivision.Eachsubdivisionrevealsadditionalringsof regularquads, representingbi-3polynomialpatches,aroundtheextraord inarypoint. ObservealsofromFigure 2-7 thattheboundaryofeachsplineringarounda Catmull-Clarkextraordinarypointcontainsasmanycorner sasthevalenceofthe extraordinaryvertex.Astheextraordinaryvalenceapproa chesinnity,theboundary consistsofcountablyinnitecorners.Ontheotherhand,in thepolarcase,theboundary issmooth:inaperfectlysymmetriccase,theboundaryappro achesacircleasthevalence approachesinnity.Weexploitthisbehaviorofthepolarco ngurationinthenew subdivisionalgorithmswedesign. 25

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2.2.4Behavioraroundandattheextraordinarypoints Sincetheinnitesequenceofsplineringsdeningthesurfa cenearextraordinary pointsandpolesarethemselves C 2 ,thesurfaceis C 2 awayfromthesevertices.The behaviorattheextraordinarypointsandpolesrequiresamo readvancedanalysisofthese splinesequences.Eventhoughsubdivisionsurfaceshavebe enaroundfor30years,only withinthelast15yearswerethemathematicaltoolsforhigh erorderanalysisdevelopedto maturity.Inalandmarkpaper, Reif [ 1995 ]setageneralframeworktoanalyzearbitrary subdivisionsurfacealgorithmsnearandattheextraordina rypoint.Subsequentwork byPrautzschandReifdevelopedsucientconditionsandpol ynomialdegreesfor C k continuitybyexaminingtheinnitesequenceofsplinering s[ PrautzschandReif 1999b a ]. Zorin [ 2000 ]insteadderivedconditionsbasedontheanalysisofcertai n \universal"surfacesthataredeterminedbythesubdivisio nschemeofinterest.Numerous papershaveanalyzedthebehaviorofsubdivisionsurfacesa roundextraordinarypoints [ PetersandUmlauf 2000 ; Sabinetal. 2003 ; PetersandReif 2004 ; Karciauskasetal. 2004 ; ReifandPeters 2005 ].Therecentbook[ PetersandReif 2008 ]summarizesand extendsthecoreresultsofthepapersaboveonthetheoryofs ubdivision. Stam [ 1998 ]derivedaconstant-timealgorithmfortheevaluationofpo intsand derivativesatparametervaluesarbitrarilyclosetotheex traordinarypoint,and Boier-MartinandZorin [ 2004 ]showedthatamorecanonicalparameterizationthan theoneusedbyStamwasneededtobeabletoalwayscomputethe derivativesatthe extraordinarypoint. Itisnowwell-knownthatCatmull-Clarksurfacescanhave unboundedcurvature nearextraordinarypointsofvalencenotequalto4eventhou ghtheyare C 1 [ PetersandUmlauf 2000 ].Ontheotherhand,bi-3polarsubdivision[ KarciauskasandPeters 2007a ; Mylesetal. 2008 ]wasderivedwithboundedcurvatureinmindandtendstogive predictableshapesinitsareasofapplicability.Thepurpo seofthisstudyistogobeyond boundedcurvatureto C 2 ,whilestillhavingasimplesubdivisionalgorithm. 26

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CHAPTER3 RADIALTAYLORSUBDIVISION(RTS) Inthischapter,wewillanalyze radialTaylorsubdivision ,aslightlymorecomplex variantofbi-3polarsubdivisionthatalsohasboundedcurv atureatthepole.Trivial modicationsofradialTaylorsubdivisionwillyieldsubdi visionalgorithmsthatare C 2 at thepoleaswell. 3.1NotationandLabeling Theunderlyingdatastructureonwhichweoperateisthe polarconguration which consistsofacentraltrianglefansurroundedbyringsoford inaryquads(seeFigure 3-1 ). Thecentralvertexofthetrianglefaniscalledthe polarvertex .The i -link ( i =0 ; 1 ; 2 ;::: ) ofapolarcongurationisthecircularchainofverticestha tis i edgesawayfromthepolar vertex.The0-linkconsistsofonlythepolarvertexitself. The i -ring ( i =0 ; 1 ; 2 ;::: ) consistsofalltheverticesthatarenomorethan i edgesawayfromthepolarvertex. Whilethesubdivisionalgorithmsusespecialrulesonlyint he1-ringofthepolarvertex, weassumeforanalysisthatapolarcongurationconstitute sthe5-ringofthepolarvertex. AsillustratedinFigure 3-1 ,thepolarcongurationisdenotedby q anditsvalence (i.e.thevalenceofitspolarvertex)is n q i :=[ q i; 0 ; q i; 1 ;:::; q i;n 1 ] T denotesthe i -link ofthepolarconguration,and q ij isthe j th controlpoint(rotatingcounter-clockwise, modulo n )onthis i -link.For j 2Zn ,the j -spoke isthevector q ;j :=[ q 0 ; 0 ; q 1 ; 0 ;:::; q 5 ; 0 ] T Countingthepolarvertex q 0 j = q 00 repeated, q has6 n vertices.Wecanenumerateall q 00 q 0 j q 1 j q 2 j q 3 j q 4 j q 5 j Figure3-1.A polarconguration consistsofatotalof6 n controlpointsdeningthe 5-ringofapolarvertex( q 00 ,whichiscounted n times)ofvalence n 27

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r r r r 0 0 1 1 1 2 2 4 4 ( G 0 q 0 )( r; ) ( G 1 q 1 )( r; ) Figure3-2.The m -timessubdividedpolarconguration q m denesacubicsplinering ( G m q m )( r; )( orange )via( 3{1 ).Theradialparametershrinksbyhalfafter eachsubdivisionsothat r 2 [2 ; 4]for G 0 q 0 ( r; )and r 2 [1 ; 2]for( G 1 q 1 )( r; ). theseverticesasthecolumnvector q :=[ q 0 ; 0 ; q 1 ; 0 ;:::; q 5 ; 0 ; q 0 ; 1 ; q 1 ; 1 ;:::; q 5 ; 1 ;:::; q 0 ;n 1 ; q 1 ;n 1 ;:::; q 5 ;n 1 ] T WewillextractthesplineringsdescribedinSection 2.2.3 insuchawaythateach controlpoint q ij hastheGrevilleabscissa j n inthecirculardirection.Thisallowsus tosimplifygeneralizationstoinnitevalencesandnon-st ationaryvalencebyusingthe valence-independentfractionalindexingnotationintrod ucedin( 2{3 ): q i; [ j n ] := q ij .Since j in q ij ismodulo n in q i; [ ] ismodulo1. Whileinpractice,eachmeshvertexliesinR3 ,ouranalysisisthesameasif q ij 2Rsincesubdivisionworksoneachcoordinateindependently. q m isthepolarconguration after m subdivisions.Omissionofthesuperscriptreferstotheini tialdata: q := q 0 n m is thevalenceof q m ,and n := n 0 Thelimitsurfaceofthesubdivisionprocedureisdenedbya ninnitesequenceof splineringsthatformadecompositionofthesurface(seeFi gures 2-7 and 3-2 ).Each splineringistheperiodicuniformsplinedenedbytheveo uterlinksof q m .More precisely, 28

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(radial) N ( m ) i ( r )isthe i th cubicB-splinebasiswithknots m [ 1 ; 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6], where := 1 2 ;and (circular)for n 3, N ( n ) j ( )isthe j th uniformperiodiccubicB-splinebasiswith knots 1 n Z n The splinering correspondingto q m isamap G m q m :[2 m ; 4 m ] R1 !R,denedby theB-splinecontrolpoints q mij ,with i 2f 1 ; 2 ; 3 ; 4 ; 5 g and j 2Zn suchthat ( G m q m )( r; ):= 5 X i =1 n X j q mij N ( m ) i ( r ) N ( n m ) j ( ) ; (3{1) where := 1 2 ,indicatingthattheradialparameterofeachsplineringsh rinksbyhalfafter everysubdivisionasillustratedinFigure 3-2 .As n m !1 ,the i -linkconvergestoa curve q mi; [ ] with 2R1 ,and G m q m simpliesto ( G m q m )( r; ):= 5 X i =1 ZR1 q mi; [ ] N ( m ) i ( r ) d: (3{2) Observealsothat G m q m islinearwithrespectto q m The polarlimitsurface x :[0 ; 4] R1 !R(R3 inpractice)inpolarparameterization isdenedpiecewiseintermsoftheseringssothat x ( r; ) r 2 [2 m ; 4 m ] :=( G m q m )( r; ) : (3{3) and x (0 ; )istheuniquelimitpoint,calledthe pole .Thedierencebetweenourpolar parameterizationandtheconventionaloneisthatourcircu lardirectionisparameterized byR1 insteadofR2 fornotationalconvenience.Theoperator L convertsapolar conguration q toitsparameterizedlimitsurface: L ( q ):= x Weproposethreealternativeconstructionsfor x inthisstudythatbuildupon eachother,andshowthatthelattertwoarecurvaturecontin uousatthepole.Toavoid ambiguity,wewillsuperscript x and L bythenameofthesubdivisionalgorithmthey represent{i.e. L RTS ( q ):= x RTS asdenedinSection 3.2 29

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Theoperator ^ G m isasimplerversionoftheoperator G m thatworksstrictlyinthe radialdirection.Forsomevector u 2R6 ^ G m u :[2 m ; 4 m ] !Risthecubicsplinedened bytheB-splinecontrolpoints u i ,with i 2f 1 ; 2 ; 3 ; 4 ; 5 g : ^ G m u ( r ):= 5 X i =1 u i N ( m ) i ( r ) : (3{4) Like G m ^ G m isalsolinearwithrespecttoitsparameter u Tosimplifynotation,weadditionallydene c :=cos(2 )and s :=sin(2 ) : Thefraction canalsoberepresentedasaratiosothat c j : n :=cos 2 j n and s j : n :=sin 2 j n : 3.2RadialTaylorSubdivision(RTS)Denition Denition1 (RadialTaylorsubdivision) RadialTaylorsubdivision ,or RTS ,renesan n -valentpolarconguration q m tothe n -valentpolarconguration q m +1 denedby q m +1 00 :=(1 a) q m00 +a n m n X h q m1 h (3{5) q m +1 1 j :=(1 ^b0 ) q m00 + n X hbh j q m1 h (3{6) q m +1 2 j :=cq m1 j +(1 c) q m2 j + n X hdh j q m1 h (3{7) q m +1 3 j := 1 2 q m1 j + 1 2 q m2 j q m +1 4 j := 1 8 q m1 j + 6 8 q m2 j + 1 8 q m3 j q m +1 5 j := 1 2 q m2 j + 1 2 q m3 j (3{8) wherebj := 1 n ^b0 + c j : n + 1 2 c 2 j : n + 1 8 c 3 j : n ; (3{9) ^b0 := 1 2 ;a:= ^b0 1 4 ;c:= 11 12 ;dj := 1 6 n c j : n : (3{10) 30

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...... ............ 1 a a=na=na=n 1 ^b0b0b1bn 1c+d0d1dn 1 1 c1 2 1 2 Figure3-3.RadialTaylorsubdivisionrules.Thesemaskssh owhowtocomputeeach vertex( )ontherenedmesh( lightgray,dashed )basedontheoldmesh( dark gray ,verticesas ).ThespecialradialsubdivisionrulesforRTSareisolated to thetrianglefanatthecenterofthepolarconguration.Fur therout,standard cubicrulesareapplied. Thelimitsurfaceisparameterizedby x RTS ,denedbythesplinerings G m q m denedin ( 3{3 ) ThesubdivisionrulesofRTSareillustratedinFigure 3-3 .Itfollowsbydenition that n m = n .Since q m +1 3 j q m +1 4 j q m +1 5 j arecomputedviauniformcubicsplinesubdivision, G m q m denesasubsetof x RTS .Therenementweightsforbi-3polarsubdivisionare identicaltothoseofRTS,exceptthatitusesuniformcubics ubdivisionfor q m +1 2 j .As wasthecaseforbi-3polarsubdivision[ KarciauskasandPeters 2007a ],weassume,that thepolarvalence n 5.Thisassumptionisnotapplicabletotheothertwosubdivi sion algorithms,RTS 1 and C 2 PS,thatwedenelater. RTScanmorecompactlybewrittenintermsofmatrixmultipli cation: q m +1 = A q m ; (3{11) where A isa6 n 6 n matrix.Sincethesubdivisionalgorithmisrotationallysy mmetric aroundthepolarvertex,ourenumerationofthecontrolpoin tsin q m allowsustowrite A 31

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inblockcirculantform,distributingevenlythecontribut ionofthepolarvertexamongst its n dierentlabels q m0 j j 2Zn A := 266666664 A 0 A 1 A n 1 A n 1 A 0 A n 2 ... ... A 1 A n 1 A 0 377777775 where A 0 := 2664 (1 a) =na=n 0000 (1 ^b0 ) =nb0 0000 0c+d0 1 c000 01 = 21 = 2000 01 = 83 = 41 = 800 001 = 21 = 200 3775 ;A j := 264 (1 a) =na=n 0000 (1 ^b0 ) =nbj 0000 0dj 0000 000000000000000000 375 ; for j 6 =0 : Onlytherst3 3blockofthesubdivisionmatrixhasnon-standardweights, whilethe restismerelytheapplicationofuniformcubicsplinesubdi visionintheradialdirection. Nevertheless,theentire6 6matrixisrequiredtodenethesplineringsforanalysis. 3.3Analysis Usingtoolssummarizedin[ PetersandReif 2008 ],weanalyzeRTSthesubdivision limitbyexaminingthelimit lim m !1 G m ( q m )=lim m !1 G m ( A m q )(3{12) ofthesequenceofsplineringsdening x RTS nearthe pole .Section 3.3.1 examinesthe spectrumof A thatmotivatesthechoiceofitsentries.Section 3.3.2 thenreformulatesRTS ineigenspacetoderive,inSection 3.3.3 ,anexpansionofthedominanttermsatthepole andconcludethatinthelimit n !1 ,thelimitsurfaceis C 2 atthepole. 3.3.1SpectralanalysisofRTS Thesubdivisionalgorithm(notnecessarilythesurface)is rotationallysymmetricand periodicsuchthat q mi;j + n = q mij .ThissuggeststhataFouriertransforminthecircular directionmayfactorouttheradialandcircularbehaviorof thesubdivisionalgorithm toaidouranalysis.Inotherwords,sincethesubdivisionma trix A isblockcirculant 32

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duetotherotationalsymmetryofRTS,wecandiagonalizeitu singthediscreteFourier transform.ThecomplexFourierblockmatrix F :=( jk n I ) j;k 2Zn = 266666666664 IIIII 1 n I 2 n I 1 n I I 2 n I 4 n I 2 n I ... ... ... ... I 1 n I 2 n I 1 n I 377777777775 (3{13) where :=exp 2 p 1 n and I isthe6 6identitymatrix.Itcaneasilybeveriedthat F isalmostorthogonal: F 1 = 1 n F = 1 n ( jk n I ) j;k 2Zn ; where F istheHermitianadjoint(conjugatetranspose)of F .Animportantproperty thatisexploitedlateristhatfor k> 0,the k th and n k th blockcolumnsof F 1 are complexconjugatesofeachother.Wecannowblockdiagonali ze A via ^ A := F A F 1 = 266666664 ^ A 0 0 0 0 ^ A 1 0 ... ... 0 0 ^ A n 1 377777775 ; ^ A k := n X j jk A j ; where ^ A 0 = 2664 1 a a0000 1 ^b0 ^b0 0000 0c+ ^d0 1 c000 01 = 21 = 2000 01 = 83 = 41 = 800 001 = 21 = 200 3775 ; ^ A k = ^ A n k = 2664 0000000 ^bk 0000 0c+ ^dk 1 c000 01 = 21 = 2000 01 = 83 = 41 = 800 001 = 21 = 200 3775 ^bk := n X j jkbj ; ^dk := n X j jkdj : Notethatthisre-denitionof ^b0 isconsistentwithitsusageinthedenition( 3{9 )ofbj Consequently, 33

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^bk = 8>>>>>>>>>><>>>>>>>>>>: 1 2 if k =0 1 2 if k 2f 1 ;n 1 g 1 4 if k 2f 2 ;n 2 g 1 16 if k 2f 3 ;n 3 g 0otherwise and ^dk = 8><>: 1 12 if k 2f 1 ;n 1 g 0otherwise ^ A k iscalledthe k th Fourierblock of ^ A andencodestheactionofRTSonthe k th frequency modewhengoingaroundthepolarvertex.Forinstance, ^ A 0 alonedeterminestheeect ofRTSwheneachcontrolpoint q ij isindependentofthecircularindex j .Thisincludes polarcongurationssampledfromaconstantfunctionorapa rabola. Sincetheeigenvaluesandeigenvectorsof A and ^ A arecloselyrelatedbytheFourier transform,weuseasimilarnotationforspectralanalysis. ` 0 ;` 1 ;:::;` 6 n 1 aretheeigenvaluesofthesubdivisionmatrix A (andalso ^ A )in non-increasingorderofmagnitude: j ` 0 jj ` 1 j ::: j ` 6 n 1 j .Equaleigenvaluesare listedonceforeachmultiplicityandtreatedseparately. k istheindexofFourierblock ^ A k contributingeigenvalue ` k ,chosensothat ` k 1 = ` k 2 and k 1
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Table3-1.Thedominantspectralbehaviorof ^ A .Theleft( ^ w k )andright( ^ v k )eigenvectors arenormalizedsothattherelatedvectors w k and v k satisfy w T k 1 v k 2 = k 1 k 2 Fouriereigenk block( k )value( ` k )vector(right)( ^ v T k )vector(left)( ^ w T k ) 001[1 ; 1 ; 1 ; 1 ; 1 ; 1] 1 3 [2 ; 1 ; 0 ; 0 ; 0 ; 0] 111 = 2[0 ; 1 ; 2 ; 3 ; 4 ; 5][0 ; 2 ; 0 ; 0 ; 0 ; 0] 2 n 11 = 2[0 ; 1 ; 2 ; 3 ; 4 ; 5][0 ; 2 ; 0 ; 0 ; 0 ; 0] 301 = 4 1 3 [ 1 ; 2 ; 11 ; 26 ; 47 ; 74][ 1 ; 1 ; 0 ; 0 ; 0 ; 0] 421 = 4 1 3 [0 ; 2 ; 11 ; 26 ; 47 ; 74][0 ; 3 ; 0 ; 0 ; 0 ; 0] 5 n 21 = 4 1 3 [0 ; 2 ; 11 ; 26 ; 47 ; 74][0 ; 3 ; 0 ; 0 ; 0 ; 0] with := 1 2 and := 2 = 1 4 .Therestoftheeigenvaluesarerealandpositivewith magnitudestrictlylessthan ` 5 .Theeigenvalues ` 0 =1and ` 3 = arefrom ^ A 0 ; ` 1 = ` 2 = arefrom ^ A 1 and ^ A n 1 ;andthenaltwo ` 4 = ` 5 = arefrom ^ A 2 and ^ A n 2 .InorderfortheseveimportantFourierblockstoexist,th evalence n mustbeat least5,justifyingthisassumption. Since A isanoperationonpolarcongurations,theeigenvectors v k of A arealso polarcongurations.The eigenspline e RTSk :[0 ; 4] R1 !Risthelimitsurface L ( v k )of thesepolarcongurations.The radialeigenspline ^ e k :[0 ; 4] !Risthelimitcurveofradial eigenvector ^ v k asdenedbythedecomposition ^ e k ( r ) r 2 [2 m ; 4 m ] := ^ G m ^ A m k ^ v k ( r ) : (3{14) Figure 3-4 illustratestherelationship,forexample,between v 3 ^ v 3 e RTS3 ,and^ e 3 Let:=diag( ` 0 ;` 1 ;:::;` 6 n 1 )beadiagonalmatrixoftheeigenvaluesof A (respectively ^ A ),and ^ V beamatrixwhosecolumnsenumeratethecorrespondingright eigenvectors(ofanyscale)of ^ A ,sothat ^ A ^ V = ^ V .Then, ^ A ^ V = ^ V )F A F 1 ^ V = ^ V ) A ( F 1 ^ V )=( F 1 ^ V ) | {z } V C := ) AV C = V C ; (3{15) 35

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A. n =6B. n !1 Figure3-4.Eigenvector v 3 isthe n -valentpolarconguration( blackmesh/curves ) deningtheeigenspline e RTS3 ( blueandredsurface ).Thecorrespondingradial eigenvector ^ v 3 ( indicatedby )denestheradialeigenspline^ e 3 ( blueandred curve ).When n !1 ,then^ e 3 ( r )= e RTS3 ( r; 0). implyingthatthecolumnsof V C are(complex)eigenvectorsof A .SincetheFourier blocks ^ A k and ^ A n k areidenticalandthecorrespondingpairsofblockcolumnso f F 1 are complexconjugate,eigenvaluesfromtheseFourierblocksa reassociatedvia V C = F 1 ^ V withpairsofcomplexconjugateeigenvectorsof A .Therealeigenvectors v k of A canhence becomputedastherealandimaginaryportionsofthesecompl exeigenvectors: ( v k ) ij :=( ^ v k ) i op k j n ; op k ( ):= 8><>: c k if k n= 2 s k otherwise (3{16) where k 2Z6 n i 2Z6 ,and j 2Zn .AccordingtoTable 3-1 ,(op 0 ( ) ; op 1 ( ) ;:::; op 5 ( ))= (1 ;c ;s ; 1 ;c 2 ;s 2 ) : Theeigenspline e RTSk inheritsthetensorednatureof v k Lemma1. Forany f :R!R,theoperator B n : f 7! P nj f j n N ( n ) j usesuniform samplesonafunction f asthecontrolpointsofaperiodicspline.Then,theeigensp line e RTSk decomposesaccordingto e RTSk ( r; )=^ e k ( r )B n op k ( )(3{17) 36

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Proof. For r 2 [2 m ; 4 m ], e RTSk ( r; ) ( 3{3 ) =( G m v m )( r; ) ( 3{1 ) = ( 3{11 ) 5 X i =1 n X j ( A m v ) ij N ( m ) i ( r ) N ( n ) j ( ) = 5 X i =1 n X j ` mk v ij N ( m ) i ( r ) N ( n ) j ( ) ( 3{16 ) = 5 X i =1 n X j ` mk ( ^ v k ) i op k j n N ( m ) i ( r ) N ( n ) j ( ) = 5 X i =1 n X j ^ A m ( ^ v k ) i op k j n N ( m ) i ( r ) N ( n ) j ( ) = 5 X i =1 ^ A m ( ^ v k ) i N ( m ) i ( r ) n X j op k j n N ( n ) j ( ) ( 3{4 ) = ^ G m ^ A m ( ^ v k ) ( r )B n op k ( ) ( 3{14 ) =^ e k ( r )B n op k ( ) Lemma 1 showsthattheblockdiagonalizationfactorstheradialfro mthecircular. Fromthisformulation,itisalsoobviousthat e RTSk isperiodicin withaperiodof 1 k whichisadirectconsequenceof ^ v k havingfrequencymode k .Eigensplinesandradial eigensplinesalsoinheritthescalingpropertyofeigenvec tors,inthatfor r = r ^ e k ( r ) r 2 [2 m ; 4 m ] =^ e k ( r ) r 2 [2 m +1 ; 4 m +1 ] ( 3{14 ) = ^ G m +1 ^ A m +1 k ^ v ( r )= ` k ^ G m +1 ^ A m k ^ v ( r ) = ` k ^ G m ^ A m k ^ v ( r ) ( 3{14 ) = ` k ^ e k ( r ) r 2 [2 m ; 4 m ] ; (3{18) implyingthat^ e k ( r )= ` k ^ e k ( r ),and,duetoLemma 1 ,that e RTSk ( r; )= ` k e RTSk ( r; ). Withthisscalingrelationship,therstsixradialeigensp linescanbewrittenout explicitly.Thesubdivisionmatrix A wasconstructedtohavethespectralbehaviorin 37

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Table 3-1 inordertosatisfythefollowingkeylemma.Forsuccinctnes sandclarity,we usethenotationB n c for(B n ( r 7! c r ))( ),wheretheoperatorB n isappliedtothe function r 7! c r :=cos(2 r ),andtheresultingsplineisevaluatedat .Similarly, (B n ( r 7! s r ))( )isshortenedtoB n s Lemma2 (ReproductionofRadialTaylorBasisFunctions) For r 2 [0 ; 4] ^ e 0 ( r )=1(3{19) ^ e 1 ( r )=^ e 2 ( r )= r (3{20) ^ e 3 ( r )=^ e 4 ( r )=^ e 5 ( r )= r 2 (3{21) ^ e k ( r )=o( r 2 ) as r 0 for k> 5(3{22) whichimply,byLemma 1 ,that e RTS0 ( r; )=1 ;e RTS1 ( r; )= r B n c ;e RTS2 ( r; )= r B n s ; e RTS3 ( r; )= r 2 ;e RTS4 ( r; )= r 2 B n c 2 ;e RTS5 ( r; )= r 2 B n s 2 e RTSk ( r; )=o( r 2 ) as r 0 for k> 5 : Proof. ( 3{19 )followssince ^ A 0 ^ v 0 = ^ v 0 and ^ G 0 ^ v 0 ( r )=1.Wecannowverify,for k 2f 1 ; 2 g ,that^ e k ( r ) j r 2 [2 ; 4] = ^ G 0 ^ v k ( r )= r j r 2 [2 ; 4] byB-spline-to-power-seriesconversion ( 2{1 ).Theadditionalpropertyfrom( 3{18 )that^ e k ( 1 2 r )= 1 2 ^ e k ( r )implies( 3{20 ).Similarly, for k 2f 3 ; 4 ; 5 g ,B-spline-to-power-seriesconversionshowsthat^ e k ( r ) j r 2 [2 ; 4] = r 2 j r 2 [2 ; 4] Hence,^ e k ( 1 2 r )= 1 4 ^ e k ( r )implies( 3{21 ). When k> 5,using r := r m ^ e k ( r ) r 2 [2 m ; 4 m ] =^ e k ( m r ) r 2 [2 ; 4] ( 3{18 ) = ` mk |{z} o( m ) ^ e k ( r ) r 2 [2 ; 4] =o( m ) r 2 [2 ; 4] =o(( m r ) 2 ) r 2 [2 ; 4] =o( r 2 ) r 2 [2 m ; 4 m ] ; proving( 3{22 ). 38

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Lemma 2 isusedtoderiveaquadraticTaylorexpansiontoshowsecond -order continuityfortwoofourproposedsubdivisionalgorithms.3.3.2ReformulatingRTSineigenspace Deningtherealmatrix V R ashaving k th column v k ,wehave(asin( 3{15 )), AV R = V R ) V 1 R AV R = ) V 1 R A = V 1 R ; (3{23) implyingthattherowsof V 1 R arelefteigenvectors w k of A .Wechoosenormalization sothat w T k 1 v k 2 = k 1 k 2 .Multiplicationwith V 1 R projectsthepolarcongurationinto eigenspaceusingtheselefteigenvectors w k .Precisely, q (respectively, x RTS )canbewritten asalinearcombinationofthe6 n righteigenvectors v k (respectively,eigensplines^ e )as follows q = V R |{z} columns v k ( V 1 R |{z} rows w k q ) ) q ij = 6 n X k p k ( v k ) ij (3{24) )L RTS ( q )= 6 n X k p k L RTS ( v k ) ) x RTS ( r; )= 6 n X k p k e RTSk ( r; ) ; (3{25) whereeach eigencoecient p k istheinnerproductof w k and q .Aswasthecasefor v k w k iscomputedfrom ^ w k (listedinTable 3-1 )usingtheinverseFouriertransformsothat p k := w T k q = 6 X i n X j 1 n ( ^ w k ) i op k j n | {z } ( w k ) ij q ij (3{26) Specically,for k 2Z6 p 0 := 2 3 q 00 + 1 3 n n X j q 1 j ; p 3 := q 00 + 1 n n X j q 1 j ; (3{27) p 1 := 2 n n X j c j : n q 1 j ; p 4 := 3 n n X j c 2 j : n q 1 j ; p 2 := 2 n n X j s j : n q 1 j ; p 5 := 3 n n X j s 2 j : n q 1 j 39

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Theeigencoecient p mk := w T k q m of q m simpliesto p mk = w T k A m q =( w T k A m ) q = ` mk w T k q = ` mk p k .Therenementequations( 3{5 ){( 3{8 )ofRTScannowberewrittenin termsof p k Lemma3. The m -timesrenedmesh q m isdenedbyeigencoecientsandpowersof eigenvaluesasfollows. q m +1 00 = p 0 m +1 3 p 3 (3{28) q m +1 1 j = p 0 + m +1 ( p 1 c j : n + p 2 s j : n )+ 2 m +1 3 ( p 3 + p 4 c 2 j : n + p 5 s 2 j : n )(3{29) q m +1 2 j = p 0 +2 m +1 ( p 1 c j : n + p 2 s j : n )+ 11 m +1 3 ( p 3 + p 4 c 2 j : n + p 5 s 2 j : n )(3{30) q m +1 3 j = p 0 +3 m +1 ( p 1 c j : n + p 2 s j : n )+ 26 m +1 3 ( p 3 + p 4 c 2 j : n + p 5 s 2 j : n )(3{31) q m +1 4 j = p 0 +4 m +1 ( p 1 c j : n + p 2 s j : n )+ 47 m +1 3 ( p 3 + p 4 c 2 j : n + p 5 s 2 j : n )(3{32) q m +1 5 j = p 0 +5 m +1 ( p 1 c j : n + p 2 s j : n )+ 74 m +1 3 ( p 3 + p 4 c 2 j : n + p 5 s 2 j : n )(3{33) Proof. q m +1 00 = 3 4 q m00 + 1 4 n m G q m X h q m1 ; [ h ] ( 3{27 ) = p m0 1 12 p m3 = p 0 m 12 p 3 = p 0 m +1 3 p 3 q m +1 1 j = 1 2 q m00 + 1 n n X h 1 2 + c h j : n + 1 2 c 2( h j ): n q m1 h additionrule forcosine&( 3{27 ) = p m0 + 1 2 ( p m1 c j : n + p m2 s j : n )+ 1 6 ( p m3 + p m4 c 2 j : n + p m5 s 2 j : n ) = p 0 + m +1 ( p 1 c j : n + p 2 s j : n )+ 2 m +1 3 ( p 3 + p 4 c 2 j : n + p 5 s 2 j : n ) Theprooffor( 3{30 ){( 3{33 )issimilartothatof( 3{29 ). Withtheradialeigenvectors ^ v 0 =[1 ; 1 ; 1 ; 1 ; 1 ; 1], ^ v 1 = ^ v 2 =[0 ; 1 ; 2 ; 3 ; 4 ; 5], ^ v 3 = 1 3 [0 ; 2 ; 11 ; 26 ; 47 ; 74],and ^ v 4 = ^ v 5 = 1 3 [0 ; 2 ; 11 ; 26 ; 47 ; 74],( 3{28 ){( 3{33 )canbe writtenas 40

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q m +1 ij = p 0 +( ^ v 1 ) i m +1 ( p 1 c j : n + p 2 s j : n ) + m +1 (( ^ v 3 ) i p 3 +( ^ v 4 ) i ( p 4 c 2 j : n + p 5 s 2 j : n )) ) q m +1 i; [ ] = p 0 +( ^ v 1 ) i m +1 ( p 1 c + p 2 s ) + m +1 (( ^ v 3 ) i p 3 +( ^ v 4 ) i ( p 4 c 2 + p 5 s 2 ))(3{34) for i 2Z6 .Inparticular, lim m !1 q mij = p 0 ) lim m !1 ( G m q m )( r; )= p 0 ; convergingtoauniquepoint p 0 ,showingthat x RTS ( r; )is C 0 atthepole. 3.3.3Eigenspaceexpansionandcurvaturecontinuity C 0 and C 1 continuitycanbeseenmoreexplicitlybyexpressing x RTS usingthe eigenprojection. x RTS ( r; ) ( 3{25 ) = 6 n 0 X k p k e RTSk ( r; ) Lemma 2 = p 0 e RTS0 ( r; )+ p 1 e RTS1 ( r; )+ p 2 e RTS2 ( r; ) + p 3 e RTS3 ( r; )+ p 4 e RTS4 ( r; )+ p 5 e RTS5 ( r; ) +o r 2 (3{35) Lemma 2 = p 0 +( p 1 r B n c + p 2 r B n s )+ p 3 r 2 + p 4 r 2 B n c 2 + p 5 r 2 B n s 2 +o r 2 ThisexpansionusingtheeigensplinesisalmostaTaylorexp ansion.[ PetersandReif 2008 ,Section5.2]showsthatthe characteristicspline ( r; ):=( e RTS1 ( r; ) ;e RTS2 ( r; ))= r (B n c ; B n s )istheonlyreparameterization,uptolineartransformati on,of x RTS that canreproducealinearTaylorexpansionatthepole.Figures 2-7 B,and 3-2 illustratethe splineringsof ,whichareregularandinjective,validatingthereparamet erization.Using ( x;y ):= ( r; ),thereparameterizedsurface x RTS ( x;y ):= x RTS ( r; )hastherst-order Taylorexpansion 41

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x RTS ( x;y )= p 0 +( p 1 x + p 2 y )+o( r ) r 2 [2 m ; 4 m ] andistherefore C 1 atthepole.Bytheconditionsin[ PetersandReif 2008 ,Section7.1], theadditionalpropertythat( ` 0 ;` 1 ;` 2 ;` 3 ;` 4 ;` 5 )=(1 ;;; 2 ; 2 ; 2 )impliesthatthelimit surfacealsohasboundedcurvature. Inordertotakethisastepfurtherto C 2 continuity,weneedeigensplines e RTS3 e RTS4 e RTS5 tobequadraticwithrespecttothereparameterization toinduceasecond-order Taylorexpansion.Preferably,theseeigensplinesshouldn otbezero,sincethatwould resultinazerosecondderivativeandavisiblerat-spotint hevicinityofthepole. ( r; )= r (B n c ; B n s )isdegree1in r anddegree3in ,whichistheminimumdegree neededtocreate C 2 splineringsaroundthepole.Consequently, e RTS3 e RTS4 ,and e RTS5 need degree(2 ; 6)tobequadraticin .Thisimpliesthatitisimpossibletocreateastationary C 2 subdivisionforpolarcongurationsbasedonuniformsplin eswithdegreelessthan6in thecirculardirection. However,inthelimit n !1 ,thesurfacearoundthepoleisnolongerasplineinthe circulardirection,butanarbitrarycurve q i; [ ] for 2R1 (Figure 3-4 B).Denotethiscase asRTS 1 ,with q i; [ ] beingthe controlcurves ofthissubdivisionalgorithm.Wenowshow thatanon-trivialsecond-orderTaylorexpansionexistsat thepoleforRTS 1 Lemma4. e RTSk ( r; ) ^ e k ( r )op k ( )=O 1 n 2 ,implyingthat e RTS 1 k ( r; ):=lim n !1 e RTSk ( r; )=^ e k ( r )op k ( ) Proof. Since^ e k ( r )isindependentofvalence,and e RTSk ( r; ) ^ e k ( r )op k ( )=^ e k ( r )B n op k ( ) ^ e k ( r )op k ( ) =^ e k ( r )(B n op k ( ) op k ( )) ; weneedonlyexaminethesplineapproximationofop k ( )viaB n .Weshowthat 1.thedistancebetweenB n op k ( )anditscontrolpolygonisO 1 n 2 ,andthat 42

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2.op k ( )anditslinearinterpolant(i.e.thecontrolpolygonofB n op k ( ))isO 1 n 2 Together,thesestatementsimplybythetriangleinequalit ythat e RTSk ( r; ) ^ e k ( r )op k ( )= O( 1 n 2 ),provingthelemma. Step1. [ LutterkortandPeters 2001 ]showedthatforuniformcubicsplineswithcontrol points[ q ij ] j 2Z n ,thedistancebetweenthecontrolpolygonandthesplineisp roportionalto theseconddierencesofthecontrolpoints: 1 6 max j fj q i;j 1 2 q ij + q i;j +1 jg .Inthecontext ofthislemma, q ij =op k j n ,and 1 6 max j fj q i;j 1 2 q ij + q i;j +1 jg = 1 6 max j 8<: op k j 1 n +op k j +1 n | {z } 2op k j n 9=; = 1 6 max j 2op k j n c k : n 2op k j n = 1 3 max j 8>>><>>>: op k j n | {z } 1 j ( c k : n 1) j 9>>>=>>>; 1 3 max j f 1 c k : n g Taylor = expan. O 2 k n 2 =O 1 n 2 Step2. Foranarbitraryfunction f :[ a;b ] 2R!R,aTaylorexpansionat a showsthat apiecewiselinearinterpolantwithdistance 1 n betweenbreakpointsapproximates f witha deviationofO 1 n 2 max [ a;b ] f f 00 g .Consequently,thepiecewiselinearinterpolanttoop k ( ) convergesO 2k n 2 =O 1 n 2 Theorem1. Inthelimit n !1 ,RTSis C 2 atthepole. Proof. Assuming n !1 andcontinuingfrom( 3{35 ), x RTS 1 ( r; ) r 2 [2 m ; 4 m ] Lemma 2 & Lemma 4 = p 0 +( p 1 rc + p 2 rs )+ p 3 r 2 + p 4 r 2 c 2 + p 5 r 2 s 2 +o r 2 43

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ChangingfrompolartoCartesiancoordinates( x;y ):=( rc ;rs ), x RTS 1 ( x;y ):= x RTS 1 ( r; )revealsthefollowingsecond-orderTaylorexpansionatth epole x RTS 1 ( x;y )= p 0 +( p 1 x + p 2 y )+ p 3 ( x 2 + y 2 )+ p 4 ( x 2 y 2 )+ p 5 (2 xy ) +o x 2 + y 2 ; proving C 2 continuityatthepole. Nevertheless,curvaturecontinuitycomesatacost:wearen olongerpolynomialinthe circulardirection.IntheChapter 4 ,weadapttheintuitiondevelopedsofartocreatea C 2 bi-3subdivisionalgorithmthatovercomesthesedisadvant ages. 3.4ApproximationviaMeshRenement Meshrenementiseasiesttodemonstrateonacontrolmeshwi thlatitude-longitude connectivityoftheearthasinFigures 2-5 Band 2-6 .Sucha sphericalmesh consists entirelyofordinaryquadsandexactlytwopolarcongurati ons.Sphericalmesheshave preciselytwodirections:A)radial,orlongitudinal,corr espondingtothe j -spokesof thepolarcongurations;andB)circular,orlatitudinal,c orrespondingtothe i -linksof thepolarcongurations.Eachradialsequenceofcontrolpo intsofthesphericalmeshis similarlycalleda spoke ,whileeach(periodic)circularsequenceisa link .Wecanperform radialsubdivision alongthespokesofasphericalmeshbyusingthespecialRTSr ulesof Denition 1 andFigure 3-3 inthevicinityofpolarvertices,whileusingunivariatecu bic renement( 2{4 )awayfromthem.Wecanalsodoublethevalenceofeachpolarv ertexby performing circularsubdivision alongeachlinkusingunivariatecubicrenement. TheRTSlimitsurfaceisdenedbycontinuallyapplyingradi alsubdivisionand interpretinglinkssucientlyfarawayfromthepolarverte xasthecontrolpointsofa uniformbi-3spline,implyingthatcircularsubdivisionma ybeappliedontheselinks. Consequently,theRTSlimitsurfaceofasphericalmeshcanb ecomputedalaFigure 2-2 (A-B)byapplyingradialsubdivisionadinnitumfollowedb ycircularsubdivisionad innitum.An m -timessubdividedapproximationthislimitsurfacecanhen cebecomputed 44

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bysubdividing m timesintheradialdirectionfollowedby m -timessubdividinginthe circulardirection,asdemonstratedinFigure 2-5 B. Ontheotherhand,thecurvature-continuousvariationRTS 1 requireseachpolar vertextohaveinnitevalencebeforeradialsubdivisionis everapplied.Thisisaccomplished byinterpretingeachlinktobethecontrolpointsofacubics plinewhichactsasacontrol curveofRTS 1 .Thecorrespondinglimitsurfacecanbecomputedbyapplyin gcircular subdivisionadinnitum(convergingtothecontrolcurves) followedbyradialsubdivision. Asaresult,an m -timesrenedapproximationiscomputedbysubdividing m timesinthe circulardirectionfollowedby m -timesintheradialdirection. A B 1 B 2 C D Figure3-5.CombiningCatmull-ClarkandRTS.A)Separating theinputmesh.B) SubdividingthepolarcongurationB 1 )radiallythenB 2 )circularlyfor boundedcurvature( redarrows ), OR B 1 )circularlythenB 2 )radiallyfor curvaturecontinuity( bluearrows ).C)Subdividingtheremainderusing Catmull-Clark.D)Joiningtherenedmeshesafterremovalo foverlapping facets. EithermeshrenementtechniquecanbecombinedwithCatmul l-Clarksubdivisionto beapplicabletoarbitraryquadmeshesaugmentedwithpolar congurations(seeFigure 3-5 ). 1. Splitopolarcongurations: Copyallpolar3-ringsandremoveeachpolarvertex fromtheinputmesh. 45

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2. Subdividepolarcongurations: Foreachpolarconguration, (a)subdivide m timesradially,andthen (b)subdivide m timesinthecirculardirection. 3. Subdividetheremainingmesh: Apply m stepsofCatmull-Clarksubdivisiontothe meshwithoutthepolarvertices. 4. Mergeresults: Droptheboundaryfacetsofthemeshessubdividedinsteps2a nd3 andjointhembyidentifyingtheresultinggeometricallyid enticalboundaryvertices. Notethatthe2-and3-linksarecopiedwiththepolarvertex, butnotremovedfrom therestofthemesh(Figure 3-5 A),andbothCatmull-Clarkandpolarsubdivisionrene thesecommonlinksusinguniformbi-3subdivisionrules.Th etransitionbetweenthe Catmull-Clarkandpolarlimitsurfacesistherefore C 2 Thedisadvantageofsucharenementschemeisthatitisnoti terative.Wecannot takethealready-renedmeshandapplyRTSradialsubdivisi ontoconvergetothesame limitsurface.Toavoidseparationofthepolarconguratio nfromtherestofthesurface, itwouldbefarbetterifthesubdivisionalgorithmoeredas imultaneousradial/circular meshrenementalgorithminthespiritofCatmull-Clarkand Figure 2-3 C.Chapter 4 describessuchanalgorithm. 46

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CHAPTER4 C 2 POLARSUBDIVISION( C 2 PS) 4.1Semi-StationarySubdivision RTScanbeadaptedtonon-stationaryconnectivitywhilekee pingtheweights dependentonlyontheconnectivityofthemesh.Inparticula r,thevalenceofthepolar vertexdoublesaftereverysubdivision.Wearenolongerlim itedbystationarysubdivision theory,whichrequiresdegree6inthecirculardirectionfo rsecond-ordercontinuity,as shownpreviously.Denition2. Denoteby C 2 polarsubdivision( C 2 PS) thealgorithmthatsubdividesan n m -valentpolarconguration q m toan 2 n m -valentpolarconguration q m +1 via(seeFigure 4-1 ) q m +1 00 :=(1 a) q m00 +a n m G q m X q m1 ; [ ] = 3 4 q m00 + 1 4 n m G q m X q m1 ; [ ] (4{1) q m +1 1 ; [ ] :=(1 ^b0 ) q m00 + 1 n m G q m X ^b0 + c + 1 2 c 2( ) q m1 ; [ ] = 1 2 q m00 + 1 n m G q m X 1 2 + c + 1 2 c 2( ) q m1 ; [ ] (4{2) q m +1 2 ; [ ] :=c~ q m1 ; [ ] +(1 c) ~ q m2 ; [ ] + 2 ^d0 n m G q m X c q m1 ; [ ] = 11 12 ~ q m1 ; [ ] + 1 12 ~ q m2 ; [ ] 1 6 n m G q m X c q m1 ; [ ] (4{3) q m +1 3 ; [ ] := 1 2 ~ q m1 ; [ ] + 1 2 ~ q m2 ; [ ] q m +1 4 ; [ ] := 1 8 ~ q m1 ; [ ] + 6 8 ~ q m2 ; [ ] + 1 8 ~ q m3 ; [ ] q m +1 5 ; [ ] := 1 2 ~ q m2 ; [ ] + 1 2 ~ q m3 ; [ ] (4{4) where ~ q m isobtainedaftersubdividing q m onceinthecirculardirection.Observethat q m +1 3 ; [ ] q m +1 4 ; [ ] q m +1 5 ; [ ] arecomputedviauniformbi-3splinesubdivision.Lettheop erator T denoteasingleapplicationof C 2 PS.Thelimitsurface x C 2 PS istheunionofsplinerings G m q m where q m := T m ( q 0 ) 47

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...... ...... a a a a 1 ^b0bn 1 2b1 2bn 3 2c dn 1 2d1 2dn 3 2 1 c1 2 1 2 Figure4-1. C 2 polarsubdivisionrules.TherulesarethesameasRTS(Figur e 3-3 ),except thattheouter i -linksrequireintermediateuniformcubiccircularsubdiv ision (intermediateverticesindicatedby 2 ).Therenedmesh( and dashedlines ) iscomputedasbeforefromtheoldmesh( solidlines )withbj anddj computedvia( 3{9 )and( 3{10 )usinghalf-integerindices.AsinRTS,bi-3rules areappliedawayfromthepolarvertex. Since C 2 PSsubdividesinboththeradialandcirculardirectionssim ultaneously,it isdirectlycompatiblewithCatmull-Clark,requiringnome shseparationforrenement: everyquadonthecoarsemeshyieldsfouraftersubdivision, andeachpolartrianglesplits intotwopolartrianglesandtwoquads,asillustratedinFig ure 4-1 .Additionally,the limitsurfaceisbi-3andcanbecomputedasaclosed-formexp ression,whichisdicultto doforRTSonaninnite-valentvertex.Eachsubsequentspli neringhastwiceasmany polynomialpatchesandcontrolpointsasitspredecessor,a ndthisexponentially-increasing orderofapproximationenablesthesplineringsequencetoc onvergetoasecond-order Taylorexpansionatthepole. 4.2Analysis Sincetheconnectivityisnolongerstationaryatthepolarv ertex,thetraditional methodofspectralanalysisdoesnotdirectlyapply.Howeve r,ifwerewrite( 4{1 ){( 4{4 ) ineigenspaceaswedidRTSinSection 3.3.2 ,wecanemployasimilaranalysistechnique. 48

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WhathelpsistheintuitionfromRTSthatthesecond-orderex pansionatthepoleis determinedbytheeigensplines e k multipliedbyeigencoecients p k denedin( 3{27 ). The eigencoecient p mk of C 2 PS isalsocomputedvia( 3{27 )on q m ,andwe abbreviate p k := p 0k .The eigensplinesof C 2 PS arethelimitsurface e C 2 PS k := L C 2 PS ( v k ) where v k isaneigenvectorofRTSwithpolarvalence n 0 .AsuperscriptofRTS n m disambiguatestheeigenspline e RTS n m k ofRTSonavalence n m polarconguration. Thefollowingsubsectionswillshowthefollowing. AsinRTS, p k 2Z6 ispreservedaftereveryapplicationof T {i.e. p m +1 k = ` k p mk = ` m +1 k p k (Lemma 5 ). T canbeapproximatedintermsof p k 2Z6 plusadeviationofO 1 8 m forpolar valence n m (Lemma 6 ). For k 2Z6 n e C 2 PS k convergesto e RTS 1 k attherateofO 1 8 m atthepole(Lemma 7 ). Thestatementsaboveyieldasecond-orderTaylorexpansion of x C 2 PS atthepole provingthatitis C 2 (Theorem 2 ). 4.2.1Preservationofeigencoecients Thefollowingsimplicationscanbeshownbyusingtheaddit ionruleforsineand cosine,andtheorthogonalityofthediscreteFourierbasis 1 2 n 2 n X j c a 1 ( g j ):2 n 1 n n X h c a 2 ( h g 2 ): n q ih = 8>>>><>>>>: 1 n P nh q ih if a 1 = a 2 =0 1 2 n P nh c a 1 ( h j 2 ): n q ih if a 1 = a 2 6 =0 0otherwise 1 2 n 2 n X j s a 1 ( g j ):2 n 1 n n X h c a 2 ( h g 2 ): n q ih = 8><>: 1 2 n P nh s a 1 ( h j 2 ): n q ih if a 1 = a 2 6 =0 0otherwise (4{5) Usingthesesimplications,weprovethefollowinglemmafo r C 2 PS. Lemma5. Forthesubdivisionalgorithm C 2 PS, p mk = ` mk p k when k 2Z6 and m 0 Proof. Thebasecase m =0oftheinductionistriviallytrue.Fortheinductivestep ,we assume p mk = ` mk p k andshowthatthispropertyholdsfor p m +1 k aswell. 49

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Case k 2f 0 ; 3 g : p m +1 3 ( 3{27 ) = q m +1 00 + 1 n m +1 G q m +1 X r q m +1 1 ; [ r ] ( 4{1 ) = ( 4{2 ) 3 4 q m00 + 1 4 n m G q m X q m1 ; [ ] + 1 2 n m G q m +1 X r 1 2 q m00 + 1 n m G q m X 1 2 + c r + 1 2 c 2( r ) q m1 ; [ ] ( 4{5 ) = 3 4 q m00 + 1 4 n m G q m X q m1 ; [ ] + 1 2 q m00 + 1 n m G q m X 1 2 q m1 ; [ ] = 1 4 q m00 + 1 n m G q m X q m1 ; [ ] = 1 4 p m3 = ` 3 p m3 = ` m +1 3 p 3 Thesequenceofstepsfor k =0isverysimilartothoseof k =3above,anditsimilarly concludesthat p m +1 0 = ` m +1 0 p 0 Case k 2f 1 ; 2 ; 4 ; 5 g : p m +1 1 ( 3{27 ) = 2 n m +1 G q m +1 X r c r q m +1 1 ; [ r ] ( 4{2 ) = 2 2 n m G q m +1 X r c r 1 2 q m00 + 1 n m G q m X 1 2 + c r + 1 2 c 2( r ) + 1 8 c 3( r ) q m1 ; [ ] ( 4{5 )& ( c r = c 0 r ) = 2 2 n m G q m X c q m1 ; [ ] = 1 2 p m1 = ` 1 p m1 = ` m +1 1 p 1 The k 2f 2 ; 4 ; 5 g casesarederivedusingaverysimilarsequenceofsteps,sho wingthatfor allsixcases p m +1 k = ` m +1 k p k ,completingtheinduction. 4.2.2Reformulationof C 2 PSintermsoftheeigencoecients Inthesameveinas( 3{28 ){( 3{33 ), C 2 PScanbereformulatedtodependonlyon p k 2Z6 plusadeviationthatdiminishesquicklyinthenumberofsub divisions. 50

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Lemma6. The C 2 PSrenementequations ( 4{1 ) { ( 4{4 ) areoftheform q m +1 00 = p 0 m +1 3 p 3 (4{6) q m +1 1 ; [ ] = p 0 + m +1 ( p 1 c + p 2 s )+ 2 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )(4{7) q m +1 2 ; [ ] = p 0 +2 m +1 ( p 1 c + p 2 s )+ 11 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (4{8) q m +1 3 ; [ ] = p 0 +3 m +1 ( p 1 c + p 2 s )+ 26 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (4{9) q m +1 4 ; [ ] = p 0 +4 m +1 ( p 1 c + p 2 s )+ 47 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (4{10) q m +1 5 ; [ ] = p 0 +5 m +1 ( p 1 c + p 2 s )+ 74 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (4{11) Duetotheinteractionwithcircularsubdivision,thederiv ationforthefourouterlinks q m +1 2 ; [ ] { q m +1 5 ; [ ] isinvolvedandrequirestheintroductionofnewabstractio ns.Theproofof ( 4{6 ){( 4{11 )ishencedeferredtotheappendixtomaintaintherowofthis discussion. Theseequationscanbereducedto q m +1 i; [ ] =( ^ v 0 ) i p 0 +( ^ v 1 ) i m +1 ( p 1 c + p 2 s ) + m +1 (( ^ v 3 ) i p 3 +( ^ v 4 ) i ( p 4 c 2 + p 5 s 2 ))+O 1 8 m +1 ; (4{12) for i 2Z6 ,dieringonlybyO 1 8 m +1 from( 3{34 )whenthevalencesareequal. 4.2.3Convergenceoftheeigensplines Hereweshowthat e C 2 PS k ( r; )convergesto e RTS 1 k ( r; )as r 0. Lemma7. e C 2 PS k ( r; ) e RTS 1 k ( r; ) r 2 [2 m ; 4 m ] =O 1 8 m Proof. SincebothRTSand C 2 PSareane-invariant, e C 2 PS 0 ( r; )= e RTS 1 0 ( r; )=1,and thelemmaholds.Assumethat k> 0,whichalsoimpliesthat j ` k j .Let v k and ~ v k be the k th eigenvectorsofRTSofvalence n m and n 0 ,respectively. E := e C 2 PS k ( r; ) e RTS 1 k ( r; ) r 2 [2 m ; 4 m ] = e C 2 PS k ( m r; ) e RTS 1 k ( m r; ) r 2 [2 ; 4] 51

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triangle inequality e C 2 PS k ( m r; ) e RTS n m k ( m r; ) r 2 [2 ; 4] + e RTS n m k ( m r; ) e RTS 1 k ( m r; ) r 2 [2 ; 4] ( 3{3 ) = j G m T m ( ~ v k )( r; ) G m A m ( v k )( r; ) j | {z } T m ( ~ v k )and A m ( v k )havevalence n m ; G m islinear + ` mk e RTS n m k ( r; ) ` mk e RTS 1 k ( r; ) r 2 [2 ; 4] = j G m ( T m ( ~ v k ) A m ( v k ))( r; ) j | {z } E 1 := + ` mk |{z} O( m ) e RTS n m k ( r; ) e RTS 1 k ( r; ) r 2 [2 ; 4] | {z } Lemma 4 ) =O 1 n 2m =O ( 1 4 m ) = E 1 +O 1 8 m Bydenition, p h = hk for h 2Z6 whencomputedoneithereigenvector ~ v k or v k ;inother words,therstsixeigencoecientsofthesetwoeigenvecto rsmatch.Since( 4{12 )and ( 3{34 )dene T m ( ~ v k )and A m ( v k ),respectively,intermsof p h 2Z6 ,andthesetwoformulae dierbyO 1 8 m ,itfollowsthat T m ( ~ v k ) A m ( v k )=O 1 8 m .Therefore, E 1 =O 1 8 m ,and E =O 1 8 m ,provingthelemma. 4.2.4Proofofcurvaturecontinuity Wecannowestablishasecond-orderTaylorexpansionatthep ole,provingcurvature continuity.Theorem2. C 2 PSis C 2 atthepole. Proof. Recallfrom( 3{24 )thatapolarconguration q 0 ofvalence n 0 canbewrittenasthe followinglinearcombinationoftheeigenvectors, v 0 k k 2Z6 n 0 q 0ij = 6 n 0 X k p k ( v 0 k ) ij )L C 2 PS ( q 0 )= 6 n 0 X k p k L C 2 PS ( v 0 k ) ) x C 2 PS ( r; )= 6 n 0 X k p k e C 2 PS k ( r; ) Weexaminethesequence x C 2 PS ( r; ) r 2 [2 m ; 4 m ] ( 3{3 ) =( G m T m q 0 )( r; )ofsplinerings thatapproachthepole.Since r 2 [2 m ; 4 m ] ) r m 2 [2 ; 4], r m isboundedawayfrom 0and 1 andhasnoimpactonasymptoticbehaviorwhenmultiplied.Th issimplies 52

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O 1 8 m =O( 3 m )=O ( r m ) 3 3 m =O( r 3 ).Thus, x C 2 PS ( r; ) r 2 [2 m ; 4 m ] = 6 n 0 X k p k e C 2 PS k ( r; ) r 2 [2 m ; 4 m ] Lemma 7 = 6 n 0 X k p k e RTS 1 k ( r; )+O 1 8 m | {z } O( r 3 ) r 2 [2 m ; 4 m ] = p 0 e RTS 1 0 ( r; )+ p 1 e RTS 1 1 ( r; )+ p 2 e RTS 1 2 ( r; ) + p 3 e RTS 1 3 ( r; )+ p 4 e RTS 1 4 ( r; )+ p 5 e RTS 1 5 ( r; ) + 6 n 0 X k =6 p k e RTS 1 k ( r; )+O r 3 r 2 [2 m ; 4 m ] Lemma 2 & Lemma 4 = p 0 + r ( p 1 c + p 2 s )+ r 2 ( p 3 + p 4 c 2 + p 5 s 2 )+o r 2 r 2 [2 m ; 4 m ] ChangingtoCartesiancoordinates( x;y ):=( rc ;rs ), x C 2 PS ( x;y ):= x C 2 PS ( r; ), x C 2 PS ( r; ) r 2 [2 m ; 4 m ] = x C 2 PS ( x;y ) p x 2 + y 2 2 [2 m ; 4 m ] = p 0 +( p 1 x + p 2 y )+ p 3 ( x 2 + y 2 )+ p 4 ( x 2 y 2 )+ p 5 (2 xy ) +o x 2 + y 2 p x 2 + y 2 2 [2 m ; 4 m ] ; givinganexplicitsecond-orderexpansionatthepolewhen m !1 .Hencethe constructionis C 2 TheexplicitTaylorexpansionatthepoleallowsonetocompu teprincipalcurvatures anddirections.Insomeconstructions,curvaturecontinui tycomesatthecostof macroscopicshapedeterioration,eventhoughthemicrosco picshapeisimproved.Chapter 5 showsempiricallythatourconstructiondoesnotsuerfrom thisdefect;itgenerates surfacesofhighvisualquality. 53

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CHAPTER5 RESULTSANDDISCUSSION Figure 5-1 showsaside-by-sidecomparisonofRTS,RTS 1 ,and C 2 PS.Toavoid curvatureructuationsintherstandsecondsplinerings(F igure 5-1 (A-B)),uniform (bi)cubicsubdivisionisappliedtocomputetherened2-li nkontherstsubdivision step(Figure 5-1 (C-E)).ForRTS,thisisequivalenttoapplyingbicubicpola rsubdivision ontherstradialsubdivisionstep,whileusingRTSonallth esubsequentones.The n -sidednessoftheRTScurvaturedistributionisobvious,wh ileRTS 1 and C 2 PSyield smoothercurvaturetransitionsinthecirculardirection. C 2 PSdistributescurvaturemore evenly,resultinginalowermaximalGaussiancurvaturetha nRTSorRTS 1 nearthe pole.Asexpected,forhighervalences,thelimitsurfaceso fthesethreealgorithmsare similar(Figure 5-2 ).Figure 5-3 tests C 2 PSagainstvariouschallengingcongurations. Thesmoothhighlightlinesattesttothesurfacequalityint hevicinityofthepole,evenon higher-ordersaddles. Itmaybepossibletodeviseabi-degree-4 C 3 polarschemeusingasimilartechnique. Thekeyingredientisthatthesplineringsconstitutingthe limitsurfacewouldneedto shrinkmorerapidlytothepole.Toseethis,observethatthe reformulationof C 2 PSin termsofeigencoecients(Lemma 6 )isprovedbysimplifyingthetreatmentofarbitrary numberofcircularsubdivisionsusingaparameterizedequi valenceclassa 4[ r ] ( q i ).The useofthisclasscontributesadeviationofO 1 8 m ,whichistheproductof andthe convergencerateO 1 n 2m =O 1 4 m ofpiecewiselinearapproximationstocosinesandsines. Section 4.2.4 showedthatfor ` 1 = ` 2 = = 1 2 ,O 1 8 m simpliestoO( 3 m )=O( r 3 ), contributingtothethird-ordertermoftheTaylorexpansio natthepole.Whileour C 2 algorithmisunaectedbythis,designinga C 3 algorithmrequiresunderstandingthe third-ordertermprecisely.Onewaytoemploythesimplicit yofourproofsistoenforce eigenvalues ` 1 = ` 2 = < 1 2 ,resultinginadeviationofO m 4 m =o( 3 m )thatconvergesto 0morequicklythan r 3 ,avoidinginterferencewiththethird-orderexpansion. 54

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RTS RTS 1 C 2 PS ABCDEF Figure5-1.ComparisonofRTS,RTS 1 ,and C 2 PS.(C)splineringsdeninglimitsurfaces (AandD)andGaussiancurvature(BandE)oftwodierentinit ialization strategiesofeachschemeonFigure 2-5 Binput.Directapplicationproduces (A)asharperbendinthesilhouetteand(B)anabruptcurvatu retransition ( darkblue meanszeroGausscurvature),whereasusingbicubicsubdivi sionto computethe2-linkfortherstsubdivisionstepimprovesth ecurvature distribution(C,D,E).(F)RTSrevealsan n -sidednessinitscurvature distribution,whilethecurvatureofRTS 1 and C 2 PSismuchmoresymmetric. A.InputB.RTSC.RTS 1 D. C 2 PS Figure5-2.RTS,RTS 1 ,and C 2 PSonapolarcongurationofvalence20showthattheir limitsurfacesandGaussiancurvaturedistributions( darkblue iszero curvature)aresimilarforlargepolarvalences. Catmull-Clarkextraordinaryverticesofarbitraryvalenc ecanbeconvertedtopolar congurations,asdemonstratedinFigure 5-4 ,additionallycreating5-valentextraordinary vertices(Figure 5-4 A)orpentagons(Figure 5-4 C).Thus,itmaybepossibletodevise C 2 algorithmsforeither5-valentextraordinaryverticesorp entagonstoconstructsurfaces thatareglobally C 2 .Thisisleftforfuturework. 55

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ABCD Figure5-3.Shapegallerydemonstratingthat C 2 PSperformswithgoodshape.A)Input, B)twicesubdividedmesh,C)Gaussiancurvatureoflimitsur face,andD) highlightlines.Zerocurvatureis green ,whilenegativecurvatureis blue and positiveis red 56

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ABC Figure5-4.B)An n -valentCatmull-Clarkextraordinaryvertexcanbeconvert edtoeither A)a2 n -valentpolarcongurationand n 5-valentextraordinaryvertices,orC) a n -valentpolarcongurationand n pentagons. 57

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CHAPTER6 CONCLUSION Forquadmeshes,wehaveintroducedthepolarconguration, whichappears naturallyattheendsofelongatedobjects,likethenoseofa planeorthetipsofngers, wherecontrollinesalongthesametensordirectionmeettof ormasingularity.We havepresentedthreepolarsubdivisionalgorithmscompati blewithCatmull-Clark [ CatmullandClark 1978 ]subdivision:RTS,RTS 1 ,and C 2 PS.WhileRTSsurfaces areonly C 1 withboundedcurvatureatitspole,RTS 1 and C 2 PShavebeenshowntobe fully C 2 .Andwhilethesecond-ordercontinuityofRTS 1 iseasiertoprove,thisalgorithm transitionsfromapolynomialsplineboundarytoanon-poly nomialsurfacethatis,in general,noteasytocomputeexactly.Moreover,asameshre nementalgorithm,RTS 1 ismorecomplextoimplement,requiringalogicalseparatio nofthepolarconguration fromtherestoftheinputmeshbeforesubdivisionisapplied toit.Incontrast,theentirely spline-based C 2 PSissimplerbothasameshrenementalgorithm,andforexpl icitly evaluatingthelimitsurface.However,since C 2 PSresultsin non-stationary connectivity, standardsubdivisiontheoryfailstoapply,andtheproofof curvature-continuityatthe poleismorecomplex.Nevertheless,wehaveshown,inthisst udythatthealgorithmis C 2 andgivenevidencethatittendstogivegoodshape. Subdivisionalgorithmsareanacceptedstandardinanimati onandaresometimesused forconceptualdesigninCAD.Thesealgorithmshavebeenbee navoidedforhigh-quality surfacesinCADpartiallyduetoshapeproblemsnearextraor dinaryvertices.Wehave goneonestepclosertoshowthatasubdivisionalgorithmmay notbecomplexandstill havegoodshapeifnon-stationaryconnectivitycanbeexplo itedtoincreasetheorder ofapproximationinthevicinityofthepole.Weoeranaddit ionalincentivetouseour methodbecausetheorydevelopedin[ Reif 1998 ]and[ Mylesetal. 2008 ]suggeststhat curvaturecontinuitymayrequireadegree6NURBSsurfacewh enmorethan4NURBS meetatapoint.Ontheotherhand,wehaveshownthatdegree3i ssucientforasimple 58

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subdivisionalgorithmexploitingnon-stationaryconnect ivity.Wehopetechniquessuchas ourshelpmakesubdivisionsurfacesmoreusefulinmainstre amCAD. 59

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APPENDIX: C 2 PSINTERMSOFTHEEIGENCOEFFICIENTS Here,wederiveindetailthereformulation( 4{6 ){( 4{11 )of C 2 PS.Acheckmark( 4 ) indicatesthatoneoftheseequationshasbeenproved.While q m +1 00 and q m +1 1 j arereadily expressedintermsoftheeigencoecients, q m +1 00 = 3 4 q m00 + 1 4 n m G q m X q m1 ; [ ] = p m0 1 12 p m3 4 = p 0 m 12 p 3 = p 0 m +1 3 p 3 (A{1) q m +1 1 ; [ ] = 1 2 q m00 + 1 n m G q m X 1 2 + c + 1 2 c 2( ) + 1 8 c 3( ) q m1 ; [ ] additionrule forcosine&( 3{27 ) = p m0 + 1 2 ( p m1 c + p m2 s )+ 1 6 ( p m3 + p m4 c 2 + p m5 s 2 ) 4 Lemma 5 = p 0 + m +1 ( p 1 c + p 2 s )+ 2 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )(A{2) ( 4{3 )&addition forcosine&( 3{27 ) q m +1 2 ; [ ] = 11 12 ~ q m; [1] + 1 12 ~ q m; [2] 1 12 n m ( p m1 c + p m2 s ) Lemma 5 = 11 12 ~ q m; [1] + 1 12 ~ q m; [2] m +1 3 ( p 1 c + p 2 s ) ; (A{3) theexpressionsforthefourouterlinks q 2 ( A{3 ), q 3 q 4 ,and q 5 areinvolveddueto circularsubdivision,andonlythedominanttermswillbesh ownandneeded.Withthe intuitionthateverypointonasplineisanane(infact,con vex)combinationofthe fourB-splinecontrolpointsthatareparametricallyclose sttoit,wedenethefollowing equivalenceclassofanecombinations.Denition3 (a 4[ r ] ) Let u beavectorof n B-splinecontrolpointsofaperiodicuniform cubicsplinewithknotsequence 1 nZn .Thecontrolpoints u = h u [ 0 n ] ;:::; u [ n 1 n ] i areindexed bytheirGrevilleabscissaesothatadjacentpairsofGrevil leabscissaeare 1 n apart.The equivalenceclass a 4[ r ] ( u ) ofalllocalanecombinationscenteredat r isdenedas a 4[ r ] ( u ):= 8>>>><>>>>: 4 X g u g u [ r g ] P 4g u g =1 ; P 4g u g r g = r; and u [ r 0 ] ;:::; u [ r 3 ] arethefour controlpointswhoseGrevilleabscissae r g areclosestto r ,orhaveweight u g =0 iftheytieforfourthplace. 9>>>>=>>>>; 60

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SinceadjacentGrevilleabscissaedierby 1 n ,theanecombinationsina 4[ r ] ( u )are suchthat j r g r j < 2 n =O( 1 n )if u g 6 =0.CubicB-splinerenementrulesensurethat allanecombinationsresultingfromarbitrarily-manycub icB-splinerenementson u belongina 4[ r ] ( u ).Thecontrolpointsineachcircularly-subdivided i -link ~ q mi areane combinationsofcontrolpointsinthe i -link q mi .Sincetheseanecombinationsresultin anarbitrarynumberofcasesinourreformulationof C 2 PS,wefocusonsimplifyingane combinationsoftrigonometricfunctions.Lemma8. If u =[op k ( g n )] g 2Zn ,thenforall ~ u [ r ] := P 4g u g u [ r g ] 2 a 4[ r ] ( u ) ~ u [ r ] = op k ( r )+O 2k n 2 : Proof.Case1: k =0(i.e.op k ( r )= c 0 r =1) For u =[1] g 2Zn ~ u [ r ] = P 4g u g u [ r g ] |{z} 1 = P 4g u g =1=op k ( r ). Case2:u r = c k r k 6 =0 ~ u [ r ] = 4 X g u g c k r g = 4 X g u g c k ( r g r )+ k r = 4 X g u g c k ( r g r ) c k r s k ( r g r ) s k r = 4 X g u g 1+O 2 k n 2 | {z } from c k ( r g r ) c k r u g k ( r g r )+O 3 k n 3 | {z } from s k ( r g r ) s k r Taylorexpan.& j r g r j =O 1 n = c k r 7 1 4 X g u g + k s k r 0 4 X g u g ( r g r )+O 2 k n 2 = c k r +O 2 k n 2 ; satisfyingthetheorem.Case3:u r = s k r k 6 =0 TheproofisalmostidenticaltoCase2andshowsthat ~ u [ r ] = s k r +O 2k n 2 ,satisfying thetheorem. EquippedwithLemma 8 ,wecannowestimate q m +1 2 ; [ ] bydescribinga 4[ ] ( q m +1 2 ) 3 q m +1 2 ; [ ] intermsof p k 2Z6 .TheboundO 2k n 2m onthetermsnotexplicitlywrittenintermsof p k simpliestoO 1 4 m since k 2f 0 ; 1 ; 2 g intherelevantcasesand n m = n 0 2 m .Foreach 61

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~ u q m +1 1 [ ] 2 a 4[ ] ( q m +1 1 ), ( A{2 )& Lemma 8 ~ u q m +1 1 [ ] = p 0 + m +1 p 1 c +O 1 4 m + p 2 s +O 1 4 m + 2 m +1 3 p 3 + p 4 c 2 +O 1 4 m + p 5 s 2 +O 1 4 m = p 0 + m +1 ( p 1 c + p 2 s )+ 2 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (A{4) Foreach ~ u q m +1 2 [ ] 2 a 4[ ] ( q m +1 2 ),thereexist ~ u q m1 [ ] 2 a 4[ ] ( q m1 )and ~ u q m2 [ ] 2 a 4[ ] ( q m2 )sothat ( A{3 )& Lemma 8 ~ u q m +1 2 [ ] = 11 12 ~ u q m1 [ ] + 1 12 ~ u q m2 [ ] m +1 3 p 1 c + p 2 s +O 1 4 m ( A{4 )& Lemma 8 = 11 12 p 0 + m ( p 1 c + p 2 s )+ 2 m 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m + 1 12 ~ u q m2 [ ] m +1 3 ( p 1 c + p 2 s )+O 1 8 m +1 = 11 12 p 0 + 5 3 m +1 ( p 1 c + p 2 s )+ 22 9 m +1 ( p 3 + p 4 c 2 + p 5 s 2 ) +O 1 8 m +1 + 1 12 ~ u q m2 [ ] (A{5) ( A{5 )describestheseta 4[ ] ( q m +1 2 )recursivelywithrespectto m .Expandingoutthe recursionshowsthatforeach ~ u q m +1 2 [ ] 2 a 4[ ] ( q m +1 2 ),thereexists ~ u q 02 [ ] 2 a 4[ ] ( q 02 )sothat ~ u q m +1 2 [ ] = 11 12 m +1 X h 1 12 h p 0 + 5 3 m +1 X h m +1 h 12 h ( p 1 c + p 2 s ) + 22 9 m +1 X h m +1 h 12 h ( p 3 + p 4 c 2 + p 5 s 2 )+O m +1 X h 1 8 m +1 h 12 h + 1 12 m +1 ~ u q 02 [ ] geom. series = 1 1 12 m +1 p 0 +2 m +1 1 1 6 m +1 ( p 1 c + p 2 s ) + m +1 11 3 1 1 3 m +1 ( p 3 + p 4 c 2 + p 5 s 2 ) +O 3 8 m +1 1 2 3 m +1 !! + 1 12 m +1 ~ u q 02 [ ] 4 = p 0 +2 m +1 ( p 1 c + p 2 s )+ 11 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (A{6) 62

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Since q m +1 2 ; [ ] 2 a 4[ ] ( q m +1 2 ),ittooisdescribedby( A{6 ),proving 4{8 .Wesimilarly deriveformulasfor ~ u q m +1 3 [ ] 3 a 4[ ] ( q m +1 3 ), ~ u q m +1 4 [ ] 3 a 4[ ] ( q m +1 4 ),and ~ u q m +1 5 [ ] 3 a 4[ ] ( q m +1 5 ), automaticallyyieldingformulasfor q m +1 3 ; [ ] q m +1 4 ; [ ] ,and q m +1 5 ; [ ] .Foreach ~ u q m +1 3 [ ] 2 a 4[ ] ( q m +1 3 ), thereexist ~ u q m1 [ ] 2 a 4[ ] ( q m1 )and ~ u q m2 [ ] 2 a 4[ ] ( q m2 )sothat ~ u q m +1 3 [ ] = 1 2 ~ u q m +1 1 [ ] + 1 2 ~ u q m +1 2 [ ] ( A{2 ) ( A{6 ) = 1 2 p 0 + m ( p 1 c + p 2 s )+ 2 m 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m + 1 2 p 0 +2 m ( p 1 c + p 2 s )+ 11 m 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m 4 = p 0 +3 m +1 ( p 1 c + p 2 s )+ 26 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (A{7) Foreach ~ u q m +1 4 [ ] 2 a 4[ ] ( q m +1 4 ),thereexist ~ u q m1 [ ] 2 a 4[ ] ( q m1 ), ~ u q m2 [ ] 2 a 4[ ] ( q m2 ),and ~ u q m3 [ ] 2 a 4[ ] ( q m3 )sothat ~ u q m +1 4 [ ] = 1 8 ~ u q m +1 1 [ ] + 6 8 ~ u q m +1 2 [ ] + 1 8 ~ u q m +1 3 [ ] ( A{2 ),( A{6 ) &( A{7 ) = 1 8 p 0 + m ( p 1 c + p 2 s )+ 2 m 3 ( p 3 + p 4 c 2 + p 5 s 2 ) + 6 8 p 0 +2 m ( p 1 c + p 2 s )+ 11 m 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m + 1 8 p 0 +3 m ( p 1 c + p 2 s )+ 26 m 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m 4 = p 0 +4 m +1 ( p 1 c + p 2 s )+ 47 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (A{8) Foreach ~ u q m +1 5 [ ] 2 a 4[ ] ( q m +1 5 ),thereexist ~ u q m2 [ ] 2 a 4[ ] ( q m2 )and ~ u q m3 [ ] 2 a 4[ ] ( q m3 )sothat ~ u q m +1 5 [ ] = 1 2 ~ u q m +1 2 [ ] + 1 2 ~ u q m +1 3 [ ] ( A{6 ) ( A{7 ) = 1 2 p 0 +2 m ( p 1 c + p 2 s )+ 11 m 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m + 1 2 p 0 +3 m ( p 1 c + p 2 s )+ 26 m 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m 4 = p 0 +5 m +1 ( p 1 c + p 2 s )+ 74 m +1 3 ( p 3 + p 4 c 2 + p 5 s 2 )+O 1 8 m +1 (A{9) 63

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BIOGRAPHICALSKETCH AshishMyleswasborninVaranasi,India.HewasawardedtheB achelorofScience degreeincomputerscienceandengineeringfromtheUnivers ityofFloridain2002.He receivedhismaster'sdegreeincomputersciencein2004als oattheUniversityofFlorida, andcontinuedonforhisPh.D.,specializingingraphicsand smooth-surfacegeometry. 68