Researh ProposalTessellation, Fairing, Shape Design, and TrimmingTehniques for Subdivision Surfae based Modeling

Dr. Fuhua (Frank) ChengDepartment of Computer SieneUniversity of KentukyLexington, Kentuky 40506-0046Tel: (859) 257-6760Email: heng�s.engr.uky.eduhttp://www.s.engr.uky.edu/~ heng/January 25, 2004

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Contents1 Projet Summary 32 Results from prior NSF support (PI: F. Cheng) 43 Projet Desription 53.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Bakground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.1 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.2 Surfae Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.3 Automati Fairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.4 Shape design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.5 Surfae Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Objetive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Approahes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.1 Surfae Tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.2 Automati Fairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.3 Shape Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4.4 Surfae Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Work Plan and Time Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 Impat (Signi�ane) of the Proposed Researh . . . . . . . . . . . . . . . . . . . . . . . 184 Biographial Skethes 22

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1 Projet SummaryThe objetive of the proposed researh is to develop neessary geometri algorithms and design tehnologiesthat support subdivision surfae based modeling in CAD/CAM appliations. We will espeially fous ondeveloping tessellation, automati fairing, shape design and trimming tehniques for subdivision surfaes. De-velopment of these algorithms/tehnologies is ritial beause they are the building bloks of many subdivisionsurfae based modeling operations and, hene, are needed by any of the CAD/CAM systems that intends toinlude subdivision surfaes as the next generation surfae representation for CAD/CAM appliations.Spei� aim has been set for eah of these four areas. The aim for surfae tessellation is to develop errorontrol tehniques and adaptive re�nement tehniques for subdivision surfaes. The error ontrol tehniquesinlude sharp, ontrol point based error prediting tehniques and preision-guaranteed subdivision level om-putation tehniques. The adaptive re�nement tehniques inlude label-driven and error-driven tessellationapproahes. These tehniques allow onurrent, non-uniform tessellation of the ontrol mesh faes while sat-isfying the rak-free requirement and the onvergene proess is subdivision sheme related, i.e., one wouldautomatially get type A limit surfae if type A subdivision sheme is used in the new node generation pro-ess. The aim for automati surfae fairing is to develop fairness indiating tehniques and automati surfaeirregularity deteting and orreting tehniques for subdivision surfaes. Control point based and shadowgraphline based fairness indiating models will be onstruted. An indexing sheme will be developed to a

eleratethe generation of these models. Loal irregularities are removed by requiring the new surfae to math thetarget shape set by the fairness indiating model. The fairing proess will be performed with knot re�nementand onstrained optimization tehniques to avoid unneessary deformation of the subdivision surfae. Theaim for shape design is to develop interpolation-based shape design tehniques and ontrol mesh manipulationtehniques for subdivision surfaes. The interpolation-based design tehniques inlude the onstrution of asubdivision surfae to interpolate a given net of parametri urves. The interpolation proess will be developedbased on existing subdivision shemes to avoid inreasing the omplexity of the surfae intersetion proess ofthe CAD system. The manipulation tehniques inlude manipulating ontrol points of the ontrol mesh anddiret manipulation of the limit surfae. The aim for surfae trimming is to develop omputation tehniques forintersetion urves (trimming urves) of two subdivision surfaes, or a subdivision surfae and a parametrisurfae, and mesh generation tehniques for trimmed subdivision surfaes for appliations in �nite elementanalysis (FEM). New trimming urve representation shemes that avoid exessive node inreasing problemand new trimming urve omputation tehniques that enhane the preision and robustness requirements willbe developed.Results of the proposed researh will provide signi�ant advanement in subdivision surfae based modeling.Results in the �rst area will not only provide us with a preise and yet eÆient way to generate input to theFEM proess, but also an opportunity to address the manufaturability issues in the surfae developmentstage. Results in the seond area will signi�antly shorten the subdivision surfae fairing proess and makeCAD more proative in the produt development proess for subdivision surfaes. Results in the third areawill provide design engineers with e�etive tools to sulpt a subdivision surfae into desired shape that hasnever been done before and, therefore, inrease the produtivity of a design engineer dramatially. Resultsin the fourth area will provide us with the needed ore tehnologies in performing Boolean operations toform more ompliated objets and, onsequently, make subdivision surfae based modeling operations morereliable. Overall, results of the proposed researh will provide the design industry with a solution set formost subdivision surfae based modeling operations and, onsequently, provide the needed support to makesubdivision surfaes the next generation surfae representation for CAD/CAM appliations.Broader impats of this projet inlude providing graduate students with utting-edge, subdivision surfaebased modeling experiene, enrihing urriulum development in geometri modeling and CAD/CAM by inte-grating researh results into graduate ourses Computer Aided Geometri Design and Freeform Solid Modeling(subdivision surfae representations and subdivision surfae based modeling are inluded in both ourses),training of female PhD students (urrently one) and minority students reruited through the department'shonor program, and tehnology transfer and publiation.The proposed researh will be arried out by the geometri modeling team at the University of Kentukyin three years. The researh is expeted to lead to two PhD theses, and nine to twelve publiations.3

2 Results from prior NSF support (PI: F. Cheng)Award Number: DMI-9400823, Amount:$270,000, Duration:8/15/94-7/31/97Projet Title: New Computation Tehniques for Shape Modeling and DesignSummary of Completed Work: New tehniques in three areas (interproximation, even-degree B-spline in-terpolation, and automati blending) of shape design/modeling have been developed. Results in the �rst areainlude: (1) new knot hoosing tehnique, (2) interpolating an arbitrary network of urves, (3) interproximat-ing the verties of an arbitrary mesh, and (4) onstrained interpolation. In the seond area we have developeda new approah to perform quadrati B-spline urve interpolation at the parameter values where the maximaof quadrati B-spline basis funtions o

ur. The work that has been done in the third area inludes: (1)blending arbitrary losed polyhedra, and (2) blending smoothing arbitrary losed polyhedra. In both ases,Varady pathes are used in the blending proess and the polyhedra an be regular or irregular. (1 Post Do,1 Ph.D., 2 MS ompleted; 5 journal and 5 onferene papers published)Award Number: INT-9722728, Amount: $102,522, Duration: 6/1/97-5/31/00Projet Title: The GEO-WEB-CAD projetSummary of Completed Work: The goal is to develop network CAD/CAM interfaes and ollaborativedesign paradigms for the following omponents of an objet-oriented geometri modeling tool kit: (1) anopen arhiteture framework for wire frame, surfae, solid and feature modeling from a ommon uni�ed dataand �le struture, (2) linear, quadri and NURBS geometry modeling, supporting manifold and nonmanifoldtopologies, (3) realizing Boolean operations, loal operations (�llet, rounding, setion, extending), Euler oper-ations, and undo/redo operations, and (4) ray traing and radiosity rendering tehniques. Requirements andproperties of a ollaborative CAD system from the system arhiteture point of view have been analyzed anda oneptual model that ful�lls these requirements has been developed. Two prototype systems have beenimplemented. One is based on X-Windows System for the UNIX platform. The other one is implementedusing JAVA whih does not have platform restrition. (1 Post Do, 4 MS and 8 BS students ompleted; 6journal and 3 onferene papers published)Award Number: DMI-9912069, Amount: $393,067, Duration: 5/1/00-4/30/04Projet Title: Constrained Design, Streamline Modeling, Automati Fairing and Automati Joining Teh-niques for Non-Uniform Rational B-Spline (NURBS) SurfaesSummary of Completed Work: The goal is aiming at developing new and eÆient tehniques that areapable of (1) addressing the manufaturability issues in the surfae development stage, (2) providing designengineers with an interative surfae design tehnique that allows diret manipulation of surfae urvatureand variation of urvature, and (3) signi�antly shortening the energy based surfae fairing and joining pro-esses and, onsequently, make CAD more e�etive in the produt development proess. Results that havebeen ahieved inlude (1) reating G1/C1 urves with presribed shape properties, (2) performing real-timesurfae design by manipulating tangent vetors instead of ontrol points, (3) deteting and removing surfaeirregularities using an energy based fairness model of the given surfae, and (4) performing smooth, indiretjoining. (1 Post Do and two MS students ompleted in 2003; 1 Ph.D and 1 MS students will omplete in 2004;6 journal and 4 onferene papers published, 2 journal papers a

epted, 1 submitted, and 1 in preparation)Award Number: DMS-0310645, Amount: $99,941, Duration: 7/1/03-6/30/05Projet Title: Subdivision Surfae based One-Piee RepresentationSummary of Completed Work: Work on this projet is urrently in progress. The goal is to developmathematial theories and geometri algorithms to support subdivision surfae based one-piee representation,i.e., representing the �nal objet in a modeling proess with only one subdivision surfae. Results that havebeen ahieved inlude (1) energy omputation tehniques for Catmull-Clark subdivision surfaes, (2) eÆientevaluation tehniques for general subdivision surfaes. (1 Ph.D, and 1 MS students will omplete in 2005; 2papers in preparation) 4

3 Projet Desription3.1 MotivationSubdivision surfaes have beome popular reently in CAD/CAM beause of their exibility, numerial sta-bility, simpliity in oding and, most importantly, their apability in modeling/representing omplex shape ofarbitrary topology [17℄. Given a ontrol mesh and a set of mesh re�ning rules (or, more intuitively, ornerutting rules), one gets a limit surfae by reursively utting o� orners of the ontrol mesh [9℄[19℄. The limitsurfae is alled a subdivision surfae beause the orner utting (mesh re�ning) proess is a generalization ofthe uniform B-spline surfae subdivision tehnique. Therefore, subdivision surfaes inlude uniform B-splinesurfaes and pieewise B�ezier surfaes as speial ases. It is also reently known that subdivision surfaesinlude non-uniform B-spline surfaes and NURBS surfaes as speial ases [53℄. Subdivision surfaes anmodel/represent omplex shape of arbitrary topology beause there is no limit on the shape and topology ofthe ontrol mesh of a subdivision surfae [17℄. See Figure 1(d) for the representation of a ventilation ontrolomponent with a single subdivision surfae. The initial ontrol mesh of the surfae and the ontrol meshafter one re�nement and two re�nements are shown in (a), (b) and (), respetively. The ventilation ontrolomponent has seventeen holes (handles). Therefore, it an not be represented by a single B-spline or NURBSsurfae. Unfortunately, subdivision surfaes have not reeived muh attention from CAD/CAM industries(a) (b) () (d)Figure 1: (a) Initial ontrol mesh, (b) ontrol mesh after one re�nement, () after two re�nements, and (d)limit surfae of a ventilation ontrol omponent.until reently beause of two reasons. First, it was not known until 1998 that subdivision surfaes an beparametrized [54℄. Without a parametri representation, it is essentially impossible for a CAD/CAM systemto inlude subdivision surfaes as a free-form surfae modeling tool beause of problems with standard opera-tions suh as piking, rendering and texture mapping [54℄. The seond problem is with hardware. Subdivisionsurfaes are typially generated through reursive meshing. The omplexity of the meshing proess grows ex-ponentially with respet to the reursive subdivision level. This made generation and rendering of subdivisionsurfaes on an ordinary workstation essentially impossible in the 80s and early 90s beause of laking enoughmemory for the reursive mesh re�ning proess.Things have hanged over the past few years. With the parametrization tehnique of subdivision surfaesbeoming available [54℄ and with the fat that non-uniform B-spline and NURBS surfaes are speial ases ofsubdivision surfaes beoming known [53℄, we now know that subdivision surfaes over both parametri formsand disrete forms. Sine parametri forms are good for design and representation and disrete forms are goodfor mahining and tessellation (inluding FE mesh generation) [1℄, we �nally have a representation shemegood for all CAD/CAM appliations. With powerful PCs that arry almost unlimited memory availableeverywhere, omputation and rendering of subdivision surfaes are no longer a problem either. The eraof subdivision surfaes is �nally here. Atually, subdivision surfaes have already been used as primitivesin several ommerial systems suh as AliasjWavefront's Maya, Pixar's Renderman, Nihiman's Mirai, andMirospae' Lightwave 3D [7℄. They are not used as a major surfae representation in CAD/CAM systemsyet beause of laking neessary geometri algorithms and modeling tehniques in surfae tessellation, surfaefairing, shape design and surfae trimming. Development of algorithms and tehnologies in these areas isimportant beause they are the building bloks of many subdivision surfae based modeling operations and,hene, are needed by any of the CAD/CAM systems that intends to inlude subdivision surfaes as the nextgeneration surfae representation for CAD/CAM appliations. In the following, we will formally de�ne theseproblems and review related works. 5

3.2 Bakground3.2.1 A Brief HistoryThe onept of generating a surfae through mesh re�nement has its root in a urve generation tehniquedeveloped by Chaikin [10℄. In his approah, a urve is generated by reursively utting o� orners of a givenpolygon. Eah reursive utting yle generates two new points on eah leg of the polygon. If there are n+1verties Pji , i = 0; 1; :::; n, after the jth reursive utting yle, then the two new points generated on thepolygon leg PjiPji+1 are de�ned as follows:Pj+12i = 34Pji + 14Pji+1; Pj+12i+1 = 14Pji + 34Pji+1:This proess generates a uniform, quadrati B-spline urve as this orner-utting proess is nothing but thequadrati B-spine subdivision proess. The onept of B-spline subdivision is atually a generalization ofChaikin's algorithm (see [51℄ for the orresponding re�nement equation).Following Chaikin's work, a variety of subdivision shemes for urves and surfaes have been proposedduring the past two deades. For instane, a 4-point subdivision sheme proposed by Dyn, Levin and Gregory[21℄ an generate a subdivision urve to interpolate given data points. New points for eah leg of the re�nedontrol polygon are de�ned byPj+12i = Pji ; Pj+12i+1 = 8 + !16 (Pji +Pji+1)� !16(Pji�1 +Pji+2)where 0 < ! < 2(p5� 1), to ensure onvergene of the re�ned mesh. The standard value is ! = 1 whih hasan order three preision.Re�ning (subdivision) shemes for subdivision surfaes an be lassi�ed into two ategories: (1) approxi-mating tehniques, and (2) interpolating tehniques. Two typial subdivision shemes in the �rst ategory areDoo and Sabin's sheme [20℄ and Catmull and Clark's sheme [9℄. Doo and Sabin's sheme generates a surfaeby reursively utting o� orners and edges of a given retangular mesh as follows:1. For every vertex Vi of the urrent mesh P , a new vertex V 0i , alled an image, is generated on eah faeadjaent to Vi.2. For eah fae Fi of P , a new fae, alled an F-fae, is onstruted by onneting the image verties V 0i sgenerated in Step 1.3. For eah edge Ei ommon to two faes Fi and F 0i , a new 4-sided fae, alled an E-fae, is onstrutedby onneting the images of the end verties of Ei on the faes Fi and F 0i .4. For eah vertex Vi, where n faes meet, a new fae, alled a V-fae, is onstruted by onneting theimages of Vi on the faes meeting at Vi.This subdivision sheme generates a uniform biquadrati B-spline surfae. Catmull and Clark's sheme [9℄ issimilar to the Doo-Sabin sheme, but is based on tensor produt biubi B-spline. The surfae generated by thissheme is C2 ontinuous everywhere exept at some extraordinary points where it is C1 ontinuous. Catmulland Clark's sheme an work on meshes of arbitrary topology. Loop [43℄ has presented a similar subdivisionsheme based on generalization of quarti three-diretion Box-splines for triangular meshes. Peters and Reif[47℄ and Habib and Warren [27℄ independently introdued shemes that generalize quadrati 4-diretion BoxSplines on irregualr meshes. Subdivision shemes that an generate surfaes with sharp features [17℄ orfrationally sharp features [29℄ have also been proposed. Reently, it is even possible to generate featuressuh as usps, reases, and darts through the introdution of non-uniform subdivision surfaes [53℄. A newsubdivision sheme that an produe triangular meshes with small number of verties is proposed by Kobbelt[37℄.The �rst interpolating sheme for subdivision surfaes was presented by Dyn, Levin and Gregory [22℄. Thetehnique, alled a buttery sheme, requires a topologially regular setting of the initial (ontrol) mesh toprodue a C1 limit surfae. Zorin et al [63℄ and Kobbelt [33℄ have both developed improved interpolating6

shemes reently. Kobbelt's sheme is a simple extension of the 4-point interpolating subdivision [21℄. Zorinet al's sheme retains the simpliity of the buttery sheme and results in muh smoother surfaes evenfrom irregular initial meshes. These interpolating subdivision shemes also �nd appliations in wavelets onmanifolds, multiresolution deomposition of polyhedral surfaes, and multiresolution editing.Some of the mathematial properties of subdivision surfaes have been studied before. For instane, Dooand Sabin have studied the smoothness behavior of their subdivision surfaes through Fourier transformationsand eigen-value analysis of the subdivision matrix [19℄. Ball and Storry [3℄[4℄ and Reif [50℄ extended Dooand Sabin' work by deriving various neessary and suÆient smoothness onditions for di�erent subdivisionshemes. Spei� subdivision shemes have also been analyzed by several other people [16℄[32℄[34℄[52℄[64℄.Nevertheless, most of the geometri algorithms and modeling tehnologies required in subdivision surfaebased modeling operations are not well studied yet. Four of these areas are espeially ritial to the designommunity.3.2.2 Surfae TessellationGiven a surfae, a major onern in both �nite element analysis (FEM) and surfae rendering is the generationof an approximating mesh of the given surfae (within a given error tolerane) with as few nodes as possible.The approximating mesh is used to analyze the physial performane of the surfae or in the san onversionproess of the surfae. Smaller number of nodes in the approximating mesh is preferred beause it makes theanalysis proess and the rendering proess both more eÆient. This proess of generating an approximatingmesh for a given surfae, alled surfae tessellation, has been extensively studied for parametri surfaes[13℄[44℄. It has not been well studied for subdivision surfaes yet.To generate a good approximating mesh for a subdivision surfae, one needs to be able to (1) estimate theerror between the ontrol mesh (or, an approximating mesh) and the limit (subdivision) surfae, (2) determinethe level (depth) of reursive subdivision needed to reah a required preision, and (3) adaptively tessellatethe faes of the initial ontrol mesh so that an approximating mesh that is just good enough for the spei�edpreision and yet satisfying the rak-free requirement an be onstruted. Existing subdivision shemes annot be used diretly in the tessellation proess beause they lak the so-alled adaptive apability; they wouldsubdivide all the faes of a mesh even if only one of them does not satisfy the preision requirement and,onsequently, would generate approximating meshes with too many nodes (see Figure 2() for exessivelygenerated nodes in at regions of a roker arm with only two levels of subdivision).The �rst adaptive sheme for subdivision surfaes is proposed by Kobbelt [33℄ for Catmull-Clark subdivisionsurfaes. The method is performed on a trial-and-error basis and only works for the so-alled balaned netswhih, in addition, have to satisfy some other onstraints suh as even ritial edges. A few more generalshemes appeared reently for interpolatory p3-subdivision surfaes [38℄, p3-subdivision surfaes [37℄, andmodi�ed buttery subdivision surfaes [11℄. But they work for triangular ontrol meshes only. Another problemwith all the above adaptive shemes is that none of them use the error riterion most ommonly used inmehanial part design, i.e., the error between the approximating mesh and the limit surfae.We have worked in all these three areas: error estimation [15℄, subdivision level (depth) omputation [14℄,and adaptive mesh generation [13℄[44℄. However, the tehniques developed for B-spline and NURBS surfaesan not be used for subdivision surfae diretly beause the parameter spae of a subdivision surfae in generalis not retangular or triangular; it an be of any shape. New tehniques have to be developed for eah of theseareas.3.2.3 Automati FairingAutomati fairing refers to the proess of deteting and removing loal irregularities of a surfae automatially.Curvature plots have been frequently used to analyze the quality of a surfae. Commonly used urvaturemeasures inlude Gaussian, mean, and prinipal urvatures as well as normal urvatures along given diretions.Isophotes [49℄, reetion lines [30, 31℄ and, more reently, highlight lines [6, 59℄ have also been used in assessingthe quality of a surfae. These tehniques prove to be more e�etive and are beoming more popular reently,espeially in automotive body surfae design, beause they are more intuitive to understand and easier toompute. The smoothness of a surfae is measured using indiators suh as parametri or geometri ontinuity.7

(a) (b) ()Figure 2: (a) Control mesh, (b) limit surfae and () an approximating �nite-element mesh of a roker arm.Several papers analyzing parametri and geometri ontinuity of subdivision surfaes have been published(see, e.g., [2, 3, 50℄). They all onentrate on analyzing the subdivision sheme, instead of the layout ofthe ontrol points, of the subdivision surfae. The latter is atually more important beause a well-designedontrol point net is likely to bring out a higher order of ontinuity.Using di�usion and urvature ow, Desbrun, Meyer, Shroder and Barr [18℄ have presented a methodfor removing undesirable noises and uneven edges from irregularly triangulated data. A problem with thisapproah is that while removing verties and edges, one might also remove important data \underneath"the \noises". For instane, the \noises" ould be introdued by numerial error in the input phase but arewithin the tolerane level, therefore, the information arried underneath the noises should still be a

eptable.A better approah would be to perturb the points or edges to ahieve the goal of shape fairing, instead ofremoving points or edges. However, no paper has been published on onstruting a new limit (subdivision)surfae with higher parametri or geometri smoothness but with minimum distane from the original limit(subdivision) surfae.Fairing tehniques based on modifying reetion or highlight lines have also been proposed [12℄[30℄[31℄[60℄.They all heavily rely on the designers to visually identify the irregular regions and to �x them manually byorreting the ontrol points of the surfae. This is an experiene-based, trial-and-error, and time-onsumingproess. The omplexity of the problem for subdivision surfaes would make the situation even worse, likely toexeed what the human being an ope with, beause the topology of a subdivision spae is usually muh moreompliated than that of a parametri surfae. One needs the apability of automati detetion and orretionof loal irregularities for subdivision surfaes. One also needs an approah di�erent from the highlight linemodel beause identifying surfae normals that interset the light soure for a subdivision surfae is too ostlya proess for an interative design environment. A newly developed surfae smoothness evaluation model byus, alled the shadowgraph line model, will be onsidered here. This model has an analytial representationfor eah shadowgraph line. Therefore, there is no ost in getting a representation for a shadowgraph line atall.3.2.4 Shape designThe design of a subdivision surfae involves (1) the design, and (2) �ne tuning of the ontrol mesh. Theonly known tehnique in the �rst area is the work of Levin [41℄ whih uses a ombined subdivision shemeto onstrut a subdivision surfae to interpolate a given net of urves. This is an important work beauseit points out a better approah for subdivision surfae shape design (a parallel work for parametri surfaesan be found in [57℄). However, properties of Levin's surfae are not known yet and it is not a good idea toinlude too many new subdivision shemes in a modeling system. It is preferred to have similar interpolationtehniques using existing subdivision shemes so that the trimming proess an be handled with eÆieny (seenext setion for the justi�ation).The only known tehnique in the seond area is the work by Miura, Wang and Cheng [45℄ whih providesthe user with a tangent manipulation tehnique to �ne tune the shape of a subdivision surfae. An example isshown in Figure 3 where a set of Doo-Sabin surfaes are deformed using the tangent vetor blending tehniqueand the resulting Doo-Sabin surfaes in non-uniform form are shown in (b). For omparison purpose, theoriginal Doo-Sabin surfaes in non-uniform form are shown in (). The advantage of this approah is that8

through the manipulation of the tangent vetors, one an diretly manipulate the urvature and variationof urvature of the surfae. The disadvantage is that it ould be too laborious for subdivision surfaes withomplex topology. Note that while it is neessary to provide the user with the apability of diret ontrolpoint or tangent vetor manipulation, it is essential that the user an manipulate the shape of the surfaediretly (suh as dragging a point of the surfae to a new loation), leaving the time-onsuming job of �ndingthe new loations of the ontrol points to the system, so that the �ne tuning proess of shape design an bearried out more eÆiently.

(a) (b) ()Figure 3: (a) Corresponding ontrol mesh, (b) �ne tuned Doo-Sabin surfaes in non-uniform form, () originalDoo-Sabin surfaes in non-uniform form.3.2.5 Surfae TrimmingNURBS surfae intersetion, even up to today, is still onsidered the most diÆult problem and one of theweaker links in even high end ommerial CAD systems [42℄[55℄. The subdivision surfae intersetion problemwould be even more diÆult beause of the irregularity of the topology of a subdivision surfae. The maindiÆulty is the development of a reliable and eÆient omputation (marhing) proess.An algorithm for alulating the trimming urves of two Loop's subdivision surfaes is proposed by Litke,Levin and Shroeder [42℄ reently. The algorithm an guarantee exat interpolation of the trimming urves.This is ahieved by introduing a new type of surfaes, alled ombined surfaes, to approximate the trimmedsurfaes. A problem with this approah is that the inlusion of a new surfae type in a CAD system with msurfae representation shemes requires m more funtions to implement the surfae intersetion problem. Itis preferred to keep the number of surfae representation shemes low in a CAD system.Biermann, Kristjansson and Zorin [8℄ have presented a new method to approximate Boolean operationson free-form solids. The result of a Boolean operation is approximated by a multiresolution surfae. Thework pays more attention to eÆieny and robustness than to preision and, onsequently, is more suitablefor appliations where preision modeling is not required, suh as animation. For appliations in CAD/CAM,however, one needs to pay more attention to preision and robustness than to eÆieny.3.3 ObjetiveThe objetive of the proposed researh is to develop neessary geometri algorithms and design tehnologiesthat support subdivision surfae based modeling in CAD/CAM appliations. We will fous espeially ondeveloping tessellation, automati fairing, shape design and trimming tehniques for subdivision surfaes.Spei� aim has been set for eah of these four areas. The sope of the proposed researh is illustrated inFigure 4 where a blok or a proessor bounded by dotted line indiates that blok or proessor will be built with9

known tehnology. Development of the proposed tehnologies for subdivision surfaes is important beauseFEMSystems Mesh Generation�� ��Mesh GenerationFor FEM�� ��Mesh GenerationFor RenderingBoolean OperationsIntersetion CurveComputation UNIONOperation� �Boolean OperationsINTERSECTIONOperation DIFFERENCEOperation����� ��I��R��R��I �����

Smoothness TestingFairness IndiatorConstrution� �Smoothness TestingAutomati IrregularityDetetion & Corretion6?6?Component Shape DesignInterpolation-basedShape Design� �Component Shape DesignMorphing BasedShape Design O�set SurfaeGeneration6?����� ��I��RSurfae TessellationErrorComputation Subdivision LevelComputation� �Surfae TessellationLabel-DrivenTessellation Error-DrivenTessellation

-����� ��I��R��R��I �����Objet RenderingShaded ImageGeneration� �Objet RenderingHighlight LineGeneration6?6?

� �� � Yes?

No?- - -6 -

�YesNoFigure 4: A subdivision surfae modeling system with modules on tessellation, fairing, shape design, rendering,Boolean operations and mesh generation.they are the building bloks of many subdivision surfae based modeling operations and, hene, are needed byany of the CAD/CAM systems whih intends to inlude subdivision surfaes as the next generation surfaerepresentation for CAD/CAM appliations.(a) Surfae Tessellation: The aim for this area is to develop error ontrol tehniques and adaptive re�nementtehniques for subdivision surfaes. The error ontrol tehniques inlude a ontrol point based numerialformula for estimating the error between the ontrol mesh and the limit surfae, and an error-tolerane drivensubdivision level omputation tehnique for the onstrution of approximating meshes of the limit surfae.The subdivision level omputation tehnique guarantees that, after the required reursive subdivision, theresulting approximating mesh is within the given error tolerane of the limit surfae. The adaptive re�nementtehniques inlude a label-driven tessellation tehnique and an error-driven tessellation tehnique. The label-driven tessellation tehnique allows parallel tessellation of the ontrol mesh faes without the possibility ofgenerating raks between adjaent faes. The error-driven tessellation proess tessellates a fae only whenit is neessary for the error tolerane requirement or the rak-free requirement. Any existing subdivisionsheme an be used to generate new nodes in the tessellation proess and the resulting mesh onverges to thelimit surfae of the utilized subdivision sheme automatially.(b) Automati Surfae Fairing: The aim for this area is to develop fairness indiating tehniques andautomati surfae irregularity deteting and orreting tehniques for subdivision surfaes. The fairness indi-ating tehniques inlude a ontrol point based approah and a shadowgraph line based approah. Fairnessindiating models based on these tehniques will be built and an indexing sheme will be used to a

elerate thegeneration of these models. Deteted loal irregularities of a subdivision surfae are removed by adjusting itsontrol points so the new surfae would math the target shape in those regions set by the fairness indiatingmodel. The fairing proess will be performed with knot re�nement and onstrained optimization tehniquesto avoid unneessary deformation of the subdivision surfae. This will be an important breakthrough in sub-division surfae fairing beause adjusting ontrol points of a subdivision surfae to improve its smoothnessis an unexpetedly diÆult proess due to both the omplexity of its topology and the omplex layout of itsontrol points.() Shape Design: The aim for this area is to develop interpolation-based shape design tehniques and ontrol10

mesh manipulation tehniques for subdivision surfaes. The interpolation-based design tehniques inlude theonstrution of a subdivision surfae to interpolate a given net of parametri urves. The interpolationproess is based on existing subdivision shemes to avoid reating new subdivision shemes and, onsequently,avoid inreasing the omplexity of the surfae intersetion proess of the geometri modeling system. Themanipulation tehniques inlude manipulating ontrol points of the ontrol mesh and diret manipulation ofthe limit surfae.(d) Surfae Trimming: The aim for this area is to develop omputation tehniques for intersetion urves(trimming urves) of two subdivision surfaes, or a subdivision surfae and a parametri surfae, and meshgeneration tehniques for trimmed subdivision surfaes for appliations in �nite element analysis (FEM) andsurfae rendering. New trimming urve representation sheme that avoids exessive node inreasing problemand new trimming urve omputation tehnique that enhanes the preision and robustness requirementswill be developed. The mesh generation tehniques for FEM appliation and for rendering appliation willa

ommodate the di�erent requirements for these appliations.3.4 Approahes3.4.1 Surfae TessellationError Estimation and Subdivision Level Calulation: The �rst task is to estimate the error between theontrol mesh (or, an approximating mesh) and the limit surfae for a given subdivision sheme. Subdivisionsurfaes would be more widely used, espeially in mehanial part design, if one knows how to estimate theerror between a part designed with subdivision surfaes and its approximating mesh. The seond task here isto estimate the level of reursive subdivision that has to be performed on a fae of the ontrol mesh so theresulting mesh would be lose enough to the limit surfae (within a given error tolerane). This tehniqueenables one to estimate the size of an approximating mesh with required preision (suh as the number ofnodes) before it is onstruted.Two approahes will be onsidered for the �rst task. The �rst one is to onsider a subdivision surfae asa group of omponents whose parameter spaes are retangular or triangular. The error in this ase an beomputed as E = max8jfEjg where Ej is the error of the jth omponent. It is suÆient to onsider the asethat the parameter spae of a subdivision surfae is omposed of retangular omponents only. Suppose thateah omponent is a C2 parametri surfae Sj(u; v) with parameter spae Dj = [0; 1℄� [0; 1℄, then Ej an beestimated using a di�erent version of Filip-Magedson-Markot's formula [24℄,Ej � M1 + 2M2 +M38 ; (1)for interpolating subdivision shemes, and the following new formulaEj � M1 + 2M2 +M38 +maxfD0;0; D0;1; D1;0; D1;1g (2)for other subdivision shemes , whereM1 = sup(u;v)2Dj

�2Sj(u; v)�2u

; M2 = sup(u;v)2Dj

�2Sj(u; v)�u�v

; M3 = sup(u;v)2Dj

�2Sj(u; v)�2v

;with D0;0 = kSj(0; 0)�P0;0k, D0;1 = kSj(0; 1)�P0;1k, D1;0 = kSj(1; 0)�P1;0k, D1;1 = kSj(1; 1)�P1;1k,and Pk;l (k; l = 0; 1) are verties of the ontrol mesh omponent. To ahieve a tighter bound than formula(1) or (2), we need to onsider more details of the subdivision sheme in the formulation proess. Sometimes,it is preferred to have an error formula expressed in terms of ontrol points diretly so that it an be easilyunderstood and used. We will analyze the relationship between the bounds of derivatives of the subdivisionsurfae and its ontrol points to get suh a formula as well.In the seond approah, the subdivision sheme will be analyzed diretly. For eah reursive subdivisionstep, new verties an be obtained from old ones through vertex-matrix multipliation : Vk+1 = MVk .Therefore, another possibility in developing the error formula is to estimate this equation diretly.11

For subdivision level alulation, there are also two possible approahes. The �rst one onsists of twosteps. First, ompute a bound on the lengths of the fae edges whih guarantees loseness of a fae to thelimit surfae (within the given error tolerane) if edges of the fae are smaller than the bound. Then, analyzethe relationship between the bound of the fae edges and the subdivision level so that after performing therequired subdivision, all the edges of the resulting mesh faes would be smaller than the bound. The seondapproah is to obtain the subdivision level from the above error estimating proess diretly. One the errorestimating formula is available, the relationship between the subdivision level and the error is studied so thata formula an be developed.Label-Driven Tessellation: Given an initial ontrol mesh, a subdivision sheme, an error tolerane, and asubdivision level for eah fae of the mesh, a parallel method for the onstrution of an approximating meshof the limit surfae will be developed. The approximating mesh is within the given error tolerane of the limitsurfae, but has muh fewer nodes than the approximating mesh onstruted using the onventional approah.The new approah, whih works on individual faes reursively, satis�es the following requirements:� New nodes are generated using the given subdivision sheme and the resulting mesh onverges to thelimit surfae of the given subdivision sheme. If a di�erent subdivision sheme is used in the new nodegeneration proess, then the resulting mesh would onverge to the limit surfae of that subdivisionsheme.� The resulting mesh is rak-free.The basi idea is to use the unbalaned subdivision sheme proposed in [13℄, oupled with the balaned subdi-vision sheme, to overome the problem of having di�erent subdivision levels for di�erent adjaent faes. Alabel will be assigned to eah vertex of the ontrol mesh �rst. The tessellation proess of a mesh fae is drivenby the labels of its verties. Verties generated on the resulting mesh of a mesh fae depend on the labels ofits verties only. The label assignment sheme follows the priniple that the label assigned to a vertex shouldbe as small as possible, but should ensure the atness of the resulting mesh (at within the error toleraneof the limit surfae). This follows from the observation that smaller vertex labels produe fewer nodes in theresulting approximating mesh.The tessellation proess needs to determine, after eah subdivision yle, whih nodes should be takenfrom the previous level and whih nodes should be generated using the given subdivision sheme. It needsto ensure that all the nodes required in the omputation of the new nodes are available for the omputationproess. A node is removed only if it is not used as a new node or it is no longer needed in the omputationproess of the new nodes.Error-Driven Tessellation: Another approah for adaptive re�nement is to drive the tessellation proessby error between the urrent mesh and the limit surfae. For eah reursive subdivision yle, those faes thatare not within the given error tolerane of the limit surfae are subdivided one more time, and the other faesare subdivided only if it is needed to avoid raks. This approah will in general produe fewer nodes thanthe approah proposed above.After the error heking proess, labels will be assigned to faes of the new mesh as follows:lf (f) = � 1; if fae f is not within the given error tolerane of the limit surfae0; otherwise :Labels are then assigned to the verties of the new mesh, subjet to the following requirements:� Labels assigned to the verties should be as small as possible (to keep the number of nodes in theresulting mesh low).� Labels assigned to the verties should guarantee a rak-free status of the resulting mesh.� Labels assigned to the verties should guarantee the validity of the new node omputation proess forthe mesh.These requirements, espeially the last one, are ritial beause verties are in general not in the same subdi-vision level after one or more reursive subdivision yles.12

3.4.2 Automati FairingParametri and geometri ontinuity of subdivision surfaes will be thoroughly analyzed. For a given subdivi-sion sheme, suÆient and neessary ondition in terms of ontrol points will be established for eah order ofsmoothness. Constrained optimization tehnique and re�nement shemes that will lead to subdivision surfaeswith higher order of smoothness within the given error tolerane will be developed.The study will also develop methods for deteting and orreting loal shape anomalies. One possibleapproah is omposed of three steps:1. Create a fairness indiating model for the subdivision surfae. The model is omposed of a family ofurves on the subdivision surfae.2. Fair urves of the fairness indiating model. This may inlude reparametrization of some of the urves.3. Adjust ontrol points of the subdivision surfae to math the faired urves of the fairness indiatingmodel.The fairness indiating model should be intuitive enough for an average user to understand and eÆientenough for an interative design environment. The shadowgraph line model reently develop by us is a possiblehoie. If C(t) is a urve in the domain of a surfae S(u; v) and P is a 3D plane, a shadowgraph line is theintersetion of the plane P with the normal of the surfae, N(u; v), at the points of the urve S(C(t)). P isalled a projetion plane and C(t) is alled a soure urve. Given a set of soure urves in the domain of asurfae S(u; v) and a projetion plane P, a shadowgraph line model for S(u; v) is the family of shadowgraphlines reated on P for the given soure urves. If the projetion plane P is de�ned by point A and vetor Zthen F an be expressed as follows:F = S(u; v) + � (A� S(u; v)) � ZN(u; v) � Z �N(u; v): (3)An example of SG lines is shown in Figure 5, with (a) being the soure urves, (b) being a shaded image withthe mapped soure urves, () being the orresponding shadowgraph line model, and (d) being a shaded imagewith a highlight line model. The light soures of the highlight line model are in the same orientation as thesoure urves.(a) (b) () (d)Figure 5: (a) Soure urves, (b) shaded image with mapped soure urves, () shadowgraph line model, and(d) shaded image with a highlight line model of an automotive hood.Several points an be made immediately about the shadowgraph line model. First, there is an analytialrepresentation for eah shadowgraph line (not available for the reetion line model and the highlight linemodel). Therefore, there is no ost in getting a representation for a shadowgraph line at all. Seond, ashadowgraph line is also sensitive to the hange of normal diretions beause it is determined by the normalof the surfae as well. Atually, sine the surfae normal passes through a more omplex soure urve, insteadof a straight line in the light soure plane, a shadowgraph line usually magni�es the variation of urvature ofa surfae more than a highlight line (see Figure 5() and (d)) and, onsequently, has a better apability inidentifying irregularities of a surfae. Third, while linear light soures are always used in the highlight linemodel to ensure the solvability of the parameter �nding proess, the soure urves in the shadowgraph linemodel an be of any shape. This is an important feature beause it allows one to use omplex soure urves13

to generate shadowgraph lines whih is not possible for the highlight line model and, therefore, it is easier fora viewer to detet irregularities of a surfae without rotating the soure urves.An indexing sheme will be used to speed up the generation of the shadowgraph line family for thisapproah. For eah node of the approximating mesh, two numbers, I and R, will be omputed as follows:D = m � I +R; 0 � R < m;where m is the distane between two neighboring soure urves. Then for eah edge, we use the followingrules to determine if there are any shadowgraph line nodes ontained in this edge:� If the end points of the edge have the same I value, then either there is no shadowgraph line nodebetween the endpoints of the edge, or the entire edge is part of a shadowgraph line.� If the end points of the edge have di�erent indiators, I1 and I2, then there are jI2 � I1j shadowgraphline nodes between its end points. These nodes an be alulated using an inremental method.We will develop new tehniques to automatially identify irregularities of shadowgraph lines. The regionsthat ontain the irregularities will be replaed with urve segments with desired shape. The smoothness of theshadowgraph lines after the replaement proess is guaranteed. The ontrol points of the subdivision surfaeare then adjusted to math the new shadowgraph lines, i.e., ontrol points of the subdivision surfae areadjusted so that the new subdivision surfae would have the modi�ed shadowgraph lines as its shadowgraphlines. This requires the establishment of formula that desribe the relationship between the ontrol points ofthe subdivision surfae and its shadowgraph lines.Another possible approah that will be onsidered is omposed of two steps:1. Create a fairness indiating model for the subdivision surfae. The model is a funtion or funtionsde�ned on the parameter spae of the subdivision surfae, but it is not made up of urves. This modelshould be intuitive to understand. One possibility, for example, is to use di�erent olors to representdi�erent urvature ranges of the surfae.2. Identify and remove anomalies of the subdivision surfae based on the above model.3.4.3 Shape DesignInterpolation-based Shape Design Tehniques: The task here is the onstrution of a ontrol meshthat guarantees the interpolation of a given net of urves by the limit surfae of the ontrol mesh for agiven subdivision sheme. We will fous on subdivision shemes for quadrilateral meshes (whih an be easilyonverted to triangular meshes). The study onsists of three steps:1. de�ne the topology of the ontrol mesh from the topology of the given net of urves so that the ontrolmesh with suh a topology would interpolate the given net of urves with a good shape;2. establish relationship between the ontrol points of the target subdivision surfae and the given urves;3. develop methods to re�ne the original mesh of the target subdivision surfae so that the given net ofurves will be interpolated by the subdivision surfae within the given error tolerane.Manipulating Control Points of Subdivision Surfaes: Manipulation of subdivision surfaes inludesmanipulating their ontrol points and manipulating points of the subdivision surfaes diretly. Our approahfor the �rst ase will be di�erent from the traditional methods.In a traditional model, ontrol points are usually onneted by springs to ensure that related ontrol pointsare moved a

ording to physis laws if a ontrol point is moved to a new loation by the user. Our approah,however, will be based on shape optimization. New shape indiators will be reated to ensure that relatedontrol points are adjusted through an optimal shape deformation if a ontrol point is dragged to a new loationby the user. This approah is an improvement of the traditional approahes in that it inludes the e�et ofstrain energy in the optimization proess. The signi�ane of the new approah is that it also onsiders theimpat of the movement of a ontrol point on the smoothness of the surfae and tries to redue the negative14

impat of suh a movement to a minimum level.Diret Manipulation of Subdivision Surfaes: It is more intuitive for a user to manipulate the subdivisionsurfae diretly, espeially for those non-interpolating subdivision shemes. Here diret manipulation of asubdivision surfae refers to the proess of dragging points of the subdivision surfae diretly, instead ofdragging its ontrol points. Mathematially speaking, dragging a surfae point V to a new loation V0 meanshanging the ontrol points of the subdivision surfae so it would interpolate V0 instead of V. This is moreappealing to the user beause user is no longer responsible for �nding the new loations of the ontrol points;it is the job of the software now. The user is allowed to manipulate only one point at a time. If several pointsof the subdivision surfae have to be manipulated, one has to do it one by one. The study onsists of threesteps:1. developing eÆient methods to alulate intersetion points of a line with the subdivision surfae so thatthe user an use a ray to identify the point to be manipulated. An alternative is to use sampled points,suh as points of the \grids", as handles for the user to manipulate the surfae. The reason that theword \grids" is quoted is to emphasize the fat that the parameter spae of a subdivision surfae is notneessarily to be retangular. The �rst approah is more natural and onvenient for a user to use, butthe alternative approah is easier to implement.2. developing formula that represents the relationship between displaement of the ontrol points anddisplaement of a point of the subdivision surfae so that by putting the displaement of a subdivisionsurfae point into the formula, one an easily �nd required displaement of the ontrol points (for thenew surfae to pass through the new loation).3. adopting the idea of the above approah (\manipulating ontrol points of the subdivision surfae") inthis approah so that through an shape optimization proess, the surfae would not only pass throughthe new loation of the moved point, but also has the best possible shape.3.4.4 Surfae TrimmingCalulating Trimming Curves of Subdivision Surfaes: This work is the ore of all Boolean operations.To allow a user to reate trimmed subdivision surfaes of his own hoie, one needs the apability of omputingintersetion urves (trimming urves) of two subdivision surfaes, or a subdivision surfae and a parametrisurfae. An intuitive approah is to follow the traditionalmarhing tehnique [5℄[55℄[58℄ for parametri surfaeswith the help of our error estimating tehnique for subdivision surfaes. However, our initial experiene showsthat in this ase the number of verties required for the trimming urve inreases exponentially with respetto the preision requirement. Hene, the traditional marhing tehnique is not a good hoie for trimmingoperation of subdivision surfaes. We will develop new tehniques to �nd and represent trimming urves ofsubdivision surfaes. The study inludes:� developing new representation sheme for trimming urves. The new sheme would take muh smallerdata size to represent a trimming urve (for a given error tolerane), and it would failitate the renderingproess and satisfy the LOD (level of details) requirement. Besides, the new representation sheme shouldsatisfy ertain requirements for the mesh generation proess beause trimming urves play an importantrole in mesh generation of trimmed surfaes as well.� developing new trimming urve omputing methods. The new methods would ensure that features ofthe new representation sheme are onsidered in the trimming urve omputation proess and would em-phasize more on preision and robustness than on eÆieny. The divide-and-onquer and the re�nementtehniques will be onsidered for the new methods. Our initial experiene with these tehniques showthat they are promising both in eÆieny and a

uray (within any arbitrarily small error tolerane).Mesh Generation for Trimmed Subdivision Surfaes: We will develop new methods to generate ap-proximating meshes for trimmed subdivision surfaes for appliations in �nite element analysis (FEM) andsurfae rendering. These two appliation areas have di�erent requirements on the input meshes. Therefore,15

di�erent mesh generation tehniques will be developed for these appliations. For FEM appliation, the studyonsists of two steps:1. developing new methods to remesh the original ontrol mesh or an intermediate mesh (half produt) sothe resulting mesh is omposed of quadrilateral or triangular elements with reasonably good shape only.2. developing new subdivision methods for the remeshed (re�ned) meshes so that the resulting mesh ap-proximates the same limit surfaes with an error that an be made arbitrarily small. The reason fordoing so is beause the remeshing proess ould hange the nodes and topology of the original mesh.Therefore, to hold the resulting limit surfae unhanged, new subdivision shemes have to be developedand used.For surfae rendering appliation, the study inludes� developing new, adaptive mesh generation methods for trimmed subdivision surfaes. For renderingappliation, shape of the mesh faes is not that important; eÆieny and robustness are more importantissues. Adaptive methods are used here beause they an speed up the mesh generation proess andredue the number of nodes in the resulting mesh.� developing new methods to generate approximating mesh for a surfae omposed of several trimmedsubdivision surfaes or a solid bounded by several trimmed subdivision surfaes. In this ase, we needto take are the rak problem between or among di�erent trimmed surfaes (for non-manifold ase).Craks may o

ur due to di�erent error toleranes or di�erent subdivision shemes on di�erent surfaes.Note that by applying di�erent subdivision shemes on adjaent surfaes, one ould produe di�erentnodes on the boundary urves shared by those surfaes.3.5 Work Plan and Time TableThree years are planned for the projet beginning with July of 2004. In the �rst year the following ativitiesare sheduled:� September 1, 2004 ! Deember 31, 2004- develop basi parallel label-driven tessellation tehniques- develop numerial formula to preisely estimate the error between ontrol mesh and limit surfae- develop remeshing tehnique subjet to FEM requirements- develop subdivision shemes for the remeshing proess that guarantee approximation of the same limitsurfae with arbitrarily small errorMilestones: tools for subdivision surfae tessellationtools for FEM mesh generation for subdivision surfaesa mesh error evaluator� January 1, 2005 ! April 30, 2005- develop tehniques to ompute subdivision level for given error tolerane- study the relationship between the ontrol points and the bounds of derivatives for subdivision surfaes- develop new label assignment tehnique to speed up the label-driven tessellation proess- develop basi parallel error-driven tessellation tehniquesMilestones: new mesh generation tools for subdivision surfaesa report on derivative bound alulationa subdivision level evaluator� May 1, 2005 ! August 31, 2005- develop onstrained optimization tehniques and re�nement shemes to obtain subdivision surfaeswith higher order of smoothness within the given error tolerane I- reate fairness indiating model for subdivision surfaes- develop fairing tehniques for urves in the fairness indiating model I16

- develop fairness indiating model based anomaly deteting tehniques for subdivision surfaesMilestones: tools for smoothing subdivision surfaestools for fairing smoothness indiating urvesa fairness indiating model for subdivision surfaesa report on anomaly detetionThe ativities for the seond year are sheduled as follows:� September 1, 2005 ! Deember 31, 2005- develop new label assignment tehnique to speed up the error-driven tessellation proess- develop onstrained optimization tehniques and re�nement shemes to obtain subdivision surfaeswith higher order of smoothness within the given error tolerane II- develop optimal rendering tehniques for the fairness indiating modelMilestones: a report on getting subdivision surfaes with higher smoothness ordera report on fairness indiating model renderinga report on optimal tessellation tehniques� January 1, 2006 ! April 30, 2006- develop automati subdivision surfae fairing tehniques by adjusting ontrol points to math fairedurves in the fairness indiating model- develop fairing tehnique for urves in the fairness indiating model II- develop anomaly deteting tehniques for subdivision surfaes by heking on the ontrol point layout- develop onstrained optimization tehniques for removing anomaly of subdivision surfaes by re-arranging ontrol pointsMilestones: tools for subdivision surfae fairinga report on fairing smoothness indiating urvesa report on anomaly detetion and removal� May 1, 2006 ! August 31, 2006- develop tehniques to obtain topology of the ontrol mesh from topology of the given net of urves- develop methods to determine desired loations of the ontrol points so the targeted subdivision surfaewould be lose to the given net of urves- develop adjustment and re�nement skills to make the targeted subdivision surfae interpolate the givenurves within a given error toleraneMilestones: a report on ontrol mesh topology onstrutiona report on ontrol mesh position alulationa report on ontrol mesh re�nementtools for performing subdivision surfae interpolation of given net of urvesThe ativities sheduled for the third year are:� September 1, 2006 ! Deember 31, 2006- develop shape optimization tehniques for diret subdivision surfae ontrol point manipulation- develop eÆient methods to alulate intersetion points of a line and a subdivision surfae- develop formula to estimate required ontrol point displaement in order to get desired subdivisionsurfae displaement- develop shape optimization method with step size ontrol for subdivision surfaesMilestones: tools for subdivision surfae shape optimizationa report on line-subdivision-surfae intersetiona report on relationship between ontrol point displaement and subdivision surfaedisplaement� January 1, 2007 ! April 30, 2007- develop intersetion urve omputation tehniques for subdivision surfaes (between two subdivisionsurfaes or between a subdivision surfae and a parametri surfae)- develop trimming urve reparametrization tehniques17

- develop adaptive mesh generation methods for trimmed subdivision surfaesMilestones: tools for performing subdivision surfae intersetiontools for mesh generation on trimmed subdivision surfaesa report on trimming urve omputation for subdivision surfaes� May 1, 2007 ! August 31, 2007- develop new methods to generate approximating mesh for the boundary of a solid omposed of trimmedsubdivision surfaes- develop UNION, INTERSECTION and DIFFERENCE operation tehniques for subdivision surfaesMilestones: tools for performing Boolean operations on subdivision surfaesa report on mesh generation for surfaes ombined of trimmed subdivision surfaesseveral reports on Boolean operations3.6 Impat (Signi�ane) of the Proposed ResearhAdvaning disovery and understanding: Results of the proposed researh will provide signi�ant ad-vanement in subdivision surfae based modeling. Results in the �rst area will not only provide us with apreise and yet eÆient way to generate input to the FEM and surfae rendering proesses, but also a betterway to understand the spatial properties of subdivision surfaes suh as onvexity and urvature distributionand, hene, provide us with an opportunity to address the manufaturability issues in the surfae developmentstage. Results in the seond area will signi�antly shorten the subdivision surfae fairing proess and makeCAD more proative in the produt development proess for subdivision surfaes. Results in the third areawill provide design engineers with e�etive tools to sulpt a subdivision surfae into desired shape and diretmanipulation of surfae urvature and variation of urvature that has never been done before and, therefore,will inrease the produtivity of a design engineer dramatially. Results in the fourth area will provide uswith the needed ore tehnologies in performing Boolean operations to form more ompliated objets and,onsequently, will make subdivision surfae based modeling operations more reliable. Overall, results of theproposed researh will provide the design industry with a solution set for most subdivision surfae based mod-eling operations and, onsequently, provide support to make subdivision surfaes the next generation surfaerepresentation for CAD/CAM appliations.Promoting teahing, training, and learning: The proposed projet will provide graduate students atthe University of Kentuky an exellent experiene to work on subdivision surfae based modeling tehniques.This unique experiene will make them invaluable researhers or engineers for the CAD/CAM industries of theUnited States. The results of the proposed researh will also enrih urriulum development at the Universityof Kentuky, partiularly in the areas of geometri modeling and CAD/CAM. The PI of this proposal hasalready integrated his researh results into two graduate ourses, Computer Aided Geometri Design (CS631)and Freeform Solid Modeling (CS630) (some subdivision surfae representation and subdivision surfae basedmodeling results have already been inluded in both ourses). The results of the proposed researh will beintegrated into these ourses as four additional topis on subdivision surfae modeling.Inreasing partiipation of minority groups: There is already a female member (Alie J. Lin) in the PhDstudent group who would partiipate in the proposed researh work. We will also reruit minority students topartiipate in the proposed projet through the department's honor program.Enhaning broad dissemination: The PI has established ollaboration with several ompanies suh asSDRC (urrently alled EDS PLM-Solution), Ford, and Honda through funded projets from these ompanies(three-year researh grant from Ford and two one-year researh grants from Honda) or required tehnologytransfer (to SDRC, required by Ford). The CAD modeling group at the Ford Researh Center and the Curvesand Surfaes Group at SDRC have already expressed their interest in reviewing the results of this projet.Therefore, in addition to publiation, tehnology sharing will also be a part of the PI's researh agenda.18

Referenes[1℄ Austin SP, Jerard RB, Drysdale RL, Comparison of disretization algorithms for NURBS surfaes withappliation to numerially ontrolled mahining, Computer Aided Design 1997, 29(1): 71-83.[2℄ Ball AA, Storry DJT, A matrix approah to the analysis of reursively generated B-spline surfaes,Computer Aided Design, 1986, 18(8): 437-442.[3℄ Ball AA, Storry DJT, Conditions for tangent plane ontinuity over reursively generated B-spline surfaes,ACM Transations on Graphis, 1988, 7(2): 83-102.[4℄ Ball AA, Storry DJT, An investigation of urvature variations over reursively generated B-spline surfaes,ACM Transations on Graphis 1990, 9(4):424-437.[5℄ Barnhill RE, Farin G, Jordan M, Piper BR, Surfae/surfae intersetion, Computer Aided GeometriDesign 1986, 4(1-2): 3-16.[6℄ Beier K-P, Chen Y, Highlight-line algorithm for realtime surfae-quality assessment, Computer-AidedDesign, 1994, 26(4): 268-277.[7℄ Biermann H, Levin A, Zorin D, Pieewise smooth subdivision surfaes with normal ontrol, Proeedingsof SIGGRAPH 2000: 113-120.[8℄ Biermann H, Kristjansson D, Zorin D, Approximate Boolean operations on free-form solids, Proeedingsof SIGGRAPH, 2001:185-194.[9℄ Catmull E, Clark J. Reursively generated B-spline surfaes on arbitrary topologial meshes, Computer-Aided Design, 1978, 10(6):350-355.[10℄ Chaikin G, An Algorithm for High-Speed Curve Generation, Computer Graphis and Image Proessing,1974, 3(4):346-349.[11℄ Cheang K, Dong W, Li J, Kuo C, Postproessing of ompressed 3D graphi data, Journal of VisualCommuniation and Image Representation, 2000, 11(1):80-92.[12℄ Chen Y, Beier K-P, Papageorgiou D, Diret highlight line modi�ation on nurbs surfaes, ComputerAided Geometri Design, 1997, 14: 583-601.[13℄ Cheng F, Jaromzyk JW, Lin J-R, Chang S-S, Lu J-Y, A parallel mesh generation algorithm based onthe vertex label assignment sheme, International Journal for Numerial Methods in Engineering, 1989,28: 1429-1448.[14℄ Cheng F. Estimating subdivision depths for rational urves and surfaes, ACM Transations on Graphis,1992, 11(2): 140-151.[15℄ Cheng F, Luken WL, Computing step sizes for the tessellation of trimmed NURBS surfaes, IBM Rep.1992: RC18499.[16℄ Dahmen W, Cavaretta AS, Mi

helli CA, Stationary subdivision, Memoirs of the Amerian MathematialSoiety, 93 (453), September 1991, Amerian Mathematial Soiety.[17℄ DeRose T, Kass M, Truong T, Subdivision Surfaes in Charater Animation, Proeedings of SIGGRAPH,1998: 85-94.[18℄ Desbrun M, Meyer M, Shroder P, Barr AH, Impliit fairing of irregular meshes using di�usion andurvature ow, Proeedings of SIGGRAPH, 1999: 317-324.[19℄ Doo D, Sabin M, Behavior of reursive division surfaes near extraordinary points, Computer-AidedDesign, 1978, 10(6):356-360.[20℄ Doo D, A subdivision algorithm for smoothing down irregularly shaped polyhedrons, in Proeedings onInterative Tehniques in Computer-Aided Design, 1978, 157-165.[21℄ Dyn N, Levin D, Gregory JA, A 4-point Interpolatory Subdivision Sheme for Curve Design, ComputerAided Geometri Design, 1987, 4:257-268. 19

[22℄ Dyn N, Levin D, Gregory JA, Buttery subdivision sheme for surfae interpolation with tension ontrol,ACM Transations on Graphis, 1990, 9(2): 160-169.[23℄ Dyn N, Levin D, Interpolating Subdivision Shemes for the Generation of Curves and Surfaes, Multivar.Approx. and Interp., W. Hausmann and K. Jetter (eds), 1990, Birkhauser Verlag, Basel.[24℄ Filip D, Magedson R, Markot R. Surfae algorithms using bounds on derivatives, Computer Aided Geo-metri Design, 1986, 3(4):295-311.[25℄ Guskow I, Sweldens W, Shroder P, Multiresolution signal proessing for meshes, Proeedings of SIG-GRAPH, 1999:325-334.[26℄ Guskov I, Vidimi�e K, Sweldens W, Shroder P, Normal meshes, Pro. SIGGRAPH, 2000:95-102.[27℄ Habib A, Warren J, Edge and vertex insertion for a lass of C1 subdivision surfaes, Computer AidedGeometri Design, 1999, 16(4), 223-247.[28℄ Halstead M, Kass M, DeRose T, EÆient, fair interpolation using Catmull-Clark surfaes, Proeedingsof SIGGRAPH, 1993:35-44.[29℄ Hoppe H, DeRose T, Duhamp T, Halstead M, Jin H, MDonald J, Shweitzer J, Stuetzle W, Pieewisesmooth surfae reonstrution, Proeedings of SIGGRAPH, 1994:295-302.[30℄ Kaufmann E, Klass R, Smoothing surfaes using reetion lines for families of splines, Computer-AidedDesign, 1988, 20: 312-316.[31℄ Klass R, Corretion of loal surfae irregularities using reetion lines, Computer-Aided Design, 1980,12(2): 73-77.[32℄ Kobbelt L, Interpolatory re�nement by variational methods, in C. Chui and L. Shumaker, editors,Approximation Theory VIII, Volume two of Wavelets and Multilevel Approximation, World Sienti�Pub. Co., 1995, 217-224.[33℄ Kobbelt L, Interpolatory subdivision on open quadrilateral nets with arbitrary topology, In ComputerGraphis Forum, Proeedings of Eurographis '96, 1996:C407-C420.[34℄ Kobbelt L, A variational approah to subdivision, Computer Aided Geometri Design, 1996, 13:743-761.[35℄ Kobbelt L, Disrete Fairing, Proeedings of the Seventh IMA Conferene on the Mathematis of Surfaes1997, 101-131.[36℄ Kobbelt L, Campagna S, Vorsatz J, Seidel H-P, Interative multiresolution modeling on arbitrary meshes,Proeedings of SIGGRAPH, 1998: 105-114.[37℄ Kobbelt L, p3 subdivision, Proeedings of SIGGRAPH, 2000, 103-112.[38℄ Labsik U, Greiner G, Interpolatory p3 subdivision, Proeedings of EUROGRAPHICS, Interlaken, Swiss,2000, 131-138.[39℄ Labsik U, Hormann K, Greiner G, Using Most Isometri Parametrizations for Remeshing PolygonalSurfaes, Proeedings of Conferene on Geometri Modeling and Proessing (GMP00), Hong Kong, 2000,220-228.[40℄ Lee A, Moreton H, Hoppe H, Displaed subdivision surfaes, Pro. SIGGRAPH, 2000, 85-94.[41℄ Levin A, Interpolating nets of urves by smooth subdivision surfaes, Pro. SIGGRAPH, 1999: 57-64.[42℄ Litke N, Levin A, Shroeder P, Trimming for subdivision surfaes, Computer Aided Geometri Design2001, 18(5):463-481.[43℄ Loop C, Smooth subdivision surfaes based on triangles, Master's thesis, Department of Mathematis,University of Utah, 1987.[44℄ Luken WL, Cheng F, Comparison of surfae and derivative evaluation methods for the rendering of NURBsurfaes, ACM Transations on Graphis 1996, 15(2): 153-178.20

[45℄ Miura KT, Wang L, Cheng F, Streamline Modeling with Subdivision Surfaes on the Gaussian Sphere,Computer Aided Design, 2001, 3(13):975-987.[46℄ Miura KT, Wang L, Cheng F, Fine Tuning: Curve and Surfae Deformation by Saling Derivatives, Pro.Pai� Graphis 2001, Tokyo, Ot. 16-18, 2001, 150-159.[47℄ Peters J, Reif U, The simplest subdivision sheme for smoothing polyhedra, ACM Transations on Graph-is, 1997, 16(4):420-431.[48℄ Peters J, Reif U, Analysis of Algorithms generalizing B-spline subdivision, SIAM Journal of NumerialAnalysis, 1998, 35(2):728-748.[49℄ Poeshl T, Deteting surfae irregularities using isophotes, Computer Aided Geometri Design, 1984,1(2): 163-168.[50℄ Reif U, A uni�ed approah to subdivision algorithms near extraordinary verties, Computer Aided Geo-metri Design, 1995, 12(2): 153-174.[51℄ Shroder P, Zorin D, Subdivision for Modeling and Animation, SIGGRAPH98 ourse note, 1998.[52℄ Shweitzer JE, Analysis and appliation of subdivision surfaes, Ph.D. dissertation, Department of Com-puter Siene and Engineering, University of Washington, 1996.[53℄ Sederberg TW, Zheng J, Sewell D, Sabin M, Non-uniform reursive subdivision surfaes, Proeedings ofSIGGRAPH, 1998:19-24.[54℄ Stam J, Exat Evaluation of Catmull-Clark Subdivision Surfaes at Arbitrary Parameter Values. Pro-eedings of SIGGRAPH 1998:395-404.[55℄ Wang X, Cheng F, Sun J, Barsky BA, Boundary Representation, Intersetion, and Evaluation of NURBSBased Non-Manifold Objets, Pro. 1996 ASME Design for Manufaturing Conferene (ASME DTC-96DFMC), August 18-22, 1996, Irvine, CA, 622-633.[56℄ Wang X, Cheng F, Surfae Design Based on Hermite Interpolation with Tension Control and OptimalTwist Vetors, Neural, Parallel & Sienti� Computations (Speial Issue in Computer Aided GeometriDesign), 1997, 5(1&2):37-57[57℄ Wang X, Cheng F, Barsky BA, Blending, Smoothing, and Interpolation of Irregular Mesh Using N-SidedVarady Pathes, Pro. Solid Modeling '99 (5th ACM Symposium on Solid Modeling and Appliation),June 9-11, 1999, Ann Arbor, Mihigan, 212-222.[58℄ Wu S-T, Andrade LN, Marhing along a regular surfae/surfae intersetion with irular steps, ComputerAided Geometri Design 1999, 16(4): 249-268.[59℄ Yong J, Cheng F, Miura KT, Chen Y, Stewarts P, Dynami Highlight Line Generation for LoallyDeforming NURBS Surfaes, Computer Aided Design 35,10 (2003), 881-892.[60℄ Zhang C, Cheng F, Removing loal irregularities of NURBS surfaes by modifying highlight lines,Computer-Aided Design, 1998, 30(12):923-930.[61℄ Zhang C, Cheng F, Fairing Spline Curves/Surfae by Minimizing Energy, Computer Aided Design, 2001,33(13):913-923.[62℄ Zhang C, Cheng F, Triangular Path Modeling Using Combination method, Computer Aided GeometriDesign 19,8 (2002), 645-662.[63℄ Zorin D, Shroder P, Sweldens W, Interpolating subdivision for meshes with arbitrary topology, Proeed-ings of SIGGRAPH 1996:189-192.[64℄ Zorin D, Smoothness of stationary subdivision on irregular meshes, Construtive Approximation, 2000,16(3):359-398.21

4 Biographial SkethesFuhua (Frank) Cheng (PI)Eduation:8/80-5/82 Ph.D., Applied Math. & Computer Siene, Ohio State University8/78-5/80 M.S., Computer Siene, Ohio State University6/77-5/78 M.S., Mathematis, Ohio State University9/69-6/75 M.S. Mathematis, Tsinghua University, Taiwan, R.O.C.Employment:7/98- Prof., Dept. of Computer Siene, Univ. of Kentuky9/93-8/94 Projet Diretor, Olympus Optial Co., Tokyo, Japan9/93-3/94 Visiting Researher, Dept. of Information Siene, Univ. of Tokyo, Japan9/93-8/94 Visiting Prof., Shape Modeling Lab., Univ. of Aizu, Japan7/89-6/98 Asso. Prof., Dept. of Computer Siene, Univ. of Kentuky8/86-6/89 Assist. Prof., Dept. of Computer Siene, Univ. of Kentuky8/82-7/86 Asso. Prof., Institute of Computer Siene, Tsinghua Univ., Taiwan, R.O.C.Researh Interest:Geometri/solid modeling, omputer graphis, ollaborative CAD, parallel omputation in graphis and geo-metri modeling, approximation theorySynergisti Ativities:1. Program Committee: CAD'06, Geometri Modeling and Proessing 2006, 24th Computer Graphis In-ternational Conferene (CGI2006), ISICS2006, IASTED Int. Conf. Computer Graphis and Imaging(CGIM 2007), IASTED Int. Conf. Graphis and Visualization in Engineering (GVE 2007), CAD'07,11th ACM Symp. Solid and Physial Modeling (2007).2. Editing: J. Information and Computational Siene, Computer Aided Design & Appliations, J. MathAnalysis & Approximation Theory, J. Computer Aided Design & Computer Graphis3. Worked with Mehanial Engineering of the University of Kentuky and researhers from three otheruniversities (University of Missouri - Rolla, University of Rhode Island, and University of Louisville)on an NSF ERC proposal (not seleted for the seond stage) and working with Mehanial Engineeringand Biomedial Engineering of the University of Kentuky on an Interative Graduate Eduation andResearh Traineeship (IGERT) proposal.4. Courses being taught at the University of Kentuky in the CAD area are (1) Free-form Solid Modeling(CS630) and (2) Computer-Aided Geometri Design (CS631).5. Work ongoing in researh lab on Subdivision Surfae based One-Piee Representation, Tessellation, Fair-ing, Shape Design and Trimming Tehniques for Subdivision Surfae based Modeling, On New Algorithmsof Curve and Surfae Modeling Based on Probabilisti Type Operators and Probability Distribution5 Publiations Most Closely Related to Proposed Researh:1. Similarity based Interpolation using Catull-Clark Subdivision Surfaes (with S. Lai), to appear in TheVisual Computer.2. Parametrization of Catull-Clark Subdivision Surfaes and Its Appliations (with S. Lai), Computer AidedDesign & Appliations, 3,1-4 (2006), 513-522.3. Subdivision Depth Computation for Catmull-Clark Subdivision Surfaes (with Junhai Yong), ComputerAided Design & Appliations, 3,1-4 (2006), 485-494.22

4. Adaptive Subdivision of Catmull-Clark Subdivision Surfaes (with J. Yong), Computer-Aided Design &Appliations 2,1-4 (2005), 253-261.5. Constrained Saling of Catmull-Clark Subdivision Surfaes (with S. Lai and S. Zou), Computer-AidedDesign & Appliations 1,1-4 (2004), CAD04, 7-16.5 Seleted Other Publiations:1. Matrix based Subdivision Depth Computation for Extra-Ordinary Catmull-Clark Subdivision SurfaePathes (with Gang Chen), Geometri Modeling and Proessing - GMP2006, Leture Notes in ComputerSiene, Vol. 4077, Springer,2. Voxelization of Catmull-Clark Subdivision Surfaes (with S. Lai), Geometri Modeling and Proessing -GMP2006, Leture Notes in Computer Siene, Vol. 4077, Springer, 595-601.3. Near-Optimum Adaptive Tessellation of General Catmull-Clark Subdivision Surfaes (with S. Lai), Ad-vanes in Computer Graphis (CGI2006), Leture Notes in Computer Siene, Vol. 4035, Springer, 2006,562-569.4. Subdivision Depth Computation for Extra-Ordinary Catmull-Clark Subdivision Surfae Pathes (withGang Chen and Junhai Yong), Advanes in Computer Graphis (CGI2006), Leture Notes in ComputerSiene Vol. 4035, Springer, 2006, 404-416.5. Texture Mapping on Surfaes of Arbitrary Topology using Norm Preserving Based Optimization (withS. Lai), The Visual Computer 21,8-10 (2005), 783-790.6. Adaptive Rendering of Catmull-Clark Subdivision Surfaes (with S. Lai), Pro. 9th Int. Conf. onComputer Aided Design and Computer Graphis (CAD/CG 2005), De 7-10, 2005, Hong Kong, 125-130.U.S. Patents:� Four U.S. Patents pending.Persons ollaborated within last 48 months:Brian Barsky, Yifan Chen, Shuhua Lai, Kenjiro T. Miura, Minetada Osano, J. Sone, Paul Stewarts, WilliamToll, Minoru Ueda, Gang Chen, Junhai Yong, Xiaoming Zeng, Caiming Zhang, Pifu ZhangGraduate Advisor:Prof. Ranko Bojani - Ohio State UniversityGraduate Student and Postdotoral Advisees (past 60 months):1. Dr. Shuhua Lai - Virginia State University, Beijing, China2. Dr. Junhai Yong - Tsinghua University, Beijing, China3. Dr. Xuefu Wang - Google, CA4. Dr. Yong Zhou - UCLA5. Dr. Huaijun Wu - MIT6. Dr. Caiming Zhang - Shandong University, Jinan, China7. Dr. Pifu Zhang - Dalhousie University, Halifax, Canada8. Dr. William Toll - Taylor University, Indiana 23