A NewParadigm for ChangingTopologyDuring Subdivision Modeling

ERGUN AKLEMAN�

VisualizationLaboratoryCollegeof Architecture

JIANER CHEN�

Departmentof ComputerScienceCollegeof Engineering

V INOD SRINIVASAN�

VisualizationLaboratoryCollegeof Architecture

Abstract

In thispaper, wepresenta new paradigmthatallowsdy-namicallychangingthe topology of 2-manifoldpolygonalmeshes.Our new paradigmalwaysguaranteestopologicalconsistencyof polygonalmeshes.Basedon our paradigm,by simplyaddingand deletingedges,handlescan be cre-atedanddeleted,holescanbeopenedor closed,polygonalmeshescanbeconnectedor disconnected.

Theseedge insertionand edge deletionoperationsarehighly consistentwith subdivisionalgorithms. In particu-lar, theseoperationscan be easily includedinto a subdi-vision modelingsystemsuch that the topological changesandsubdivisionoperationscanbeperformedalternativelyduringmodelconstruction.

We demonstratepracticalexamplesof topologychangesbased on this new paradigm and show that the newparadigmis convenient,effective, efficient,and friendly tosubdivisionsurfaces.

1 Intr oduction

Parametricrepresentationssuchas Bezier surfaces,B-splinesandNURBShave long beenpopularfor designingsmoothshapes[16, 4]. The greatestpower of parametricrepresentationsis that they allow real time smoothshapedesign[4]. Unfortunately, thesewidely usedtensorproductparametricrepresentationsdonot providea largevarietyoftopologiessincethecontrolmeshesof tensorproductpara-metricsurfacesmustbeorganizedasa regular rectangularstructure[29].

�Address:216 LangfordCenter, College Station,Texas77843-3137.

email:ergun@viz.tamu.edu.Supportedin partby theTexasA&M, Schol-arly & Creative Activities Program.�

Address: Departmentof ComputerScience,College Station, TX77843-3112.email: chen@cs.tamu.edu.Supportedin partby theNationalScienceFoundationunderGrantCCR-9613805.�

Address:216 LangfordCenter, College Station,Texas77843-3137.email: vinod@viz.tamu.edu.Supportedin partby theTexasA&M, Inter-disciplinaryActivities Program.

Subdivisionmethods[7, 13, 28, 22, 36, 35, 12] solve thefundamentalproblemof tensorproductparametricsurfaceswithoutsacrificingthespeedof shapecomputation.Unliketensorproductsurfaces,in subdivisionsurfaces,thecontrolmeshesdo not have to have a regularrectangularstructure.Subdivision algorithmscan smoothout 2-manifold (or 2-manifoldwith boundary)polygonalmeshes[44].

An important property of subdivision surfacesis thatthey area generalizationof parametricsurfaces.If thecon-trol meshis organizedasaregularrectangularstructure,anyparametricsurfacecanberepresentedby subdivision algo-rithms.For instance,theDoo-Sabinsurface[13] is agener-alizationof quadricB-splinesandtheCatmull-Clarksurfaceis a generalizationof cubicB-splines[7] andNon uniformrationalsubdivision surfaces[36] arethegeneralizationofNURBS.

Themainproblemwith subdivisionschemesis thattheydonotsupporttopologychange.Thisrestrictionmeansthatwith subdivision algorithmsthedesignerscanonly changetheshapeof theobjects.They cannotaddor deletehandles,openand closeholes,connector disconnecttwo objects.Sincethesetopologychangingoperationsareessentialfordesigningunusualandinterestingshapes,it is importanttocombinetheseoperationswith subdivisionalgorithms.

In thispaper, wepresentapurely”polygonal” and”non-implicit” approachchanging the topology of polygonalmeshes.The topologicalchangeswe demonstrateincludeopeningandclosing holes,creatinganddeletinghandles,connectingtwo disjoint meshesinto one, and separatingone meshinto two disconnectedones. Thesetopologicalchangesaresimplydoneby insertingor deletingedgesto apolygonalmesh.During theseoperations,we alsoguaran-teethatpolygonalmeshesremainvalid 2-manifolds.There-fore,unlikeimplicitly basedtopologicaloperations,ourop-erationsare subdivision friendly, i.e., subdivision opera-tions can always smoothout the meshesobtainedby ouroperations.

Our approachto handlingtopologicalchangesmakesanaturalpartnerwith the subdivision approachwithin a 3Dmodelingsystem. Our approachis not only suitableformodelinginitial 2-manifoldpolygonalmeshes,but alsouse-

ful for changingthe topology of a 2-manifold polygonalmeshthat aresmoothedby subdivision operations.Sincesubdivisionoperationsareessentialfor theimprovementofthe quality of meshes,performingthe topologicalchangeoperationsandthesubdivisionoperationsalternatively pro-videsa powerful shapemodelingapproachthat supportsahierarchyof topologychangesandquality improvementsatdifferentlevelsof details.

Our approachis basedon graphrotation systems. Agraphwith a rotation systemis embeddedto a unique2-manifold shape[14]. It alwaysguaranteesthe representa-tion of valid 2-manifoldpolygonalmeshes.Moreover, it iseasyto changethetopologyof meshesby simply insertingor deletingedgesin the correspondinggraphrotationsys-tem. An edgeinsertoperationcaneither(1) combinetwofacesby insertinga hole,(2) combinetwo facesby addinga handle,(3) combinetwo facesby joining two separate2-manifold meshes,or (4) subdivide a single faceinto two.Conversely, anedgedeleteoperationcaneither(1) separateonefaceinto two by closinga hole, (2) separateonefaceinto two by deletinga handle,(3) separateone faceintotwo by disconnectingone2-manifoldobjectinto two, or (4)combinetwo facesinto one. In eitheroperationsthe firstthreecaseschangethe topologyof a mesh. The last onedoesnot changethe topology, it only changesthe numberof polygons.

In order to representgraph rotation systemsand effi-ciently implementedgeinsertionand deletionoperations,we have developedthe Doubly Linked Face List (DLFL)datastructure.Using DLFL, we demonstrateexamplesoftopologychangesbasedonthisnew paradigmandshow thattheparadigmis convenient,effective,andefficientfor topo-logicalchangesfor subdivisionsurfaces.

Wefirst introducetheconceptsbehindsubdivisionmeth-odsand topologicalrequirementsimposedby subdivisionmethods.Wethenintroducethepreviousworksin graphro-tationsystem.Then,wedemonstratewhy andhow edgein-sertandedgedeleteoperationswork by illustratingeachofthesecaseswith figures. Thediscussionsin thesesectionsareformal andabstract.The readerswho areinterestedinonly implementationissuescanskipthemandgodirectlytothesectionthatcoversimplementationissuesandexamples.

2 Topological Consistency and SubdivisionMethods

With the introductionof subdivision surfacesandwiderusageof implicit surfaces,topologyhasbecomeanimpor-tant elementof computergraphicsresearch,developmentandproduction.Therehasbeenvariousstudiesin topologi-calmodelingduringthelastdecade[17, 38, 37, 19].

Subdivision surfacesassumethat the usersprovide anirregularcontrolmesh.Theseinitial controlmeshescanei-

therbecreatedby directmodelingor obtainedby scanninga sculptedreal object. A smootherversionof this initialmeshwithoutchangingtheoriginal topologyis obtainedbysubdivisionoperations.All subdivisionschemescanbeex-pressedby a setof linear differenceequations.More for-mally, eachnew point is computedasa linearcombinationof a setof pointsin a local topologicalregion. Theschemecanbewrittenasa linearsystem

������ ������

where��� and ������ arethe vectorsof respectively the oldpoints and the new points in the local topologicalregionand � is thetransformationmatrix [44]. Notehere,the lo-cal topologicalregion shouldcorrespondto a simpledisk(topologically).This impliesthatunderlyingstructuremustbea valid 2-manifold.

Sincethequalityandtopologyof thesmoothsurfacere-sultingfrom subdivision rulesdependgreatlyon theinitialcontrolmesh,theoreticalassuranceof thequality of initialcontrolmeshesis extremelyimportant.In otherwords,theprocessof obtainingtheinitial controlmeshmustberobustandguaranteevalid 2-manifolds. Unfortunately, setoper-ations,which are the most commonlyusedoperationsinmeshmodeling,canresultin non-manifoldsurfaces.More-over, the existing data structuresin meshmodeling arespecificallydevelopedin suchawaythatthey canrepresentnon-manifoldsurfacesresultingfrom thesetoperations.Inparticular, they do not guaranteevalid 2-manifoldsurfaces.Becauseof this fundamentalproblem,in theprocessof ob-taining the initial control mesh,unwantedartifactscanbegenerated.Theseartifactsincludewrongly-orientedpoly-gons,intersectingor overlappingpolygons,missingpoly-gons,cracks,andT-junctions. Therehave beenrecentre-searchefforts to correcttheseartifacts[3, 33].

Besides guaranteeingtopological consistency, datastructuresfor meshmodelingshouldalsosupporttopologi-caloperationsefficiently.

The classicalview of meshrepresentationis basedonadjacency relationshipsbetweenpoints, edgesand faces.For instance,the vertex-edgeadjacency relationshipspec-ifies two adjacentverticesfor eachedge. Thereexist ninesuchadjacency relationships,but it is sufficient to maintainonly threeof theorderedadjacency relationshipsto obtaintheothers[41].

In most practical computer graphics applications,meshesareoften representedwith oneadjacency relation-ship. The datastructureis generallyorganizedas an un-orderedlist of polygonswhereeachpolygonis specifiedbyan orderedsequenceof vertices,andeachvertex is speci-fied by its � , � , and � coordinates[3]. Let uscall this datastructurea vertex-polygonlist. Vertex-polygonlists do notalwaysguaranteetopologicalconsistency. In addition,theycanevencreatedegeneraciessuchascracks,holesandover-

laps[3, 33].Thesedegeneraciescanbe partly eliminatedby adding

an additionaladjacency relationship:edgelists to vertex-polygonlists [3]. In a vertex-polygon-edge list structure,alist of vertices,a list of directededges,anda list of poly-gonsare described. Verticesare specifiedby their threecoordinates,directededgesare specifiedby two vertices,andpolygonsarespecifiedby anorderedsequenceof edges.Eachpolygonis orientedin aconsistentdirection,typicallycounter-clockwisewhenviewedfrom outsideof themodel.Becauseof thelastcondition,vertex-polygon-edgelistsaremorepowerful thanvertex-polygonlists. However, therep-resentationdoesnot guaranteevalid manifold surfacesei-ther. It is still possibleto specifya non-manifoldsurfaceintermsof thevertex-polygon-edgelist.

Oneof theoldestformalizeddatastructuresthatsupportsmanifold surfacesis the winged-edge representation[5].Baumgartalsosuggestedusingawinged-edgestructureandEuleroperatorsin orderto obtaincoordinatefreeoperations[6]. Winged-edgedatastructuressupport2-manifoldsur-faces,andstartingfrom a valid 2-manifoldmesh,winged-edgecan only createvalid 2-manifoldswith Euler opera-tors. However, like vertex-polygon-edgelists winged-edgestructurescan also acceptnon-manifoldsurfaces[5, 40].To representVoronoidiagramsandDelauney triangulation,GuibasandStolfi introducedthe quad-edge datastructureandtwo topologicaloperators,make-edgeandsplice[21].Like winged-edgestructure,quad-edgestructurecreatesvalid 2-manifoldswith make-edgeandspliceoperationsifit startsfrom a valid 2-manifold.Unfortunately, quad-edgedatastructurescanstill supportnon-manifoldsurfaces.

When using set operations,resulting solids can havenon-manifoldboundaries[24, 25]. It is worthwhileto notethatalthoughthedatastructures,suchaswinged-edge,canhandlesomenon-manifoldsurfaces,they actuallycompli-catethealgorithmsfor solid modeling[25, 32]. Therefore,it is interestingto note that datastructuresthat can sup-portawiderrangeof non-manifoldsurfaceshavebeenlaterinvestigated.Examplesof suchwork areWeiler’s radial-edgestructure[42], Karasick’sstar-edgestructure[31], andVanecek’sedge-baseddatastructure[39].

In thecurrentpaper, weproposeto returnbackto theba-sic conceptof coordinatefree operationsover 2-manifoldsurfacesby ignoring set-operations. Similar to our ap-proach,insteadof setoperationstheusageof Morseopera-torsthatdescribethechangesof cross-sectionalcontoursatcritical sections(peaks,passesandpits) hasrecentlybeeninvestigated[38, 19]. We usetopologicalgraphoperationswhich aresimilar to Euler’s operationsandbasedon graphembeddings.Thebiggestadvantageof ouroperationsis thatthey areextremelysimpleandalwaysguaranteetopologicalconsistency. Only two operations,Insert edge andDeleteedge, areenoughto changethe topology. If an insertedor

deletededgechangethe topology, we can efficiently findthenew topologyby usinggraphembeddings.Thisefficientcomputationis dueto ourDoublyLinkedFaceList (DLFL)datastructure[10]. We proposeto usethisdatastructuretosupporta representationin which thebasictopologicalop-erationsrelatedto computergraphics,suchassurfacesub-division,addingor removing a handle,canall bedoneveryefficiently.

DLFL not only supportsefficient computationson 2-manifolds,but also alwaysguaranteestopologicalconsis-tency, i.e. it alwaysgivesa valid 2-manifold.

DLFL structureis basedontheclassicaltheoryof graphrotation systems. An extensive researchhasbeendoneinthe studyof graphrotationsystems.Among the extensiveresearchin the areaby mathematicians,the most remark-ableresultis thatgraphrotationsystemsprovidenecessaryandsufficientconditionsin representingvalid 2-manifolds.OurDLFL structureis anefficient implementationof graphrotationsystems.

3 Graph Rotation Systems

In this section,we introducehistoricalbackgroundandsomemathematicalfundamentalsfor graph rotation sys-tems(see[20] for moredetaileddiscussion).

The conceptof rotation systemsof a graphoriginatedfrom thestudyof graphembeddingsandit is implicitly dueto Heffter [23] who usedit in Poincaredualform. A graphembeddingin anorientablesurfacecorrespondsto anobvi-ousrotationsystem,namely, theonein whichtherotationateachvertex is consistentwith thecyclic orderof theneigh-boring verticesin the embedding.Edmonds[14] was thefirst to call attentionexplicitly to studyingrotationsystemsof agraph.

Let � be a graph. A rotation at a vertex � of � is acyclic permutationof theedge-endsincidenton � . A rota-tion systemof � is a list of rotations,onefor eachvertex of� . Givena rotationsystemof a graph � , to eachorientededge ������� � in � oneassignstheorientededge ������!"� suchthatvertex ! is the immediatesuccessorof vertex � in therotationatvertex � . Theresultis apermutationonthesetoforientededges,that is, on thesetin which eachundirectededgeappearstwice, oncewith eachpossibledirection. Ineachedge-orbitunderthispermutation,theconsecutiveori-entededgesline up headto tail, from which it follows thatthey form a directedcycle in thegraph. If thereare # ori-entededgesin anorbit, thenan # -sidedpolygoncanbefittedinto it. Fitting a polygonto everysuchedge-orbitresultsinpolygonson both sidesof eachedge,andcollectively thepolygonsform a 2-manifold.

Edmonds[14] hasshown thatevery rotationsystemof agraphgivesa uniqueorientable2-manifold. Moreover, thecorrespondingorientable2-manifoldis constructible,aswe

describedabove. For any 2-manifold $ , thereis a graph �anda rotationsystem% of � suchthat % correspondsto anembeddingof � on $ [8].

Therefore,theexistenceof thebijective correspondencebetweengraphembeddingson orientable2-manifoldsandgraph rotation systemsenablesus to representtopologi-cal objectsby combinatorialones. In particular, every 2-manifoldcanberepresentedby arotationsystemof agraph,andevery rotationsystemof a graphcorrespondsto a valid2-manifold. In consequence,the presentationof graphro-tation systemsalways guaranteestopologicalconsistency.Recently, Chen,GrossandRiperhave developeda veryef-ficient algorithmthat, given a rotationsystemof a graph,constructsthecorresponding2-manifold[9].

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Figure 1. A graph drawn in a rotation sys-tem and the corresponding embed ding on thesphere .

Basedon graphrotationsystems,a graphthat is drawnin 2D canuniquelyrepresenta 2-manifoldshape.For ex-ample,considertwo graphsgivenin Figures1 and2. Thesegraphsare drawn in sucha way that the rotation at eachvertex can be tracedby traversing the incident edgesincounter-clockwiseorder. Theseare two different rotationsystemsfor thesamegraph.Whenweconsidertherotationorderthey correspondto two different2-manifoldshapes:thespherein Figure1 andthetoroid in Figure2.

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Figure 2. The same graph as in Figure 1 drawnin a diff erent rotation system and the corre-sponding embed ding on the tor oid.

The graphrotationsystemalsoprovidesa conciseandsimpleinternaldatarepresentationasa setof verticesanda rotationorderfor eachvertex. For example,the rotationsystemof thegraphin Figure1 is givenby theorderedlists:

� �"& �(')�*+�,-� �* & �,.� � �('-��, & � � �*+�('/� �(' & � � �,0�*�

andtherotationsystemof thegraphin Figure2 is givenbyorderedlists:

� �"& �(')�,+�*-� �* & �,.� � �('-��, & �(')�*+� � � �(' & � � �,0�*1

Rotation systemscan be easily implementedas a set oflinked lists. We shouldalsopoint out that eachedgeof agraphappearsexactly twice in any of its rotationsystems.Therefore,the amountof computermemoryusedfor thisrepresentationis small.

Basedona rotationsystem,eachfacecanbeeasilycon-structedby traversingthe faceboundary, following the ro-tation order of the verticeson the boundary, as shown inFigure3. By a face-cornerof a face,wereferto avertex onthefaceboundary, plusthetwo neighboringedges.

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Figure 3. Rotation order of faces is providedby rotation system.

4 Changing Topology with Graph RotationSystems

We have observed that graphrotation systemprovidesa greatconveniencein topology changes[2]. Only edgeinsertandedgedeleteis enoughto changethe topologyofa2-manifoldmeshsupportedby graphrotationsystems.

4.1 Inserting or Deletingan Edgein a Graph Ro-tation System

Any rotation order of two verticesof a graphrotationsystemcanbegivenby thefollowingrepresentationwithoutlossof generality:

� � & � , � *.2 � � ' & �3+�405where 2 and 5 areany sequenceof vertices. Let an edgebe insertedbetween� � and �(' asshown in Figure4. Thenew graphrotationsystemcanbeeasilyobtainedby simplyupdatingtherotationordersof � � and �(' asfollows:

� � & � , � ' � *.2 � � ' & �3+� � �4+5

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Figure 4. Inser ting an edge.

Although this insert operationis extremely simple, itchangesthe facesof the 2-manifoldobjectrepresentedbythe graphrotationsystem.It eitherconnectstwo facesby“merging” theminto onesinglefaceor subdividesa singlefaceinto two faces.Thesetwo casesareillustratedin moredetailasfollows:

6 The face-corners7��,-��� � �8�*)9 and 7�� 3 ���('-��� 4 9 do notbelongto the sameface. In this case,the insertop-erationconnectsthe two facesby merging themintoonesinglefaceasshown in Figure5.

6 Theface-corners7�� , ��� � ��� * 9 and 7��3�8� ' �8�409 belongtothesameface.In this case,the insertoperationsubdi-videsthefaceinto two asshown in Figure6.

Theedgedeletionoperationcanbeelaboratedin a simi-lar manner. Basically, if anedgehasits two sidesin differ-ent faces,thendeletingtheedgemergesthe two facesintoa largerone.On theotherhand,if thetwo sidesof theedgeare in the sameface,thendeletingthe edgeseparatesthefaceinto two faces.

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A single facev1v2v3v1v4v5v6v4

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More intuitive representations of single facev1v2v3v1v4v5v6v4

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Figure 5. Inser ting an edge between two dif-ferent faces merges the two faces.

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Figure 6. Inser ting an edge between two ver-tices of the same face divides it into two faces.

4.2 TopologyChangewith EdgeInsert and EdgeDelete

Let therebe two facesthat aredescribedby a rotationsystem.We assumethat thesetwo facesarenot thesame.However, they cansharea vertex, anedgeor severaledges.They may also belong to two disconnected2-manifolds.Basedontherotationsystem,any edgeinsertedby connect-ing thesetwo facesconvertstheminto onesinglefaceaswehaveshown in Figure5.

If thesetwo facesbelongto two disconnected2-manifoldshapes,aftertheinsertedgeoperationthesetwo 2-manifoldshapesbecomeconnectedby thenew insertededge.If thefacesbelongto thesame2-manifold,theedgeinsertoper-ation increasesthe genusof the 2-manifoldby one,i.e., ahandle(or a hole)is added.

In bothcases,a new facecreatedby anedgeinsertoper-ation is, in fact,a handleasshown in Figure7. Therefore,wecall this typeof face“face-handle”. It is not easyto vi-

sualizethis face-handle,howeverthestructureof thehandlebecomesapparentwhenthefaceis smoothedby subdivisionoperationsasrepresentedin Figure7.

If this handlegoesthroughthe insideof a 2-manifold,it becomesa hole. If the handlegoesoutsideof the 2-manifold, it becomesa handle. The hole and the handleareautomaticallycreatedor deletedbasedongeometryandedgeinsertion/deletionruleswe describedin previoussec-tion.

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Subdivision

Figure 7. Handle becomes apparent after sub-division operations.

Thehandlethatresultsfrom a subdivision operationin-cludesan extraordinaryvertex with a high valance. (Thetermvalanceis usedto denotethenumberof edgesincidentona vertex. Valanceof anextraordinaryvertex on a handleis at leastequalto : ). In mostcases,theresultingsubdivi-sionsurfacewill notbe ; � continuousat thisextraordinaryvertex becauseof theunusualstructureof our face-handles.

Fortunately, it is possibleto improve the quality of thehandlesimplyby insertingnew edges.As shown in Figure8if a secondedgeis insertedto connectany two verticesoftheface-handle,thenew edgeseparatestheface-handleintotwo faceswithout changingthe topologyof the mesh.Ontheotherhand,thequalityof thehandleisusuallyimproved.

5 Implementation

As we have statedearlier, graphrotationis a mathemat-ical concept.Although, it canbe easily implementedasaset of linked lists, it doesnot supportcomputergraphicsimplementationsefficiently. Oneof themainproblemsus-ing sucha graphin an interactive systemis that it requiresconstructionof facesin eachrenderingstep.

5.1 DLFL Structur e

In our implementation,we usea Doubly Linked FaceList (DLFL), a datastructurewe have proposedtheoreti-cally earlier[2]. A DLFL structurealwayscorrespondsto

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Figure 8. Improving the quality of handles byinser ting more than one edge.

a graphrotation system. Therefore,it always guaranteesvalid topologicalconsistency [2]. ThemainreasonthatweuseDLFL is that it supportslogarithmictime edgeinser-tion anddeletionoperations“with faceconstruction”.WiththeDLFL datastructure,subdivisionoperationscanalsobeimplementedeasily.

In DLFL structure,eachfaceis givenby a sequenceofverticescorrespondingtoaboundarytraversingof theface.1

Thevertex appearancesin thesequencewill becalledver-tex nodes. Note that two consecutive vertex nodesin thesequencecorrespondto anedgesidein therotationsystem.Thesequenceis representedby a cyclically concatenatabledatastructure.

Formally, let %��<�=� bea rotationsystemof a graph � �<>?�8@A� with faceset B . A doubly-linked-face-list(DLFL)for the rotation system %��<�=� is a triple C 7EDF�HGI�8J�9 ,wherethe face list D consistsof a set of K BLK sequences,eachis givenby a linkedlist andcorrespondsto thebound-ary walk of a facein the rotationsystem%M�E�=� . Moreover,theselinkedlists areconnectedby a circulardoubly linkedlist. Thevertex array G has K >NK items.EachGPO �-Q is a linkedlist of pointersto the vertex nodesof � in the facelinkedlists in D . Theedge array J has K @NK items.Each GRO �-Q alsoincludesthe3D positionof therelatedvertex. Each J?O S0Q isdoublylinkedto thefirst vertex nodesof thetwo edgesidesof theedgeS in thefacelinkedlist in D . Figure9 givesanillustrationof theDLFL datastructurefor a tetrahedron.

1To simplify the implementationwe uselinked lists for the faces.Tofurther improve efficiency of our system,balancedtreesfor boundarywalksof facescanbeusedinsteadof linkedlists [2].

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Figure 9. An illustration of the DLFL datastructure for a tetrahedr on (Only details forver tex � � and edge S 4 are sho wn).

5.2 Examples

To show thefeasibilityof ourapproach,wehavecreatedvarious3D modelsthat show the connectionof several 2-manifoldsandthecreationof handlesandholes.In all ourexamples,wefirst convertirregularmeshesinto meshesthatconsistof only quadrilateralsto obtain initial mesh. We,then,smoothout this initial meshby usingCatmull-Clarksubdivisionscheme[7].

6 Figure10 shows the connectionof four disconnectedtetrahedra. All facesof the tetrahedraare first sub-divided to make eachfacea quadrilateral.Then, theinside facesof every two neighboringtetrahedraareconnectedby aninfinitely smalledge.It is interestingto notethatthese3D shapesmay“look” non-manifold.They areactually3D representationsof a rotationsys-temwhich is a representationof a 2-manifold.There-fore, we can apply subdivision algorithms. On theotherhand,if they hadbeenrepresentedby setopera-tions,eventheinternalrepresentationwouldhavebeennon-manifold.

6 Figure11 shows the connectionof two disconnectedtoroids. This figure alsogivesan exampleof handleimprovementby insertingadditionaledges.As canbeseenclearly in Figure11, whenonly oneedgeis in-sertedit createsa ; � discontinuousextraordinaryver-

Figure 10. Connecting four tetrahedra.

tex. Insertingextraedgesremovesthis ; � discontinu-ity.

6 Figure12 shows anexampleof thecreationof a han-dlefor acupanddemonstrateshandleimprovementbyinsertingadditionaledges.

6 Figure13 shows anexampleof thecreationof a hole.As it is clear in this figure, eachadditionaledgeim-provesthequalityof theholein thecube.

More figuresareprovided in the color pages.TheFig-ures14, 15, 16 and 17 shows two examplesof topologi-cal changes.TheFigures14 and15 shows the creationofvarioushandlesfor a small stellateddodecahedron[43] (aregularpolyhedrondiscoveredby Kepler). TheFigures16and17 shows the creationof several holesin a greatdo-decahedron[43] (A regularpolyhedrondiscoveredby De-Poinsot,100yearsafterKepler. It is thedualof smallstel-lateddodecahedron).

We offer a few implementationremarksbeforewe closethissection.

The topological changesare implementedin our ap-proachby connectingfacesin a polygonalmesh.Thetypeof thetopologicalchangesis closelyrelatedto thevisibilityof thecorrespondingfaces.Generallyspeaking,if thefaces

Figure 11. Connecting two tor oids: the samecolored faces connected with edge inser tion.

arevisible from eachotherfrom the“outside”of themesh,thenaddinganedgeto connectthetwo facescreatesa han-dleto themesh.If thefacesarevisiblefromeachotherfrom“inside” of themesh,thenaddinganedgetoconnectthetwofacesopensa holein themesh.Whenthetwo facesarenotvisible from eachother, eitherfrom outsideor from inside,addinga straightedgeto connectthetwo facesstill givesatopologicallycorrectmeshbut the meshwill be geometri-cally self-intersected.Thus,theuserscaneasilyavoid suchself-intersectionsbasedon thefacevisibility properties.

Usingmorethanonestraightedgeto connecttwo facescreatesanextraordinarypointwith veryhighvalancein thefacehandlesinceeachedgeaddse to thevalance.Suchanextraordinarypoint with a high valancewill mostlikely be; � discontinuous.It is easyto avoid extraedgesby extrud-ing thefacesbeforeinsertinganedgeasshown in Figure12

Figure 12. Creation of handle for a cup.

asanexampleof thecreationof ahandle.

6 Conclusionand Future Work

In thispaper, wepresentedanew paradigmfor changingtopology of 2-manifold mesheswithout using an implicitapproach. Our new paradigmguaranteesthe 2-manifoldpropertyfor meshesduringthesetopologicalchanges.Thenew paradigmfor changingtopology is highly consistentwith the subdivision approachin a 3D modelingsystem.Our approachcanbe usedfor modelinginitial 2-manifoldpolygonalmeshes,as well as for changingthe topologyof 2-manifold polygonal meshesthat are smoothedbyCatmull-Clarksubdivision operations. Note that smooth-ing operationsprovidedby subdivisionapproachareessen-tial for improving the quality of 2-manifolds. Thus, thetopologicalchangeand subdivision operationsperformedalternatively duringmodelconstructionprovidea powerfulshapemodelingparadigmthatsupportsahierarchyof topol-ogy changesandquality improvementin differentlevelsofdetails.

Our approachcanalsobeextendedto includetopologi-

Figure 13. Creation of a hole in a cube .

cal changesinto progressive meshes[26] andmultiresolu-tion representationsof meshes[15, 11]. Two majoropera-tions in progressive meshestechniquearevertex-split andedge-collapse.It can be shown that theseoperationsdonot changethe topology[20]. The samecanalsobe eas-ily shown for multiresolutionrepresentationsof meshes.Infact,oneof themajorproblemin multiresolutionmeshmor-phing is that sourceandtarget mustsharethe sametopol-ogy[27]. By includingtheedgeinsertionandedgedeletionoperations,it is possibleto changethe topology for pro-gressivemeshesandmultiresolutionrepresentations.Thus,thisapproachmaypotentiallyaddnew powerto theexistingmorphing,compressionandsimplificationschemes.

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Figure 14. Creation of various handles forsmall stellated dodecahedr on.

Figure 15. Handles impr ovement for for smallstellated dodecahedr on.

Figure 16. Creation of various holes for greatdodecahedr on.

Figure 17. Hole impr ovement for great dodec-ahedron.