Is This A Quadrisected Mesh?

Transcription

1 I Thi Quadrieted Meh? Gariel Tauin alifornia Intitute of Tehnology Tehnial Report STR Deemer 2000 STRT In thi paper we introdue a fat and effiient linear time and pae algorithm to detet and reontrut uniform Loop udiviion truture, or triangle quadrietion, in irregular triangular mehe Intead of a naive equential traveral algorithm, and motivated y the onept of overing urfae in lgerai Topology, we introdue a new algorithm aed on gloal onnetivity propertie of the overing meh We onider two main appliation for thi algorithm The firt one i to enale interative modeling ytem that upport Loop udiviion urfae, to ue popular interhange file format whih do not preerve the udiviion truture, uh a VRML, without lo of information The eond appliation i to improve the ompreion effiieny of exiting lole onnetivity ompreion heme, y optimally ompreing mehe with Loop udiviion onnetivity Extenion to other popular uniform primal udiviion heme uh a atmul-lark, and dual heme uh a Doo-Sain, are relatively trightforward ut will e tudied elewhere R ategorie and Sujet Deriptor: I35 [omputer Graphi]: omputational Geometry and Ojet Modeling - urfae, olid, and ojet repreentation General Term: Sudiviion urfae, 3D Geometry ompreion, lgorithm, Graphi 1 INTRODUTION Sudiviion urfae are eoming a popular multi-reolution repreentation in modeling and animation [18, 17] For example Figure 2- how the reult of applying Loop triangle quadrietion heme [6] to a triangular meh Sine the mot popular interhange file format, uh a VRML [15], do not preerve the udiviion truture, a prolem exit if the model i aved uing one of thee file format and further editing i required at a later time lternatively, a proprietary file format with upport for udiviion urfae an e ued, ut limiting the ditriution of the ontent The method introdued in thi paper to detet uniform quadrietion onnetivity and to reontrut the udiviion truture olve thi prolem Mot 3D geometry ompreion tehnique por polygonal mehe preerve the onnetivity information without lo [13] Lole onnetivity ompreion heme are important for example, when a meh i arefully ontruted y an artit uing a modeling or animation pakage In thi framework, hanging the onnetivity may detroy important feature uh a reae line In mot interative modeling ytem polygonal mehe are ontruted and refined y alifornia Intitute of Tehnology, Department of Eletrial Engineering, MS , Paadena, 91125, On aatial from IM TJWaton Reearh enter, POox 704, Yorktown Height, NY Figure 1: Example with omplex topology : oare triangular meh "!$# %'&)'*+,-/'/) ;:=<$>?@ ) i not a quadrieted meh eaue it overing meh i onneted : Quadrietion []\_^`aa of oare meh EDGFFIHJK4LM=N9O'OP4QRTS U9VXWZY ) The lutering meh of thi quadrieted meh ha tw o onneted omponent Rendering of edge ha een turned of q'qr4tuwvyxxz {~}= )ƒ f : Firt omponent of lutering meh dfehgg4ikjl4mnpo ) D: Seond omponent of lutering meh equivalent to oare meh reurively applying a equene of operator to a relatively imple ae meh Some of thee operator, uh a uniform udiviion, hange the onnetivity, while other, uh a moothing, hange the geometry One example of thi proe i a meh ontruted y reurively applying a mall numer of uniform udiviion and moothing tep to a oare meh [10] When a 3D polygonal meh i large and generated y overampling a relatively mooth urfae with imple topology, uh a thoe produed y 3D anning ytem, loy onnetivity ompreion heme an e ued Simplifiation algorithm [3] an e regarded a loy onetivity ompreion tehnique nother very effiient heme to ompre thi kind of data i aed on remehing, ie, on approximating the geometry of the given polygonal meh y a emi-regular udiviion urfae within ertain tolerane, and uing wavelet-aed oding tehnique to ompre the D

2 Ö ½ ¾ ½ À D Ó Ô Õ T /T , T /T 4,096 1, ,384 3, ,536 8, ,144 18, Tale 1: ot of enoding the onnetivity of a tetrahedron and eight mehe ontruted y reurive triangle quadrietion with the MPEG-4 3D Meh oder T: numer of meh triangle, : ot to enode onnetivity in it, /T average ot to enode onnetivity, in it per triangle ÜÝ åæç âãáä Þ)ßáà Ú)Û ô'õ ü'ýþ îkï ùúáû èké êë ìí öt áø òó ØTÙ ð4ñ Figure 3: Notation ued for triangle quadrietion : triangle ÿ with edge edge! #"$ %,,- /0#123 &')+*, and 46587:9 ;< =>?@ 8 : The quadrieted triangle with V6W:XZY [\]^#_`ade8f8g -vertie E F G, HI, and JK, L -vertie M NO G, PQR, and ST8U, and fae 6ŠŒ m Ž 8 # G, hikjml n ö pq rt#uv8w8x, yz {m} ~ ƒ ˆ G, and Figure I 4 ~ š œ w=ž~ÿ 2: lgorithm overview : oare meh ˆ Š)Œ''Ž ª«= ± 9² ³Eµ ) : Quadrieted meh ~ ' ' T = ) The lutering meh of thi quadrieted meh ha two onneted omponent : Firt omponent of lutering meh ¹»º)k½¾ À=ÁœÂÃ'ÂIÅƻǚȜÉwÊ Ë=ÌÍÎ'ÏÐ ) Note the large numer of hole and fae overlap D: Seond omponent of lutering meh equivalent to oare meh geometry information [5] Thi method doe not produe good ompreion reult when the topology i not imple, though, and replaing the onnetivity of the meh i not alway aeptale Uniform udiviion heme an e regarded a optimal Ñpr o- greive onnetivity ompreion heme, eaue the ot of enoding eah udiviion tep i ontant [11] Unfortunately, urrent geometry ompreion heme [13] do not detet udiviion onnetivity, and a a reult, the ot of enoding a uniform udiviion tep i normally funtion of the ize of the oare meh For example, Tale 1 how the ot of enoding the onnetivity of a tetrahedron and eight mehe ontruted y reurive triangle quadrietion with the MPEG-4 3D Meh oder [8] in inglereolution mode [12] Note that, the total ot of enoding a quadri- D eted š œ:žÿ meh i aout twie the ot of enoding the original meh, while if optimally enoded, the inremental ot hould e ontant ª«: ± ² ³µ ¹ #º», orreponding to the numer of it ued to repreent the intrution peifying the udiviion operation in the ompreed ittream Other ingle reolution heme [14] are more effiient at ompreing thee quai-regular mehe, ut till the inremental ot of enoding a quadrieted meh i funtion of the ize of the oare meh The method introdued in thi paper, to detet uniform udiviion onnetivity and to reontrut the udiviion truture, an e ued to minimize the ot of enoding the onnetivity information of a fine meh with uniform udiviion onnetivity y repreenting the onnetivity information a a oare meh followed y one or more uniform udiviion tep, rather than a a fine meh ompreed with a ingle-reolution or progreive heme 2 LGORITHM OVERVIEW polygonal meh i defined y the poition of the vertie geometry), y the aoiation etween eah fae and it utaining vertie onnetivity); and optional olor, normal and texture oordinate propertie) In thi paper, we are primarily onerned with the onnetivity of triangular mehe Figure 3 how the notation we ue to repreent a triangle and the reult of quadrieting it We all tile et a group of four onneted triangle with the ame onnetivity a the reult of quadrieting one triangle, ie, four triange onneted a in Figure 3- The enter triangle of a tile et i onneted to three orner triangle through regular edge The orner of the tile et are the vertie of the orner triangle not hared with the enter triangle In a naive traveral algorithm to olve our prolem tile et are equentially ontruted while the meh i travered, ay in depthfirt order, trying to over it avoiding tile overlap, ie, every fae i allowed to elong to at mot one tile et If when the meh traver-

3 Î Á Ú Ó Ç Æ Å Ó t Ø ý o v p Figure 4: The overing meh of a triangular meh : quadrieted meh : overing meh with olor-oded onneted omponent The onneted omponent are artifiially diplaed in pae The quadrietion of the purple onneted omponent i equivalent to the input meh al proedure top, not all the fae are overed y tile et, a new traveral mut e tarted from a tile et not viited during the previou traveral Sine eah triangle i overed y up to four tile et, we may need to retart the traveral up to four time to deide if the fine meh ha udiviion truture or not Non-manifold ituation are diffiult to handle, and may require aktraking Intead of thi equential algorithm, we propoe an alternative gloal approah, where all the traveral i avoided, and replaed y a parallelizale algorithm to ontrut the overing meh of a triangular meh The overing meh of a triangle meh i ompoed of triangular fae alled tile upported on the ame et of vertie The tile are in ÂÃÆÅ orrepondene with all the tile et that an e ontruted in the original meh, and when quadrieted, eah one ha the ame onnetivity a the orreponding tile et Our algorithm, motivated y the onept of overing urfae in lgerai Topology [7], i aed on a theorem that tate that a triangular meh i a quadrieted meh if and only if it i equivalent to the quadrietion of one onneted omponent of it overing meh Figure 4 illutrate thi ontrution for a imple quadrieted meh The onneted omponent of the overing meh are painted in different olor, and diplaed in pae vertex poition are irrelevant) Note that the quadrietion of the purple onneted omponent i equivalent to the input meh There i a annonial mapping etween the overing meh and the orreponding triangular meh, that aign vertie to vertie and fae to fae Etalihing whether or not the quadrietion of a given onneted omponent of the overing meh i equivalent to the original meh redue to imple and linear ounting algorithm that determine if the annonial mapping retrited to the onneted omponent i ÈÉ¹Ê and onto or not Ë 3 POLYGONL MESHES In thi etion we introdue ome definition, notation, and fat aout polygonal mehe that we will need in uequent etion to formulate our main reult more preiely It an e kiped on a firt reading onnetivity The onnetivity of a polygonal meh Ì i defined y the inidene relationhip exiting among it Í vertie, edge, and Ï fae We will alo ue the ymol Ð G, Ñ G, and to denote the et of vertie, edge, and fae fae with Ô orner i a equene of Õ ÖÙØ different vertie If ÛÆÜmÝ Þß à!á!á!á8àâãåä i a fae, all the ylial permutation of it orner are onidered idential, ie, æmçéè êëì!í!í!í8ìdîïðñ ò ó!ô õ!õ!ö ø ùûú8ü ý þ ÿ for Multiple onneted fae fae with hole) are not repreentale proedure onnetedomponent # initialize partition to et of ingleton!" # traveral for eah regular edge # onneting $ % and & ' #if)+* and,- are in different partition if / find :9 find;<=?> ) # join orreponding partition uniondfefg H I # JLKNMOPRQTSTUNVXWNY[Z\] ^ _ `?a find et of upporting vertie # return umehe return degf h i i j?kml n Figure 5: Proedure to ontrut the onneted omponent of a meh Vertie not ontained in any fae are alled iolatednedge q i an un-ordered pair of different vertie that are oneutive in one or more fae of the meh We will denote y r the et of fae inident to the edge oundary edge ha exatly one inident fae u regular edge ha exatly two inident fae ingular edge ha three or more inident fae eaue the edge are derived from the fae, we write a meh a wyx{z ~}5 We aume the reader i familiar with the onept of manifold, orientale, oriented and non-manifold meh The algorthm introdued in thi paper do not make ue of thee onept, though e ay that two fae ƒ and onneted omponent W are onneted if we an find fae ˆ?Š Š Š m Œ3Ž uh that eah fae hare and edge with it ueor? in the equene note that š œ and Ÿž are valid hoie) Thi i an equivalene relation on the et of fae that define a partition into dijoint onneted omponent Ç N? 5 Eah one of thee onneted omponent i a maximal uet of onneted fae, ie, a uet of fae that atifie the following property: given a fae in the uet, a eond fae i onneted to the firt one if and only if it alo elong to the uet Together with it uet of upporting vertie ª «, eah onneted omponent R± define a umeh ² ³~µ Ŗ¹º5»N¾½ Note the uet of vertie ÁÀ  à à Ã?Âm are not neearily dijoint, ie, different onneted omponent may hare vertie We all a meh onneted if it i ompoed of only one onneted omponent It i uffiient to know how to olve our prolem for onneted mehe: if the meh i not onneted, firt deompoe it into onneted omponent, and then olve the prolem for eah omponent n algorithm aed on Tarjan fat union-find data truture [9] an e ued to partition the et of fae of a meh into it onneted omponent It i deried in peudoode in Figure 5 It firt initialize the partition to one ingleton per fae, and then for eah edge of the meh, and eah pair of different fae haring the edge, replae the uet orreponding to the two fae y their union ÈÊÉNËÍÌÏÎÑÐÍ from Mapping mapping a firt meh ÓÕÔ into ááâ a eond meh Ö i defined y a vertex funtion ÙRÚ:ÛÝÜÞàß and a fae funtion ãåä æ[çéèëêíì[î that atify the following additional property: for every fae ï ðòñôó õ ö ø ù?útûürýÿþ of the firt meh, the equene of vertie of the eond meh defined y the vertex funtion applied to the orner of the fae, i modulo ylial permutation) equal to the fae of the eond meh that the fae of the firt meh i mapped to y the fae funtion ie,! #" $&%')+*,* -/021

4 P ½ { G ý ² P Û Ú ìhíâî ñóò éêâë!"$# % ßÕàâá ãhäâå ôõõ ÜÝÞ ïwð ö øùûúü ýhþâÿ æ)çè Figure 6: Example of triangle meh quadrietion : oare meh; : quadrieted meh with 3 -vertie red) orreponding to vertie of the oare meh, and 4 -vertie lak) Equivalene Two mehe 576 and 879 are alled equivalent if ajlknm mapping :<; = >@?7 t exit uh that oth DFE and GIH are and onto funtion In uh ae the mapping O i alled a meh equivalene Note that ine the et of vertie and fae are finite, the mapping Q i an equivalene if and only if the vertex and fae funtion are onto, and the numer of vertie and fae in oth mehe are the ame: RIS TVUW and XZY\[N]_^ nd a imple linear time and pae algorithm to ount the numer of element of et `aled fhgjilknmporq an e ued to determine if a funtion utwvvxzy etween two finite et i onto or not reate a inary }~ array with element in orrepondene with the element of and initialized to ƒ Then, for eah element ˆ et the element orreponding to 2Š eœ to Finally, add all the value of the array The funtion i onto if and only if the um i equal to the numer of element of Ž Quadrietion Figure 6 how an example of a fine meh ) with 24 triangle reulting from quadrieting a oare meh ) with 6 triangle The vertie of the oare meh are a uet of the vertie of the fine meh We all thee vertie the -vertie of 2 u the fine meh The remaining vertie of the fine meh are in orrepondene with the edge of the oare meh We all thee vertie the -vertie of the fine meh Sine we are only onerned with onnetivity here, the poition of the -vertie in pae i irrelevant, ut for illutration purpoe, we draw them a the mid-edge point of the oare meh edge in Figure 6- Eah triangular fae of the oare meh i replaed y four triangle in the fine meh One triangle onnet the three inident -vertie, and eah of the other three triangle onnet one -vertex and two -vertie In general, the quadrietion operator š tranform a triangular meh œÿžz # into a new triangular meh N ª «# Z±, and define a mapping ²³µẃ z¹»º, whih aign eah vertex of into a ½ -vertex of ¾V G, and eah fae of À into the enter fae of the orreponding quadrieted fae oth funtion are Á2Âà ut not onto With repet to the numer of vertie, edge, and fae, the following relation hold: Å Æ Ç È ÉNÊ Ë ÌÎÍ Ï ÐÑN ÓÕÔ 1) Ö Ø ÙhÚ The quadrietion operator i one of many udiviion heme that introdue new vertie along the edge of the oare meh, and replae the oare fae y fine fae upported on the new et of vertie In general, eaue of limitation of the moothing Figure 7: Notation ued for tile ontrution : tile et : The orreponding tile proedure overingmeh &'*) # initialization +-,/0 # ontrut tile for eah fae $8:9<;$=?><@?EDGF # if inident edge are regular if HI J?KML N OPMQSRT are regular) append U7V$W X?Y<Z$[ \]<^_`a -d to egfihgjkml<n-omp return qr # ontrut tile and append to et of fae Figure 8: Proedure to ontrut the overing meh of a triangular meh Notation a in Figure 7 operator aoiated with thee udiviion method, mehe are required to e manifold without oundary, and peial moothing rule an e deigned for manifold mehe with oundarie hole) [1] ut ine the onnetivity refinement rule an e applied to non-manifold mehe, and our algorithm to detet and reontrut udiviion onnetivity alo work on non-manifold mehe, we allow our mehe to e non-manifold Note that the quadrietion operator preerve and reflet onneted omponent, ie, the onneted omponent of a meh t are alway in uwvyx orrepondene with the onneted omponent of the quadrieted meh z { 4 ONSTRUTING TILES } of a meh ~ ƒ M - are The tile et defined y the algorithm ued to ontrut them, whih we will derie with the notation introdued in Figure 7 Eah fae ŷ gš7 Œ:<Ž$?<? E G, with three regular edge, š œ, and :žÿ ha three neighoring triangular fae G,, and S Eah one of thee fae ª «ha a vertex? oppoite to the orreponding edge M ± The tile orreponding to i defined y thee three vertie ³ µ 7¹º»?¾½ ÀÁM ÃÅ Note that a a meh the quadrieted tile i equivalent to the umeh of Æ defined ÎSÏÐ y the fae Ç G, it three immediate neighor È:É Ê, G, and their uporting vertie, whih we all a tile et of Ñ G ËÌ Í, and 5 THE OVERING MESH The overing meh of a triangular meh ÓÕÔ ÖØmÙMÚ-Û i a new triangular meh ÜÝ/Þ5ßà-áâ<ã-ämå defined y the vertie and tile of æ Figure 8 illutrate the algorithm to ontrut the overing meh of a meh in peudoode

5 t T Š I ˆ Ç Å Ð ^ proedure IQuadrietion # ontrut overing meh % overingmeh ^ ) # partition šœ into onneted omponent Ÿž 7 =onnetedomponent ^ % `ª # for eah onneted omponent for «* ²±±³ # determine if equivalent to µ if IEquivalene º¹»g½>¾ return Ãœ return Æ ÀÁ% ) Figure 10: Peudoode of proedure to determine if a meh ha quadrietion onnetivity, and to reover the udiviion truture Figure 9: In general, four tile defined y the lue orner) over eah triangle red) Note that thee four tile have neither ommon vertie nor ommon edge in the overing meh Note that even if the meh i manifold without oundary, the overing meh may e non-manifold and with oundary lo, a illutrated in Figure 9, eah fae may elong to up to four different tile et of a triangular meh, where the given fine triangle oupie either the enter poition, or one of the three orner in eah one of the four tile et The numer of tile et overing a fae may e le than four, and even zero, though, uh a when a fine triangle or ome of it immediate neighoring triangle are inident to oundary or ingular edge èêéìëímîmï-ð The overing meh operator ç G, that aign a triangular meh to a new triangular meh ñóòôöõ -øúùmû-ü ý define a annonial mapping þ ÿ, from the quadrietion of into If we partition the overing meh into onneted omponent, and apply the quadrietion operator to eah one of them, we otain a partition of into onneted omponent!"""#%$ & The annonial mapping ' retrited / 0 to the onneted omponent alo define mapping *), Note that in general, the oreponding vertex and fae funtion of thee mapping are neither nor onto For <>= example, not all the fae of : may e overed y fae of ;? G, and up to four fae may e overing the ame fae G H of D With repet to MONQP vertie, retrited to the E -vertie of F, the mapping JLK i G, ut not neearily o when retrited to the R vertie; V>W and ome vertie of S may not orrepond to any vertex of U X However, the following theorem, whih ontitute the main reult of thi paper, hold: Theorem 1 onneted triangular meh Y Z\[^]`_ad ha m npo q i a quadrietion onnetivity, if and only if egf,hji%k l meh equivalene for ome r The proof of the uffiieny i trivial We refrae the neeity a follow: Theor the annonial mapping ƒ ` g ˆ for ome em 2 For every onneted triangular meh uwvyx^z {~}d, ŒŽ i a meh equivalene ÑÓ proedure IEquivalene È ÉËÊ ÌgÍÎ Ï # firt hek the eaiet neeary ondition if ÔÖÕ ØÚÙÜÛ Ý ) return fale if Þàß á âäã åçæéèüê ë ì ) return fale # initialize inary array í î7ïqðäñóò,ôöõ ˆø7ù9údûgüöýöþ # travere fae, udivide, and ount for eah tile ÿ for eah vertex of the tile et overed y for eah fae of the tile et overed y if!"$#&%' ì ) return fale if )+*,-0/132 ) return fale return true Figure 11: Proedure to determine if the mapping 465&7689: ; <>= i an equivalene Proof 2 Eah fae of? define one tile et in and a orreponding JLKNM tile in DEF Let GIH e the et of all thee tile in orrepondene with O Sine the vertie of thee tile are upported on P -vertie of Q, the et of vertie RTS of thee tile i in UVXW orrepondene with the et of vertie Y We have ontruted a meh equivalene etween Z [ \]_^ `a_edgfhji and the umeh of klm, whih an extended to an equivalene etween the orreponding quadrieted mehe Sine udiviion alo no preerve oneted omponent, we only need to how that i a onneted omponent of prq tu i learly onneted, eaue it i equivalent to v, the reult of udividing w i xry, whih i onneted, and the quadrietion operator doe not hange the numer of onneted omponent It only remain to e hown that no other tile are onneted to z { ut the tile in }~ that are not memer of ƒ [ are upported on -vertie, while tile in are all upported on -vertie, and o, dionneted Theorem 1 i the ai of our algorithm to detet uniform quadrietion onnetivity and to reontrut the udiviion truture, deried in peudoode in Figure 10 To determine if the mapping ŠŒƒŽ3 q i an equivalene or not it i not neeary to ontrut the quadrieted onneted omponent 3 It i uffiient to ount all the vertie and [ fae of tile et overed y tile in š œ Figure 11 how uh an algorithm in peudoode

6 Æ È É È 6 IMPLEMENTTION ND RESULTS polygonal meh i normally peified only y it vertie and fae, uh a in the ž Ÿ3 3 ª «node of the VRML tandard [15] Neither the edge, whih ontain the inidene relationhip among fae, nor the onneted omponent of the meh are expliitly repreented n expliit repreentation of edge i needed oth to partition the et of fae into it onneted omponent, and y our tile ontrution algorithm deried in etion 4 Effiient data truture to repreent edge of oriented manifold mehe, uh a the half-edge data truture [16] or the quad-edge [2] data truture are well known For non-manifold mehe, thee data truture need extenion [4] We will aume that the data truture ued to repreent the et of edge effiiently implement e ae funtion ± ² ³g µ ±¹ º»½¾, Á Â, the edg whih given two vertie À and return the et of inident fae whih may e empty if the two vertie do not orrepond to an edge of the meh) In our implementation, we ue a hah tale to implement the edge ae funtion Thi data truture an e populated ontruted) in linear time y viiting the fae in equenial order The full algorithm deried y the peudoode method hown in figure 5, 8, 10, and 11 ha een implemented in ++ Figure 2, and 1 how example where the algorithm ha een run on mehe of moderate ize with imple and omplex topology à 7 ONLUSIONS In thi paper we have introdued a oneptually very imple and effiient algorithm to detet quadrietion onnetivity in triangular mehe, and we have demontrated it in a numer of example explained in the introdution, thi algorithm ha important appliation in modeling ytem, and onnetivity ompreion heme In a uequent paper we plan to extend thi algorithm to other uniform udiviion heme, uh a atmull-lark, and Doo- Sain; and to ome adaptive udiviion heme REFERENES [1] H ierman, Levin, and D Zorin Pieewie mooth udiviion urfae with normal ontrol In Siggraph 2000 onferene Proeeding, page , July 2000 [2] LJ Guia and J Stolfi Primitive for the manipulation of general udiviion and the omputation of voronoi diagram M Tranation on Graphi, 42):74 123, 1985 [3] P Hekert oure 25: Multireolution urfae modeling Siggraph 97 oure Note, ugut 1997 [4] L Kettner Deigning a data truture for polyhedral urfae In 14th nnual M Sympoium on omputational Geometry, page , 1998 [5] Khodakovky, P Shröder, and W Swelden Progreive geometry ompreion In Siggraph 2000 onferene Proeeding, page , July 2000 [6] Loop Smooth udiviion urfae aed on triangle Mater Åthei, Dept of Mathemati, Univerity of Utah, ugut 1987 [7] WS Maey ai oure in algerai topology Springer-Verlag, New York-erlin, 1991 ISN X [8] Mpeg4-3dm-oder [9] RE Tarjan Data Struture and Network lgorithm Numer 44 in MS-NSF Regional onferene Serie in pplied Mathemati SIM, 1983 [10] G Tauin ignal proeing approah to fair urfae deign In Siggraph 95 onferene Proeeding, page , ugut 1995 [11] G Tauin, Guézie, W Horn, and F Lazaru Progreive foret plit ompreion In Siggraph 98 onferene Proeeding, page , July 1998 [12] G Tauin and J Ç Roigna Geometry ompreion through Topologial Surgery M Tranation on Graphi, 172):84 115, pril 1998 [13] G Tauin and J Roigna oure 38: 3d geometry ompreion Siggraph 2000 oure Note, July 2000 [14] Touma and Gotman Triangle meh ompreion In Graphi Interfae onferene Proeeding, Vanouver, June 1998 [15] The Virtual Reality Modeling Language ISO/IE , Septemer 1997 Ê˱ËÍÌÏÎgбÐeѱѱÑÓÔÑÕ Ö ØÚÙÛÝÜÞ ß [16] K Weiler Edge-aed data truture for olid modeling in urvedurfae environment IEEE omputer Graphi and ppliation, 51):21 40, January 1985 [17] D Zorin and P Shröder oure 23: Sudiviion for modeling and animation Siggraph 2000 oure Note, July 2000 [18] D Zorin, P Shröder, and W Swelden Interative multireolution meh editing In Siggraph 97 onferene Proeeding, page , ugut 1997