NURBS rendering in OpenSG Plus

Transcription

1 NURS rering in OpenSG Plus F. Kahlesz Á. alázs R. Klein Universiy of onn Insiue of Compuer Science II Compuer Graphics Römersrasse onn, Germany Absrac Mos of he indusrial pars are designed as rimmed NURS. Today s graphics hardware suppors he high-speed display of polygonal objecs consising of riangles. This means ha analyical surfaces (like NURS) need o be riangulaed so hey can be displayed by 3D graphics cards. To be able o conrol he qualiy of he esselaed model, he riangulaion should be carried ou wih a given error orelance. Currenly available mehods make i possible o achieve approximaion of he surface in 3D space wih a given error and o polygonize he rimming curves wih anoher given error, bu only in he parameer space of he surface. From his fac follows ha here exiss no error conrol of he rimming curves in Euclidean space, which can resul in cracks along he rimming curves in case of models consising of more han one surface - and models usually consis of several hundreds or housands of surfaces. Our new mehod inroduced in his paper akes care of his problem, i.e. i can guaranee he geomeric error along he rimming curves. This makes also possible o sew hese surfaces ogeher along heir borders. Several examples of indusrial daa 1 demonsrae he applicabiliy of our new mehod. The inroduced echniques will also be included ino he OpenSG scenegraph API as he basic ool for NURS rering. Keywords: NURS esselaion, error olerance 1 Inroducion Designing machine pars using CAD ools needs geomeric modeling from he designer. The mos popular and de faco sandard way of geomeric modeling in he indusry is o use rimmed NURS surfaces. NURS provide a convenien way o describe surfaces of almos any shape and rimming hem gives he designer he possibiliy o hrow away he uneeded areas of ha surface; combining housands or even en-housands of rimmed surfaces makes i possible o describe very sophisicaed objecs like cars, airplanes or submarines. The analyical represenaion of rimmed NURS canno be used direcly on curren graphics pipelines, herefore every NURS surface of a complex objec mus be esselaed before i can be feed o he graphics hardware. The riangulaion of he surfaces needs wo approximaions: firs, he surface should be approximaed wih a given error in space and second, he rimming curves are approximaed wih polylines. This polygonizaion of he rimming curves in curren mehods is carried ou in he parameer plane of he surface hey are applying o. This can resul in arifacs along he common borders of surfaces of an objec as polylines are subsiued ino he surface wihou error conrol on he geomeric posiion of he 3D border poins. 1 The images of objecs presened in his paper are kindly provided by DaimlerCrysler. Our new mehod overcomes his problem: he inroduced esselaion algorihm ensures ha he verices on he border polygons are in a given proximiy o he original rimming curve in space. This, besides eliminaing he effec of a wrongly chosen error olerance parameer for he approximaion of he rimming curves resuling in arifacs, also raises he possibiliy of sewing he surfaces of an objec along he rimmings curves based on a geomerical disance. A shor overview of our mehod is he following: firs we conver he surface and is rimming curves from Spline represenaion ino èzier represenaion. Afer his is done, we approximae he surface wih a quadree. The rimming curves are racked and subdivided no o cross quadree leaves. This is followed by performing he rimming and riangulaion. The final sep is he sewing of differen surfaces. Firs, we briefly discuss previous work abou NURS esselaion a he of his secion. Secion 2 inroduces our daa-srucure used in various seps of our mehod. Secion 3 discusses he mahemaical background of NURS curves and surfaces. Secion 4 shows how he approximaion of he surfaces is carried ou using a quadree. Secion 5 is abou deermining he inersecions of èzier curves and sraigh lines, his mehod is used in he algorihm discussed in Secion 6, which resrics he pars of he èzier curve represenaion of he rimming curves o he leaves of he quadree, which was generaed in Secion 4. This resricion enables us o approximae he rimming curves wih a given error in 3D Euclidean space. Secion 7 describes rimming and riangulaion. Secion 8 oulines a possible sewing algorihm. In secion 9 we presen resuls and conlusions. There is considerable lieraure on he opic of rering NURS curves and surfaces. Differen approaches are based on ray racing [11], [3], scan-line generaion [12], [13], pixel level subdivision [14], [15]. Due o recen advances in graphics hardware, algorihms based on polygon esselaion are increasingly popular, and are generally much faser han oher approaches. Such algorihms, based on uniform and adapive subdivision have been published [16], [17], [18], [19]. However, a major drawback of mos curren approaches is ha hey deal wih each curve/surface individually, and make no aemp o consruc one mesh ou of several paches. [9] presens a novel approach using super-surfaces o decrease he number of riangles in he esselaion, bu his approach needs aprioriconneciviy informaion. Anoher recen publicaion is [10], bu i only deals wih very specific configuraion of NURS surfaces ha are sacked on op of each oher. 2 Daa-srucures Our main daa srucure is a 3D mesh srucure, which is able o handle boh manifold and nonmanifold cases. The mesh daa-srucure consiss of he following hree componens:

2 ffl Verices. The verices are sored in a STL se, in order o be able o make sure ha no duplicae verices can be insered. For his we use a lexicographic soring on he 3D coordinaes of he verices in he mesh, hus guaraneeing uniqueness. For each verex he following informaion is sored: 3D coordinaes for his verex. Poiners o edges ha are conneced o his verex. Poiners o faces ha his verex is par of. Unique id of his verex, for file I/O. ffl Edges. The edges are also sored in a STL se, for similar reasons as in he case of verices. For each edge he following informaion is sored: Poiners o he wo verices. Poiners o he faces his edge is par of. Orienaion: an edge may be oriened. Unique id of his edge, for file I/O. ffl Faces. The faces are simply sored in a STL vecor. For each face he following informaion is sored: Poiners o he verices of his face. These need no be in any paricular order. Poiners o he edges of his face. These need no be in any paricular order eiher. The norm associaed wih his face, during he quadreegeneraion phase. (This will be needed laer o approximae he rimming curves ha go hrough his face.) The èzier represenaion of he rimming curves ha go hrough his face. Poiners o he enry and exi verices of he rimming curves ha go hrough his face. Unique id of his face, for file I/O. The following operaions are suppored on he mesh: addverex, addedge, addface, addtriangle, addquad, addquadtreeleaf (special funcion used by he quadree generaor o add a leaf of he quadree o he mesh), spliedge (splis an edge ino wo edges a a specified poin, insering he necessary verex, and keeping he mesh consisen). Our second special daa-srucure is a direced graph. This is used when raversing along he already subdivided rimming curves. I has wo main componens: ffl Nodes. The nodes of he graph are sored in STL vecors. Each node conains a vecor, which holds he indices of he edges ha connec o his edge. I also has an (emplaed) arbirary node-info field. We use i o sore he 2D parameer space coordinaes associaed wih he node. ffl Edges. The edges of he graph are also sored in STL vecors. Each edge conains he following informaion: A flag which decides wheher he edge is direced or no. The indices of he wo nodes of his edge. (The order is only significan when he edge is direced.) A flag which decides wheher he edge is valid or no. Invalid means he edge is logically deleed from he graph. Arbirary emplae edge-info. The following operaions are suppored on he graph: AddNode, addedge, deleeedge, geedges (ge all edges ha go hrough a given node) genode, geedge (ge a reference o a node/edge wih a given index), geedgedirecion, seedgedirecion, changeedgedirecion, read, wrie. We build our daasrucures on he C++ Sandard Templae Library [5], [6]. This gives us boh flexibiliy and robusness. 3 Geomeric descripion of rimmed NURS surfaces A rimmed NURS surface can be defined by a se of rimming loops ogeher wih he surface iself. Each rimming loop mus consis of a se of NURS curves which are defined over he parameer space of he surface. We only give he basic definiions of NURS curves and surfaces here, he defailed mahemaical background can be found for example in [1] or in [2]. A NURS surface of degree (p; q) can be defined by: S(u; v) = nx mx i=0 j=0 R i;j(u; v)p i;j where n +1, m +1are he number of conrol poins in he u and v direcions, respecively. P i;j are he conrol poins and R i;j(u; v) are he piecewise raional basis funcions defined as N i;p(u)n j;q(v)w i;j R i;j(u; v) = P n k=0p m l=0 N k;p(u)n l;q (v)w k;l where w i;j are he weighs and N i;p(u) and N j;q(v) are he nonraional -spline basis funcions defined on he kno vecors U = a; :::; a; u p+1; :::; u r p 1; b; :::; b V = c; :::; c; v q+1; :::; v s q 1; d; :::; d where r = n + p +1and s = m + q +1. Since we require he rimming loops o consis of ses of NURS curves, each curve of he rimming loops can be represened in NURS form: C(u) = nx i=0 R i;p(u)p i where P i are he conrol poins, and R i;p(u) are he raional basis funcions defined as: N i;p(u)w i R i;p(u) = P n j=0 Nj;p(u)wj where N i;p are he ph-degree -spline basis funcions defined over he kno vecor: U = e; :::; e; u p+1; :::; u m p 1; f; :::; f where m = n + p +1. The number of conrol poins is n +1. Since our algorihms operae on curves and surfaces given in he èzier represenaion, firs we have o decompose each NURS curve ino a se of èzier curves and each NURS surface ino a se of èzier paches. This is done via simple kno-inserion [1], [2].

3 4 Surface approximaion using a quadree A quadree is used o riangulae he NURS surface wih a given error. The iniial leaves of he quadree are he èzier paches convered from he original NURS using kno-inserion. We approximae he paches wih wo riangles laid on he corner poins of he èzier surfaces. The parameric error of his approximaion is deermined for each pach and if his error is greaer han a given " hreshold, a midpoin subdivision is carried ou on he èzier surface. This midpoin subdivison is done recursively unil all approximaion errors are smaller han ". In our mesh daa srucure we implemened a funcion o subdivide recangular faces which auomaically preserves he neighbourhood informaion of he subdivided paches; wih he help of his mehod, he surface approximaion algorihm becomes very simple: foreach ezier pach while approximaion error() >ffldo midpoinsubdivision( ) compue bilinear norm( ) The approximaion error of he riangulaion of a èzier pach is compued using he following lemmas from [8]: Lemma 1. Four poins are given: b 00, b 01, b 10, b 11. If a bilinear surface f(u; v) = P 1 i=0p 1 j=0 bij1 i (u) 1 j (v) over [0; 1] 2 is approximaed hrough wo piecewise linear funcions over wo riangles ((0; 0); (1; 0); (1; 1)) and ((0; 0); (1; 1); (0; 1)) or ((0; 0); (1; 0); (0; 1)) and ((1; 0); (1; 1); (0; 1)), hen he resuling error is indepen of he selecion of riangles and is value is ffl = 1 kb00 b01 + b11 b00k: (1) 4 P m Lemma 2. Le f(u; v) = i=0p n j=0 bijm i (u)j n (v) be aèzier pach wih conrol poins b ij 2 R d and le g(u; v) = P 1 i=0p 1 j=0 bin;jm1 i (u)j 1 (v) be a bilinear inerpolaion surface of he corners b 00, b 0m, b n0, b nm, hen he following can be saed: where ( sup kf(u; v) g(u; v)k» 0»u;v»1 (m i)(n j) b 00 + mn kf gk 1 = (2) c ij = b ij (m i)j b 0m + mn max kcijk; i=0;:::;mj=0;:::;n i(n j) ij bn0 + mn mn bnm) From he previous lemmas i follows ha an upper bound of he approximaion error for a èzier pach can be compued by summing 1 and 2. The compue bilinear norm() funcion calculaes an approximaion of he norm of he bilinear funcion which ransforms he recangle in he parameer space of he original NURS surface corresponding o he appropriae èzier pach o he 3D Euclidean space. This norm is laer needed, when he allowed approximaion error of he rimming curves over he quadree leaves is compued. If bl : R 2 7! R 3 is a bilinear funcion and P i, i =1:::4, are he four corners of he ransformed parameer space in R 3 wih P 1 corresponding o (0,0) and P 3 o (1,1), we can esimae he norm: kp3 P1k kblk = max fkp2 P1k; p ; kp 4 P 1kg; (3) i=2::4 2 i.e. we simply measure he norm based on how much he uni square sreches during ransformaion. 5 Inersecion of èzier curves and sraigh lines In order o approximae he rimming èzier curves wih he given error, we have o do his approximaion individually on each quadree leaf. Thus, he rimming curves have o be subdivided, since hey do no generally or begin exacly on he borders of quadree leaves. To achieve his, we have o find he inersecion poins a èzier curve and a sraigh line (in he parameer space of he èzier curve). This is done in wo seps. Given a èzier curve C wih conrol poins C i and an arbirary line (L) we firs conver he èzier curve ino he explici (or non-parameric ) form D(u) = nx i=0 D i n i (u): where D i =(u i;d i) are he èzier conrol poins of he new ( explici ) curve, wih evenly spaced conrol poins u i = i n and signed disances d i from he ih original conrol poin o L. This form has he following imporan propery: he new èzier curve crosses he x axis a he same parameer values as he original èzier curve crosses he line. This conversion process is shown in deail in [3]. Having his new form, we simply apply recursive subdivision a = 0:5 for each new curve, eiher unil he curve is compleely below or above he x axis (his can be checked very efficienly), or he area of he curve s bounding box is smaller han a predefined ffl value. In his case we record a hi a he curren parameer value. Noe ha his way we are guaraneed o find all inersecions. For a more deailed descripion of his mehod see [2], [4]. 6 Guaraneeing a given error along he rimming curves in 3D space As saed in he inroducion, we wan o guaranee ha no verices along he rimming borders deviae farher han a P given error from he analyical rimming curve boundaries. For his, we have o be n able o approximae a rimming curve. If p() = i=0 bin i () is aèzier curve and l is a linear approximaion of p() wih l(0) = p(0) and l(1) = p(1), hen he following over esimaion holds for he approximaion error: sup j p() l() j» 2n 1 1 2[0;1] 2 n 1 max i=0;:::;n kbi n i b 0 i i n bnk ased on his error esimaion a simple recursive midpoin subdivison algorihm can ensure he desired linear approximaion of he èzier curve. Le f beaèzier ensor produc surface, c aèzier curve and l a linear approximaion of c in he parameer plane. Now kf ffi c f ffi lk clearly means he 3D Euclidean error if we subsiue l insead of c ino he èzier ensor produc surface. Denoing he bilinear approximaion of f wih bl, his error can be esimaed: kf ffic f ffilk»kf ffic blffick+kblffic blffilk+kblffil f ffilk (4)

4 1 3 OVER_FACE IN_VERTEX 4 2 Figure 1: The sae machine, which implemens he èzier chain racking over he mesh. Assuming ha f was creaed during he quadree algorihm, i is clear, ha he firs and hird ag of he sum on he righ side of (4) are smaller han " (he error hreshold of he quadree). Facoring ou bl and using he noaion ffi := kc lk,wegekblffic blffilk» kblkffi. As kblk canno be changed, he only way o conrol he error is o decrease ffi,i.e.over differen quadree leaves (he èzier paches ) we have o approximae he rimming curve wih differen errors, based on he bilinear norm of ha pach. If we wan o guaranee a» error on he boundaries, we can achieve his by choosing general " =» 3 for he quadree algorihm and a differen ffi =» 2ffl kblk for approximaing over differen quadree leaves. To be able o apply he described mehod we need o solve wo problems: ffl As we need èzier curves, we have o conver he original Spline curves ino chains of èzier curves. This can be easily done using kno inserion. ffl In order o be able o calculae he ffi values, we have o resric each curve over jus one quadree leaf, as here is no guaranee ha he creaed èzier curves do no cross quadree leaf borders. This resricion of he rimming curves is realized by racking each chain. If we find a curve which crosses a leaf border, we cu i ino wo èzier curves a he corresponding inersecion poin. The racking is realized via a sae machine wih wo saes: OVER FACE and IN VERTEX (see figure 1). These names indicae wheher we are over a face of he mesh (quadree) or we are in a verex of i during he following of he èzier chains. The nex sae deps on he acual sae and on he currenly racked èzier curve. If he algorihm begins o follow a new chain of curves, is sae is inialized based on he firs conrol poin of he firs curve of he new chain. The sae ransiions can be described as follows: 1. Sae ransiion 1 happens when inersecs one side of he curren face (only he inersecion wih he smalles parameer value is of ineres). If his inersecion poin is no already a verex in he mesh, a new verex is creaed (via SpliEdge operaion). This verex will be he curren verex. is subdivided ino wo curves, he firs par of will be sored o he lef face and he second par will be he curren. Ifhe curve coincides wih he face border, is inerpreed as a subdivison a = 1:0 and is he nex èzier curve of he racked chain. 2. leaves he curren verex over a face (in conras o ransiion 4). This face will be he curren face. 3. s over he curren face. is sored ino he curren face and he nex èzier curve of he racked chain becomes. 4. leaves he curren verex along an edge of he mesh. If s before i reaches he nex verex along he edge, a new verex is creaed a he poin, his will be he curren verex and he nex èzier curve of he racked chain becomes. If reaches he nex verex along he edge, i is subdivided, is firs par is deleed and he oher par becomes. The edge beween he previous and he curren verex will be oriened according o he rimming curve as i wen over i. Le us summarize wha has been achieved unil now: we sored he appropriae pars of he rimming curves in èzier form in he quadree leaves (he faces of he mesh). This allows us o approximae hem in he parameer plane guaraneeing a 3D error under he given hreshold. New verices were insered ino he edges of he mesh (on quadree leaf borders) where he rimming curves inerseced hem. The edges which were coinciden wih pars of he rimming curves have been oriened correcly. 7 Trimming and Triangulaion y insering he approximaion of rimming curves ino he mesh as direced edges (he orienaion is he same as he direcion of he rimming curve) and by deleing all he faces from he mesh we essenially creae a (semi-direced) graph. This graph spans he whole parameer space of he surface, and has direced edges exacly where he rimming curves are in he parameer space. The rimming is performed by ravelling along hese direced edges. The pseudo-code of he raversal is: finddirecededge() while here are direced edges lef do while we have a valid edge do sore sar node handleedge() genexedge() if we are back a he sar node hen handleedge() riangulae() geougoingedge() fi finddirecededge() The funcions used in he pseudo code are he following: ffl finddirecededge() Find a direced edge in he graph. ffl handleedge() If his edge was direced, delee i from he graph. Oherwise, make i a direced edge, wih opposie orienaion we raversed his edge. ffl riangulae() Given a sequence of nodes defining a polygon, riangulae i. (See below.) ffl genexedge() Given a node and an edge, find he lefmos edge which is no equal o he given edge. ffl geougoingedge() Given a node, find he ougoing edge: ha is, he edge which is direced and is poining ou of his node. Noe ha he consrucion of he graph guaranees ha here can be a mos one such edge. Whenever he graph raversal algorihm finds a closed polygon, we have o riangulae i. The polygon may no be convex, bu i mus be closed. The riangulaion produced is a consrained Delaunay [20] riangulaion. The following pseudo-code illusraes he algorihm used:

5 foreach edge of he polygon do Find a hird poin so ha he riangle made up of his 3 poins saisfies he Delaunay crieria. if here exiss such a poin hen Record his riangle. Subdivide he polygon ino wo new polygons o he lef and o righ of his new riangle, and call he riangulaor recursively wih hese wo new polygons. else Take he nex edge. fi Noe ha i is guaraneed ha here exiss a leas one edge, for which a suiable hird poin can be found. More deails and a proof can be found in [8]. 8 Ouline of he sewing algorihm The sewing algorihm sews ogeher appropriae pars of he differen mesh borders, if hese borders are closer o each oher han he sewing disance d s. efore describing he algorihm we should noe ha we have o calculae he disance of each boundary verex P o each boundary # edge. This means ha we have o carry ou i=1p # j=1 #Vi#Vj, where # denoes he number of boundaries and #V k he number of verices in he kh boundary. To handle his large number of operaions, we use a 3D grid; he boundary edges are scan-convered ino his grid, so he disance of a boundary verex is calculaed only o he edges in he grid cell of he verex and neighbouring cells. The daa srucures used in he sewing algorihm are: verex f mesh ID m; boundary ID b; bool original; sew o lis fverex v,...g; g Here he mesh ID field conains he informaion abou o which mesh his verex belongs o; he boundary ID field idenifies he boundary (see below) he verex belongs o; he original field is rue if he verex is from a mesh and false when he verex is insered ino he a boundary during he sewing; finally he uples of he sew o lis conains he closes verex v (on boundary boundary ID of v ) if v is closer o he verex han d s. The inpu of he sewing algorihm are he boundaries exraced from he meshes of each surface: boundary f verex lis g fverex ID v,...g; The verex lis is an ordered lis of idenifiers of he verices of he boundary. The ordering is based on how he verices follow each oher along he border. During he boundary exracion, he mesh field of all verices is se appropriaely, he original flags are se o rue and heir sew o lis liss are cleared. The oupu are o sew daa srucures. A o sew daa srucure holds he following informaion: o sew f along lis g f[v IDsar;v 1 IDsar], 2... [v ID;v 1 ID]g; 2 v v v_new Figure 2: Examples for find or inser closes verex(). In he upper case v NEW is no creaed, beacuse i was already a verex of. Inhelower example v NEW is creaed and insered ino. The field along lis conains which verex of 1 should be sewn o which verex of 2 and is ordered according o he raversal of he verices along he boudaries. 1 and 2 areimplicilysored by soring he verices in he along lis. Noe ha he along lis can conain verices which have no been par of he original mesh, bu were creaed during he sewing process. We use he following funcions wihin he sewing: v_new ffl find or inser closes verex( verex v, boundary b ): This funcion projecs v ono each edge of b (see he noe abou using a 3D grid a he begining of his secion) and calculaes he disance of his projecion, i.e. he Euclidean disance of v o he basepoin v P of he projecion. If no d P <d s is found, he funcion reurns. Oherwise he v P wih he smalles d P is insered ino b, preserving he ordering of b. Ifv projecs onoanalreadyexisingverexofb, no verex duplicaion occurs. We refer o he newly creaed (or already exising) verex as v NEW. The following operaions are carried ou on v and v NEW (see Figure 2): v NEW.original is se o false, if i was newly creaed, oherwise i is no changed he ID of v is insered ino v NEW.sew o lis he ID of v NEW is insered ino v.sew o lis ffl ge preliminary o sew lis( boundary ): This funcion ieraes hrough he verices of and examines heir sew o lis fields. If i finds ha a verex v START ges close o anoher boundary, i begins o rack ha proximiy unil he borders move away from each oher a verex vi END of. The funcion is capable of racking muliple proximiies a he same ime (his can happen when 3 or more surface are o be sewn); for each proximiy i finds, he funcion creaes a o sew srucure and fills in he along lis wih he corresponding verices (hey will be he uples of along lis). The corresponding verices reference o each oher is erased from he sew o liss of boh verices o avoid duplicaed deecion of proximiy segmens when processing he neighbouring boundary. The funcion reurns all he o sew srucures i creaes. These are called preliminary o sew liss, as foldings can happen. Such a case is depiced in Figure 3 a). No disincion is made beween original and newly creaed verices in his funcion. ffl process foldings( o sew prets ): This funcion resolves he folding problems a he sar and a he of prets creaed wih he previous funcion. Noe ha - as we laer carry ou a reparamerizaion of he common border pars (see below) - we do no have o ake care abou he folding verices wihin he common borders, jus a he sar and of he common inerval. We deec he folding by running along boh boundaries and regisering where we leave he common par. These

6 leaving verices will be valid corresponding sar and verices of he muual boundaries. Figure 3 b) shows an example of his operaion. Finally he funcion ses he original field of he valid sar and verices o rue and delees all verices wih false original value from boh boundaries of prets. ffl reparamerize( o sew TS ): This funcion reparamerizes he wo boundaries of he common border segmen in TS. The boundary pars of he muual inerval sored in TS are referred o as 1 and 2. The lenghs of boh border pars are calculaed by simply summing up he lengh of he border edges, hen, scaling hese lenghs o 1, he T 1 = f 11; :::; 1mg and T 2 = f 21; :::; 2ng parameer values are compued for he m and n verices of 1 and 2. T 1 and T 2 are merged in T = T 1 [ T 2. Processing he 2 T in increasing order, a new verex is insered ino eiher 1 or 2 if his verex does no already exiss (his avoids duplicaing verices). Having compleed his verex inserion, boh 1 and 2 will have less han m + n 1 verices (verices a = 0 and = 1 are always presen in boh borders) and hese verices will be in a 1:1correspondence o each oher. ased on hese known correpondences he uples of TS.along lis are filled appropriaely. In Figure 3 c) a simple reparamerizaion is shown, where ( 12) is insered ino 2 as a new verex. The hick line shows ha he border verices are moved ino he middle poins of he connecing segmens of he corresponding verices. Wih he help of he above funcions, he pseudo-code of sewing is as follows: foreach boundary do foreach verex v of do foreach boundary oher 6= do find or inser closes verex( v, oher ) sl = creae empy o sew lis() foreach boundary do sl.add o lis(ge preliminary o sew lis()) foreach o sew TS in sl do process foldings( TS ) foreach o sew TS in sl do reparamerize( TS ) Afer his code is execued, sl will conain o sew srucures describing he correc muual border pars wih 1:1verex correspondences. Of course, for all new verices in he borders, appropriae edge spliing operaions should be carried ou on he meshes, inroducing new riangles. Afer his is done, he border edges of he meshes can be moved o heir new posiions and hen he acual sewing can be execued by incremenally adding he meshes o a coninuously growing saring mesh. For a deailed descripion on he sewing process see [7]. 9 Resuls and conclusions We have presened a sysem ha is capable of esselaing rimmed NURS surfaces wih a given error olerance in Euclidean space. This approach makes i possible o sew he surfaces ogeher ino c) a) b) ( ) START END Figure 3: In a) examples for folding in he preliminary o sew liss can be seen. b) shows he resul of calling process foldings on he same common boundary, he verices indicaed wih black circles are he valid sar and poins of he common border, he arrows show he correspondences. c) reparamerizaion one mesh, making i suiable for simplificaion and LOD echniques. Our curren implemenaion runs on Linux worksaions. I reads a file in Open Invenor forma (.iv) conaining rimmed NURS surface objecs and esselaes hem using he algorihms oulined in his paper. Figure 4 shows a highly complex model esselaed wih our mehod. The following able shows he running imes (in seconds) of he various algorihms for differen models, excluding file I/O. The imings were done on a Penium III 866Mhz PC running Linux. Number of surfaces spline! èzier conversion Quadree generaion Quadree raversal Trimming and riangulaion Sum

7 [6] Sandard Templae Library Programmer s Guide. June, hp:// [7] F. Kahlesz, Á. alázs and R. Klein. Muliresoluion rering by sewing rimmed NURS surfaces. Submied o Solid Modeling [8] R. Klein. Nezgenerierung implizier und paramerisierer Kurven und Flächen in einem objekorienieren Sysem. PhD hesis, Universiy of Tübingen, [9] S. Kumar, D. Manocha, H. Zhang and K. E. Hoff. Acceleraed Walkhrough of Large Spline Models. Symposium on Ineracive 3D Graphics, 1997, pages , 190. [10] R. Kumar, P. Srinivasan, K. G. Shasry and. G. Prakash. Geomery based riangulaion of muliple rimmed NURS surfaces. Compuer-Aided Design 33 (2001) pages [11] J. Kajiya. Ray racing parameric paches. ACM Compuer Graphics, 16(3), 1982, (SIGGRAPH Proceedings) pages [12] J. M. Lane, L. C. Carpener, J. T. Whied and J. F. linn. Scan line mehods for displaying paramerically defined surfaces. Communicaions of ACM, 23(1), 1980, pages [13] J. T. Whied. A scan line algorihm for compuer display of curved surfaces. ACM Compuer Graphics, 12(3), 1978, (SIGGRAPH Proceedings) pages [14] E. Camull. A Subdivision Algorihm for Compuer Display of Curved Surfaces. PhD hesis, Universiy of Uah, Figure 4: Tesselaion resuls. The lower image shows an enlarged par of he upper image. The imings indicae ha on currenly available hardware he preprocessing required by our mehod is fas enough o be done during load ime. References [1] L. Piegl and W. Tiller. The NURS ook. Springer, [2] G. Farin. Curves and Surfaces for Compuer Aided Geomeric Design: A Pracical Guide. Academic Press Inc., [3] T. Nishia, T. Sederberg and M. Kakimoo. Ray Tracing Trimmed Raional Surface Paches. ACM Compuer Graphics, Vol.24, No.4, , pages [4] G. Farin and D. Hansford. The Essenials of CAGD. A K Peers Ld., [5]. Srousrup. The C++ Programming Language. Addison-Wesley, [15] M. Shanz and S. Chang. Rering rimmed NURS wih adapive forward differencing. ACM Compuer Graphics, 22(4), 1998, (SIGGRAPH Proceedings) pages [16] J. H. Clark. A fas algorihm for rering parameric surfaces. ACM Compuer Graphics, 13(2), (SIGGRAPH Proceedings) pages [17] A. Rockwood, K. Heaon and T. Davis. Realime rering of rimmed surfaces. ACM Compuer Graphics, 23(3), 1989, (SIGGRAPH Proceedings) pages [18] D. R. Forsey and V. Klassen. An adapive subdivision algorihm for crack prevenion in he display of parameric surfaces. In Proceedings of Graphics Inerface, 1990, pages 1-8. [19] R. Klein and W. Sraber. Large mesh generaion from boundary models wih parameric face represenaion. In Proc. of ACM SIGGRAPH Symposium on Solid Modeling, 1995, pages [20] F. P. Preparaa and M. I. Shamos. Compuaional geomery: an inroducion. Springer-Verlag New York, Inc