Interactive Rendering of Deforming NURBS Surfaces

Transcription

1 EUROGRAHICS 97 / D. Fellne and L. Szmay-Kalos Volume 6, (997), Numbe 3 (Guest Edtos) Inteactve Rendeng of Defomng NURBS Sufaces Fedeck W. B. L Rynson W. H. Lau Mak Geen Compute Gaphcs and Meda Laboatoy Depatment of Computng The Hong Kong olytechnc Unvesty, Hong Kong {cswbl, cswhlau}@comp.polyu.edu.hk Depatment of Compute Scence Unvesty of Albeta Albeta, Canada mak@cs.ualbeta.ca Abstact The non-unfom atonal B-splnes (NURBS) has been wdely accepted as a standad tool fo geomety epesentaton and desgn. Its ch geometc popetes allow t to epesent both analytc shapes and fee-fom cuves and sufaces pecsely. Moeove, a set of tools s avalable fo shape modfcaton o moe mplctly, object defomaton. Exstng NURBS endeng methods nclude de Boo algothm, Oslo algothm, Shantz s adaptve fowad dffeencng algothm and Slbemann s hgh speed mplementaton of NURBS. Howeve, these methods consde only speedng up the endeng pocess of ndvdual fames. Recently, Kuma et al. poposed an ncemental method fo endeng NURBS sufaces, but t s stll lmted to statc sufaces. In ealtme applcatons such as vtual ealty, nteactve dsplay s needed. If a vtual envonment contans a lot of defomng objects, these methods cannot povde a good soluton. In ths pape, we popose an effcent method fo nteactve endeng of defomable objects by mantanng a polygon model of each defomng NURBS suface and adaptvely efnng the esoluton of the polygon model. We also look at how ths method may be appled to mult-esoluton modellng.. Intoducton Non-unfom atonal B-splnes (NURBS) [,2] have become a de-facto ndusty standad fo epesentng cuves and sufaces. They eceve wdespead acceptance and populaty because they posses many mpotant advantages. Fo examples, they povde a common mathematcal fom fo epesentng and desgnng standad analytc shapes and fee-fom cuves and sufaces. A vaety of shapes can be poduced smply by manpulatng the contol ponts and weghts. In addton, they have a poweful geometc toolkt ncludng knot nseton, knot efnement and knot emoval, whch can be used thoughout the desgn pocess. Applcatons of NURBS nclude fee-fom suface modellng n CAD systems, vtual ealty and compute anmaton. In fee-fom suface modellng and vtual ealty, nteactve endeng of defomng NURBS sufaces s mpotant. In vtual ealty, n patcula, we may want to model scenes wth many defomable objects movng aound and the effcent endeng of these objects s ctcal. Even n compute anmaton, t s an advantage to be able to pevew a defomng object as some of ts paametes ae beng modfed. Because most hadwae gaphcs acceleatos can only handle polygons, the standad way to ende NURBS sufaces s polygonsaton that s tessellatng a NURBS suface nto polygons fo endeng. Theefoe, the most obvous appoach to addess the poblem s to acceleate the polygonsaton pocess. In ths pape, we pesent an effcent method fo nteactve endeng of defomng NURBS sufaces. The method acceleates the endeng speed wth two technques. Fstly, we consde the fame-to-fame changes of the polygon models geneated n consecutve fames as the object defoms. Instead of executng the polygonsaton pocess n evey fame, we mantan a polygon model of the defomng The Euogaphcs Assocaton 997. ublshed by Backwell ublshes, 08 Cowley Road, Oxfod OX4 JF, UK and 350 Man Steet, Malden, MA 0248, USA.

2 L et al. / Rendeng of Defomng NURBS Sufaces NURBS suface and ncementally update the polygon model to epesent the defomng NURBS suface at each ndvdual fame. Secondly, as the NURBS suface defoms and the polygon model s updated ncementally, some pats of the polygon model may become too coase to epesent the NURBS suface due to the ncease n local cuvatues. Ths wll esult n a polygonal appeaance of the suface. On the othe hand, othe pats of the polygon model may become too fne to epesent the suface due to the decease n local cuvatues. Ths, on the contay, wll esult n a waste of pocessng tme. To ovecome ths poblem, we suggest a technque fo ncementally efnng the esoluton of the pe-geneated polygon model. We also dscuss how ths technque can be used n mult-esoluton modellng. The est of the pape s ogansed as follows. Secton 2 descbes pevous wok on endeng NURBS sufaces. Secton 3 pesents the ncemental technque fo endeng defomng NURBS sufaces and secton 4 pesents the technque fo esoluton efnement of the pegeneated polygon model. Secton 5 dscusses how ou method can be used n mult-esoluton modellng. Secton 6 shows some expemental esults and dscusses the advantages and lmtatons of ou method. Fnally, secton 7 pesents conclusons of ou wok and dscusses some possble futue wok. 2. Related Wok Many methods have been poposed to speed up the pocess of endeng NURBS sufaces [3,4,5,6,7,8,9,0]. In ode to make use of the polygon endeng capabltes of exstng hadwae gaphcs acceleatos, the majoty of the methods developed ae based on polygonsaton. Fo example, the ecusve de Boo algothm [6] evaluates a NURBS suface n paamete space. Boehm n [4] poposed a subdvson method usng knot nseton. Cohen et al. [5] also poposed a smla subdvson method called the Oslo algothm. Ths method dvdes a suface nto polygons by smultaneously nsetng multple knots nto the suface. Howeve, t pefoms a lage numbe of knot nseton opeatons, whch ae computatonally vey expensve. A bette method s to convet the NURBS suface nto Béze patches by nsetng multple knots nto each nteo knot of the suface followed by ethe de Casteljau s [] o Hone s [2] Béze subdvson method. Hee, fewe knots ae needed to be nseted. The Mesa 3D gaphcs lbay [3] s an mplementaton of such technque. Othe methods poposed to acceleate the polygonsaton pocess, such as Shantz s adaptve fowad dffeencng algothm [9] and Slbemann s hgh speed mplementaton of NURBS [0], ae based on smplfyng the evaluaton of the NURBS equatons. All the methods mentoned above concentate on acceleatng the polygonsaton of ndvdual NURBS sufaces wthout concen fo the ncemental evoluton n successve fames. Hence, the complete polygonsaton pocess s executed n evey fame. Kuma et al. n the ecent pape [4] poposed an ncemental method fo dsplayng NURBS sufaces. The method geneates a polygon model fom the NURBS suface by convetng the suface nto Béze segments followed by a Béze subdvson pocess. The method uses the vewpont as a functon of coheence between successve fames fo the polygonsaton of the NURBS suface. Snce thee s usually only a small change n vew pont between two consecutve fames, the method mnmses the numbe of tangles geneated wthn the gven tme fame by ncementally polygonsng segments that wll become vsble n the next fame and deletng those that wll become nvsble. Howeve, ths method does not consde the stuaton when a NURBS suface s contnuously defomng. In such stuaton, the complete polygonsaton pocess stll needs to be executed. As such, ths method s moe sutable fo dsplayng statc NURBS sufaces. 3. Incemental NURBS Suface Defomaton As mentoned eale, exstng NURBS endeng methods do not consde the coheence of a defomng NURBS suface between two successve fames, and the complete polygonsaton pocess s executed n evey fame. If thee ae a lot of defomng objects n the scene and n patcula, f a fne tessellaton s equed, these endeng methods may be too costly to execute n eal-tme. The objectve of The Euogaphcs Assocaton 997

3 L et al. / Rendeng of Defomng NURBS Sufaces ths wok s to speed up the endeng of defomng NURBS sufaces. The man dea s to mantan two epesentatons of the NURBS suface, the suface model tself and the polygon model appoxmatng the suface model. As the suface defoms, the polygon model s not egeneated. Instead, all the polygon vetces n the ognal polygon model ae updated, such that the updated model appoxmates the defomed suface model. 3. Updatng the olygon Model A NURBS suface s a atonal genealsaton of the tenso poduct non-atonal B-splne suface defned as follows: n m j w R ( u, S( u, () u, whee 0 0, j, j, j n m w j j R 0 0,, j ( w, j ae the weghts,, j R, j ( u, v net, and ) ae the bass functons. fom a contol A polygonal epesentaton of the NURBS suface can be obtaned by evaluatng equaton () wth some fxed u and v paametes. If a contol pont s, t s moved to s, t, the dsplacement vecto can then be defned as * s. The ncemental dffeence V, t s, t between the two polygonal epesentatons of the suface befoe and afte the contol pont movement can be shown as follows: S( u, S( u, ( ) w R s, t s, t s, t s, t n m j w j R 0 0,. j * α V s, t ( u, ( u, whee S ( u, and S ( u, ae the polygonal epesentatons of the suface befoe and afte the contol pont movement espectvely. α s a constant fo each patcula set of u and v paamete values and defned as: w R ( u, α (2) ( u, s, t s, t s, t n m j w j R 0 0,. j If the subdvson esoluton emans the same befoe and afte the defomaton, all the scala coeffcents α can be pe-computed. Theefoe, s, t s, t the cost of updatng a polygon vetex fom the cuent fame to the next s only two opeatons, one multplcaton and one addton. In addton, wth the local contol popety of NURBS, t s known that the tanslaton of a patcula contol pont, of a NURBS suface would only affect s t the egon [ u s, us+ q ) [ vt, vt+ q ) whee q s the ode of the NURBS suface. Hence, we only need to update the polygon vetces wthn ths local egon to mnmse the polygon model update tme. 3.2 The Incemental Rendeng ocess Ou endeng pocess can be dvded nto two phases, the ntalsaton phase and the polygon model update phase. Intalsaton hase: Dung ths phase, we pefom a polygonsaton pocess to poduce the ntal polygon model of the NURBS suface. We also calculate the scala coeffcents fo all the contol ponts and stoe them fo subsequent defomaton. Howeve, f memoy s tght, the computaton of the scala coeffcents may be pefomed on demand upon the movement of one o moe contol ponts and the computed scala coeffcents ae then stoed fo subsequent defomaton. olygon Model Update hase: Dung ths phase, we update all the polygon vetces affected by the dsplaced contol pont by multplyng the dsplacement vecto wth the pe-computed set of scala coeffcents. When the defomaton nvolves multple contol ponts, we may consde each of them n sequence. 4. Resoluton Refnement Wth the technque poposed n Secton 3 above, we may now ende a defomng NURBS suface vey effcently. Howeve, snce the polygon model epesentng the NURBS suface s pegeneated, the esoluton of the polygon model s fxed thoughout the defomaton. The poblem wth ths s that f the cuvatue of the suface s nceased by a lage amount dung a defomaton pocess, the esoluton of the polygon model may become too coase to epesent the defomed suface and polygonal appeaance may esult. On The Euogaphcs Assocaton 997

4 L et al. / Rendeng of Defomng NURBS Sufaces the othe hand, f the cuvatue of the suface s deceased by a lage amount dung a defomaton pocess, the esoluton of the polygon model may become hghe than necessay to epesent the defomed suface. In ths stuaton, many polygons may be pojected to a sngle pxel esultng n a waste of pocessng tme. To ovecome ths poblem, we popose a technque fo nsetng o deletng polygons ncementally accodng to the change n local cuvatue of the suface. Inteactve fame ates can theefoe be mantaned by avodng complete e-evaluaton fo the new polygon vetces. The method s based on the de Casteljau algothm []. 4. Inseton and Deleton of olygons To ncease the esoluton of a egon, we dvde a polygon (o moe pecsely, a squae) nto fou smalle polygons. The majo computatonal cost of ths opeaton s to ceate a new set of scala coeffcents fo the new vetex. Howeve, calculatng the scala coeffcents wth equaton (2) s expensve. Hence, we deve a new technque whch geneates new scala coeffcents ncementally fom the exstng scala coeffcents nstead of e-calculatng them wth equaton (2). Ths technque mnmses the ovehead fo esoluton efnement. Afte a NURBS suface s decomposed nto Béze patches, a polygon model can then be obtaned by applyng the de Casteljau subdvson method [2] on each of the Béze patches. Fo smplfcaton, we deve the fomula by consdeng a atonal Béze cuve. The devaton of the fomula fo atonal Béze patches goes n a smla way. 4 Let,, R and 0 u, whee 0, n n 0,,, ae the homogeneous Béze ponts of a atonal Béze cuve of degee n. The Béze cuve can be defned by the de Casteljau Algothm as follows:,, n ( u) + u + 0,, n (3) whee 0. Then n ( ) s the pont wth 0 u paamete value u on the Béze cuve n. Now, let us assume a Béze cuve segment has - level of efnement and a futhe subdvson step s appled on t wth u t (whee 0 t ). Then, we wll obtan two Béze cuve segments: ( t) + t, and ( t) n n + + t n whee and n ae the Béze ponts of the fst and second cuve segments espectvely. These two cuve segments behave the same n the efnement pocess except that they have dffeent paamete anges. Afte the suface defomaton, we have 0,,, n as the Béze ponts of the ognal Béze cuve and the new postons of and n can be calculated as: ( t) + t, and + n ( t) n + tn The movement of and n ae then expessed as: ( t) and n ( t) ( ) + t( + + ) n ( n n ) + t( n ( t) n ( t) ) We can eplace the movement of contol ponts wth the scala coeffcents and dsplacement vectos as follows: * * * α V ( t) α V + tα + V, and * * * α V ( t) α V tα V n n + n Fnally, we may genealse the above two equatons fo the scala coeffcents to the de Casteljau fom: α ( u) α + uα +,, n 0,, n (4) Theefoe, the set of new scala coeffcents can be calculated by the combnaton of adjacent The Euogaphcs Assocaton 997

5 L et al. / Rendeng of Defomng NURBS Sufaces scala coeffcents stoed at exstng vetces. Instead of applyng the de Casteljau scheme, n ou mplementaton, we e-expess equaton (4) n monomal fom by the Hone s subdvson scheme as follows to obtan bette pefomance. n 0 n 0 α ( (( 0 u )( u) 0 α + α n u α 2 )( u) + n n ) u α n + The eason fo ths s that de Casteljau scheme eques O(N 2 ) computaton whle Hone s scheme eques O(N) only. In contast, when the esoluton of a egon s to be educed because of the decease n local cuvatue, we can apply the de Casteljau algothm n evese to constuct the egessed Béze segments. We cannot smply delete the exta vetces, because t wll affect the ntegty of the Béze segments. Wth ths technque, we can now modfy the esoluton of the pe-geneated polygon model dynamcally. In pactce, due to the fame-tofame coheence, the numbe of polygon vetces needed to be nseted o deleted fom the polygon model between two consecutve fames s expected to be small most of the tme. 4.2 The Cuvatue Test Because we use a pe-geneated polygon model to appoxmate a defomng NURBS suface, we should theefoe make sue that the appoxmaton s pecse enough at any tme whle the suface s defomng. To ncease the pecson of the appoxmaton, egons of hghe cuvatue should be epesented wth moe polygons. On the othe hand, to save pocessng tme, egons of low cuvatue may be epesented wth fewe polygons. To detemne the cuvatue of a suface, we measue the devaton of the suface fom a plana polygon. The esult of ths cuvatue test s then used as an ndcato to quantfy the pecson of ou appoxmaton. It may also be used as a stoppng cteon fo subdvdng the NURBS suface. In ou subdvson pocess, the NURBS suface s fstly conveted to Béze patches, whch ae then adaptvely subdvded nto quadtee nodes based on the local cuvatues. The cuvatue test 0 n s caed out n the followng steps statng fom the oot patch of the quadtee: ) We check the flatness of each bounday cuve of the Béze patch by detemnng whethe the contol ponts of the cuve le wthn a gven toleance fom the lne segment fomed by jonng the two endng contol ponts. If a cuve s found to fal n the test, we subdvde the patch and epeat the check on the subpatches. 2) If the flatness of all the bounday cuves s wthn the specfed toleance, we then check the flatness of the Béze patch by detemnng whethe the total dstance of all the contol ponts of the patch fom the plane defned by any thee cone ponts of the patch s wthn a cetan toleance. If t s, the patch s consdeed as suffcently flat and no futhe subdvson s equed. If t s not, we subdvde the patch and epeat the check on the sub-patches. 3) If the patch passes both tests and 2 above, we output the appoxmatng plana ectangle fomed by the fou cone vetces. 4.3 Cack eventon When adaptvely efnng the esoluton of a polygon model, t s possble that neghboung polygon patches may not be at the same esoluton level as shown n Fgue (a). The addtonal vetces nseted at the boundaes between hgh and low esoluton polygon patches can cause cacks to appea between the patches. To solve ths poblem, we adopt Basky s cack peventon method [5] to ensue the vsual contnuty of these patches. The dea s that cacks can be pevented f the shaed boundaes of neghboung polygon patches become flat at the same esoluton level, and they can use the same lne segment to epesent the bounday. Hence, ths lne segment s passed down to any sub-patches shang the bounday. When a polygon patch s to be subdvded, the flat bounday segment s used to coect the locatons of the nseted vetces to pevent cackng as shown n Fgue (b). The Euogaphcs Assocaton 997

6 L et al. / Rendeng of Defomng NURBS Sufaces (a) cacks (b) Fgue. Cack peventon. 5. Mult-Resoluton Modellng Because a dstant object occupes a elatvely smalle sceen aea than a neaby object, most of the detals n the dstant object ae not vsble to the vewe. Mult-esoluton modellng s a technque to optmse the endeng pefomance by epesentng a neaby object wth a detaled model, but a dstant object wth a smple one. To detemne the esoluton of an object model, ou ecent wok [6] dentfes two mpotance factos: the statc vsual mpotance (SVI) and the dynamc vsual mpotance (DVI). SVI efes to the vsual mpotance of a pont on an object calculated accodng to ts geometc mpotance. Fo example, a pont at a cone of an object has a hghe SVI value than a pont on a flat suface. DVI efes to the vsual mpotance of a pont on an object calculated accodng to some dynamc vewng and anmaton paametes. Fo example, a egon of an object has a hghe DVI value f t ntesects the vewe s lne of sght. Although thee ae many methods fo geneatng mult-esoluton models [7,8,9,20, 2], most of them ae slow and some of them may not even peseve the geomety of the object model. To geneate accuate models wthout sacfcng pefomance, a common method s to pegeneate a few key models of the object at dffeent esolutons. The un-tme object dstance detemnes whch model to use fo endeng. Although ths method s fast and smple, t has two majo lmtatons. Fstly, when the object cosses the theshold dstance, thee s a sudden change n model esoluton and an objectonable vsual dscontnuty effect can be obseved. In [2], Tuk poposed to have a tanston peod dung whch a smooth ntepolaton between the two successve models s pefomed to poduce models of ntemedate esolutons. Ths method, howeve, futhe nceases the computatonal cost dung the tanston peod because of the need to pocess two models at the same tme. Secondly, because the mult-esoluton models ae pe-geneated, the object's DVI s not consdeed. The esult of ths s a unfom ncease o decease n model esoluton, and hence the geneated models ae not optmsed fo the gven vewng and anmaton paametes of ndvdual fames. Such optmsaton s mpotant when an object coves a lage depth ange, fo example, a pece of landscape o a lage buldng. Even though thee may only be a small egon of the object lyng close to the vewe o nsde the vewe's lne of sght, we stll need to use the hgh esoluton model fo endeng so that the detals n the closest pat of the object wll not be lost. Sometmes, a small object may also have smla poblems f, fo example, t s close to the vewe. To ovecome these lmtatons, elated methods fo managng lage tean models can be used [22,23,24]. These methods bascally dvde a lage tean suface nto squae blocks. All the polygons nsde a block ae aanged n a quadtee stuctue wth the oot node epesentng the whole block and the leaf nodes epesentng ndvdual polygons. Each successve lowe level of the tee epesents a fou-tme ncease n the numbe of polygons and two-tme ncease n esoluton. Wth such a data stuctue, t s possble to calculate the DVI values of ndvdual hgh level nodes dung un-tme. A node wth hgh DVI value can be endeed usng ts lowe level subnodes,.e., wth moe detals, whle a node wth low DVI value can be endeed usng ts hghe level nodes. Howeve, ths knd of method eques the object to be egulaly subdvded. In [25], Hoppe poposed an dea called pogessve meshes. Ths method can be used to pefom lmted selectve efnement by consdeng the DVI of the object. Ou ecent wok [6] subdvdes each object nto egons to allow moe flexble selectve efnement wth bette pefomance. Howeve, both of these methods do not consde defomng objects. To apply ou method fo endeng defomng NURBS sufaces to mult-esoluton modellng, we need to be able to detemne the esoluton of the defomng suface accodng to ts SVI and DVI. The mpotance of a Béze patch,, can be calculated as follows:, mp, SVI, DVI The Euogaphcs Assocaton 997

7 L et al. / Rendeng of Defomng NURBS Sufaces whee, SVI and DVI, ae the SVI and DVI values of patch espectvely. Ths patch mpotance value,, can be used to, mp detemne the actual subdvson level, and hence the esoluton, of patch. The hghe ths value s, the moe subdvsons wll need to make to the patch. When the suface s not defomng,, SVI emans unchanged. When the suface s defomng, the SVI value of the Béze patch,, can be updated fom the esult of the cuvatue test, R, descbed n secton 4.2 as follows:, SVI K SVI R whee K SVI s a constant. The DVI value of the patch can be calculated accodng to some ealtme vewng and anmaton paametes. Fo example, f we consde the dstance of the patch fom the vewe and the dstance of the patch fom the vewe s lne of sght, the DVI value of the patch can be calculated as follows:, DVI I dst I sght whee I dst and I sght ae the mpotance values (between zeo and one) based on the dstances of the patch fom the vewe and fom the vewe s lne of sght espectvely. The actual calculaton of these and othe paametes can be found n many efeences [26,6,27]. 6. Results and Dscussons We mplemented the new method n OpenGL and wth the Mancheste NURBS lbay [28]. We compaed the pefomance of the new method wth two othe polygonsaton methods on a SGI Indgo 2 wokstaton wth a 200MHz MIS 4400 CU and Exteme gaphcs acceleato. The two polygonsaton methods ae the Oslo algothm and the method used n the Mesa gaphcs lbay, whch s convetng the NURBS suface nto Béze segments followed by Hone s subdvson. Fgue 2 shows the esults of the compason. Cuve (a) shows the pefomance of the Oslo algothm. Cuve (b) shows the pefomance of the method used by the Mesa gaphcs lbay. Cuves (c) and (d) show the pefomances of ou method wth and wthout esoluton efnement espectvely. Compaed to the method used by the Mesa gaphcs lbay, ou method wthout esoluton efnement s oughly 5 tmes faste, and s oughly 3 tmes faste wth esoluton efnement when the total numbe of polygons used to epesent the suface changes by about 2% due to changes n efnement. The exta computatonal cost s due to the cuvatue tests and the calculaton of the scala coeffcents fo the nseted vetces. Note that the pefomance shown n fgue 2 does not nclude the polygon endeng tme. Tme (seconds) Cuve (a): Knot Refnement (by Oslo Algothm) Cuve (b): Convetng to Béze patches followed by Hone's Subdvson Cuve (c): Ou method wth esoluton efnement Cuve (d): Ou method wthout esoluton efnement ,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 00,000 Numbe of olygons Fgue 2. efomance compason of dffeent NURBS endeng. The Euogaphcs Assocaton 997

8 L et al. / Rendeng of Defomng NURBS Sufaces Tme (seconds) Cuve (a): Insetng olygons 0.05 Cuve (b): Deletng olygons ,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 00,000 Numbe of olygons Fgue 3. efomance of polygon nseton and deleton. Tme (seconds) ,000 20,000 30,000 40,000 50,000 Numbe of atches Fgue 4. efomance of the cuvatue test Fgue 3 shows the pefomance of nsetng polygons and deletng polygons fom the model. Fom the dagam, we can see that t s moe expensve to nset polygons than to delete polygons fom the model. Ths s because moe vetces ae nvolved n calculatng the sets of scala coeffcents when nsetng vetces than when deletng vetces. Fgue 4 shows the pefomance of the cuvatue test wth espect to the numbe of Béze patches beng tested. Fgue 5 shows some mages geneated wth ou method. Fgues 5(a) and 5(b) ae geneated wthout esoluton efnement. Whle the flat egons have too many polygons, the egons of hgh cuvatue may have too few. Fgues 5(c) and 5(d) ae geneated wth esoluton efnement and contan the same total numbe of polygons as fgues 5(a) and 5(b). We can see that fgue 5(d) has a much smoothe appeaance than fgue 5(b). Fgues 5(e), 5(f), 5(g) and 5(h) ae geneated when the suface s at some dstance away fom the vewe. Fgues 5(e) and 5(f) gve a close look of the suface whle fgues 5(g) and 5(h) show the actual sze of the suface. In pactce, as the suface defoms, we need to update the postons of all affected polygon vetces and pefom the cuvatue test on all affected patches dung evey fame tme. Howeve, we may not need to nset o delete polygons n evey fame. Even f we need to modfy the esoluton of a patch, thee ae usually only a small numbe of nsetons o deletons n a sngle fame. As such, ou method can be vey effcent. The majo lmtaton of ou method s that, because the polygon model s aanged n a quadtee stuctue, cacks may appea at the boundaes between patches of dffeent esolutons, and exta pocessng tme s needed to pevent ths to happen. We ae now wokng on anothe subdvson method whch can elmnate cack peventon opeaton. 7. Conclusons We have pesented a method fo nteactve endeng of defomng objects modelled by NURBS sufaces. The method makes use of the ncemental popety of the defomng suface n successve fames by updatng the geomety and efnng the esoluton of a polygon model pegeneated fom the defomng NURBS suface nstead of executng the polygonsaton pocess n evey fame. Results show that ou method s moe effcent than exstng methods. The Euogaphcs Assocaton 997

9 L et al. / Rendeng of Defomng NURBS Sufaces We ae cuently developng a set of tools to be ncopoated nto the MR Toolkt [29] to suppot the endeng of defomable objects. We ae also consdeng addng some anmaton suppot based on the method descbed n ths pape. 8. Refeences. L. egl, On NURBS: A Suvey, IEEE Compute Gaphcs & Applcatons, pp. 55-7, Jan L. egl and W. Tlle, The NURBS Book, Spnge-Velag, S. S. Ab-Ezz and L. A. Shman, Tessellaton of Cuved Sufaces unde Hghly Vayng Tansfomatons, Euogaphcs 9, pp , W. Boehm, Insetng New Knots nto B-splne Cuves, Compute Aded Desgn, 2(4), pp.99-20, E. Cohen, T., Lyche and R. F. Resenfeld, Dscete B-splnes and Subdvson Technques n Compute-Aded Geometc Desgn and Compute Gaphcs, Compute Gaphcs and Image ocessng, 4, Octobe C. de Boo, On Calculatng B-splnes, Jounal of Appoxmaton Theoy, 6(), pp.50-62, July J. M. Lane, L. C. Capente, J. T. Whtted and J. F. Blnn. Scan Lne Methods fo Dsplayng aametcally Defned Sufaces, CACM, 23(), pp.23-24, T. Nshta, T. W. Sedebeg and M. Kakmoto. Ray Tacng Tmmed Ratonal Suface atches, In ACM SIGGRAH 90, 24(4), pp , M. Shantz and S. L. Chang, Rendeng Tmmed NURBS wth Adaptve Fowad Dffeencng, In ACM SIGGRAH 88, pp.89-98, August M. J. Slbemann, Hgh Speed Implementaton of Nonunfom Ratonal B-Splnes (NURBS), SIE Vol. 25 Cuves and Sufaces n Compute Vson and Gaphcs (990), pp , aul de Casteljau, Coubes et Sufaces à ôles, S. A. Andé Ctoen, as, G. Fan. Cuves and Sufaces fo Compute Aded Geometc Desgn: A actcal Gude, Academc ess Inc., Ban, The Mesa 3-D Gaphcs Lbay, URL 4. S. Kuma, D. Manocha and A. Lasta, Inteactve Dsplay of Lage-Scale NURBS Models, oceedngs of 995 Symposum on Inteactve 3D gaphcs, pp. 5-58, B. A. Basky, T. D. DeRose and M. D. Dppé, An Adaptve Subdvson Method Wth Cack eventon fo Rendeng Beta-splne Objects, Repot UCB/CSD 87/348, Dept. of Compute Scence, U.C. Bekeley, R. Lau, D. To and M. Geen, An Adaptve Mult- Resoluton Modelng Technque Based on Vewng and Anmaton aametes, IEEE VRAIS 97, pp.20-27, Mach M. DeHaeme and M. Zyda. Smplfcaton of Objects Rendeed by olygonal Appoxmatons, Computes & Gaphcs, 5(2), pp.75-84, H. Hoppe, T. DeRose, T. Duchamp, J. McDonald and W. Stuetzle. Mesh Optmzaton, In ACM SIGGRAH 93, 27, pp.9-26, August J. Rossgnac and. Boel. Mult-Resoluton 3D Appoxmatons fo Rendeng Complex Scenes, Techncal Repot RC 7697 (#7795), IBM Reseach Dvson, T.J. Watson Reseach Cene, W. Schoede, J. Zage and W. Loensen. Decmaton of Tangle Meshes, In ACM SIGGRAH 92, 26, pp.65-70, July G. Tuk, Re-tlng olygonal Sufaces, In ACM SIGGRAH 92, 26, pp.55-64, July J. Falby, M. Zyda, D. att and R. Mackey. NSNET: Heachcal Data Stuctues fo Real- Tme Thee-Dmensonal Vsual Smulaton, Computes & Gaphcs, 7(), pp.65-69, B. Hezen and A. Ba. Accuate Tangulatons of Defomed, Intesectng Sufaces, In ACM SIGGRAH 87, 2, pp.03-0, July Lndstom, D. Kolle, W. Rbasky, L. Hodges, N. Faust and G. Tune. Real-Tme, Contnuous Level of Detal Rendeng of Heght Felds, In ACM SIGGRAH 96, pp. 09-8, August H. Hoppe. ogessve Meshes, In ACM SIGGRAH 96, pp.99-08, August T. Funkhouse and C. Séqun. Adaptve Dsplay Algothm fo Inteactve Fame Rates Dung Vsualzaton of Complex Vtual Envonments, In ACM SIGGRAH 93, 27, pp , August T. Ohshma, H. Yamamoto and H. Tamua. Gaze-Dected Adaptve Rendeng fo Inteactng wth Vtual Space, In IEEE VRAIS 96, pp.03-0, July W. T. Hewtt and D. Yp. The NURBS ocedue Lbay. Techncal epot, Compute Gaphcs Unt, Unvesty of Mancheste, Unvesty of Albeta, MR Toolkt, URL The Euogaphcs Assocaton 997

10 L et al. / Rendeng of Defomng NURBS Sufaces Fgue 5. Images geneated wth ou method. The Euogaphcs Assocaton 997