Interactive Animation of Cloth including Self Collision Detection

Transcription

1 Interactve Anmaton of Cloth ncludng Self Collson Detecton Arnulph Fuhrmann Fraunhofer IGD Fraunhoferstr Darmstadt, Germany afuhr@gd.fhg.de Clemens Groß Fraunhofer IGD Fraunhoferstr Darmstadt, Germany cgross@gd.fhg.de Volker Luckas Fraunhofer IGD Fraunhoferstr Darmstadt, Germany luckas@gd.fhg.de ABSTRACT We descrbe a system for nteractve anmaton of cloth, whch can be used n e-commerce applcatons, games or even n vrtual prototypng systems. In order not to restrct the shape of the cloth we use a trangulated mesh, whch serves as bass of a mass-sprng system. For the anmaton of the partcles nternal and external forces are consdered. Snce cloth s a very rgd materal when stretched, extremely large forces occur n such a system. Several methods have been descrbed n the recent years to solve the underlyng dfferental equatons effcently. We descrbe an algorthm whch replaces the nternal cloth forces by several constrants and therefore can easly take large tme steps. These constrants can be parameterzed to generate dfferent behavor of the smulated cloth. Furthermore we provde an algorthm for an effcent avodance of self collsons. For these reasons very plausble anmatons of cloth at nteractve rates are possble wth our system. Keywords real-tme anmaton, cloth modelng, physcally based modelng, nteractve applcaton 1. INTRODUCTION Several cloth models have been proposed n the last decade. Some of them am to reproduce the physcal behavor of cloth [Bre94, Ebe96, Vol95, Bar98, Vol01]. Even more accurate ones are [Ebe00, Hau01, Cho02]. Others are able to generate approxmate, but plausble anmatons n the context of nteractve VR applcatons or games [Mey01, Kan01, Jak01, Kan02]. Some recent work [Cor02] combnes fast geometrc wth physcal methods for nteractve anmaton of dressed humans. The model presented n ths paper belongs to the category of approxmate methods, snce the computatonal efforts needed for accurate smulatons are too large for real-tme or nteractve applcatons. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Journal of WSCG, Vol.11, No.1., ISSN WSCG 2003, February 3-7, 2003, Plzen, Czech Republc. Copyrght UNION Agency - Scence Press In partcular, when the pece of cloth should be able to move freely through 3D-space, some smplfcatons, whch are possble, when anmatng dressed humans, can not be made. One problem of cloth modellng s the effcent soluton of stff dfferental equatons. These equatons result from the formulaton of nternal forces, whch are very large for most cloth, snce t ressts strongly aganst stretch. Wth explct ntegraton methods, lke the one descrbed n [Ebe96], the tme step must be very small for the system not to dverge. Ths results n a large computatonal effort. Larger tme steps can be taken by usng an mplct method lke the ones descrbed n [Bar98, Hau01, Vol01]. But these methods are not able to compute real-tme anmatons of cloth wth a suffcent number of partcles, because they requre the soluton of a lnear system. Although ths system s usually sparse, ts soluton wth a modfed conjugate gradent method takes nevertheless about O(n 1.5 ) tme where n denotes the number of partcles [Bar98]. In [Mey01] an approxmate mplct method s descrbed, whch s able to anmate systems composed of several hundred partcles. The method uses a precomputed flter matrx of sze O(n 2 ) whch s appled

2 at every tme step. Although extremely fast for a small number of partcles the advantage of ths method aganst mplct methods vanshes for a larger number of partcles. Whle expermentng wth ths method we notced that after movng the cloth t lasted several seconds for t to reach ts fnal poston. Ths means, that ther method s perfectly stable but the evoluton of cloth moton s below real-tme. Ths may be due to dampng caused by the approxmate mplct ntegraton scheme. Smlar experences were also made by [Kan00, Kan01]. Therefore, ths research group developed a method, whch mproves the work of [Mey01] by replacng the flter matrx wth an update formula, whch only consders neghborng partcles for the flterng. Ths solves the problem of the O(n 2 ) matrx, but snce only drect neghbors are consdered only a few mass ponts can be used. Therefore the authors added wrnkles and more detals by usng so-called wrnkled cubc splne curves for calculatng n-between nodes. The wrnkles generated by ths method are not physcally motvated. Recently, an nnovatve method was presented, whch uses two layers of meshes a sparse mesh for global moton and a fne mesh for more realstc appearance and detals [Kan02]. Ths method s able to anmate thousands of partcles n real-tme. But t does not handle self-collsons, whch s an mportant part of cloth anmaton. Addtonally, snce t s an mprovement of [Kan01], t can handle only several hundred partcles n the sparse mesh. The partcles n the fne mesh are nfluenced only by nternal forces but not by external ones. In our approach nternal and external forces are calculated for all partcles. Furthermore, snce there are only a few partcles n the sparse mesh, t wll be dffcult to anmate a hghly curved pece of cloth, whch s needed when the cloth falls onto the ground and ples up or when the user grabs t on one end only. The later s shown n fgure 1. Another knd of optmzaton s possble when anmatng dressed humans snce most parts of a garment do not change poston relatve to the movng body [Cor02, Rud02]. In [Cor02] a system s descrbed whch allows real-tme anmaton of complex dressed humans. To acheve ths tremendous performance, the partcles of the garments are dvded nto three layers. The ones belongng to the frst and the second layer are treated mostly geometrcally and only partcles of the thrd layer are anmated by physcal methods. To get a further speed-up, no tests for self collsons are done. The technque for anmatng garments descrbed n [Rud02] reles also on the presence of a human body. The partcles are moved manly accordng to the moton of the human body. Addtonally, an explct euler ntegraton scheme s used for descrbng the nteractons between partcles. Because most (a) (b) (c) (d) Fgure 1: A round table top s hold at one end. In (a) and (b) the self collson detecton s enabled. In (c) and (d) t s dsabled. Self ntersectons are clearly vsble. The vew of (b) and (d) s from below. of the moton of a partcle s determned by geometrc crtera, the system remans stable even for large tme steps and consequently t s capable to anmate garments n real-tme. As n the system above, no selfcollsons are handled. Snce our system shall be able to anmate cloth wthout any underlyng body or other shape, both technques can not be appled here. Untl now we dd not dscuss the problem of collson detecton between the cloth tself and between the cloth and rgd bodes. In the area of cloth-object collson detecton several methods were proposed n the last years. The most sutable ones for nteractve applcatons are mage space or voxel-based technques [Zha00, Vas01, Mey01]. They tend to be much faster than classcal approaches usng boundng volume herarches [Bar98]. Recently an algorthm for robust treatment of collsons whch also models cloth thckness was presented [Br02]. The anmatons shown n the paper are really mpressve and demonstrate the power of the algorthm. Unfortunately such a proper handlng of self collsons s computatonally too expensve for nteractve applcatons. As mentoned above, none of the nteractve cloth anmaton systems dd mplement self collson handlng routnes yet. Although some speedup technques basng on the curvature of the cloth surface are known [Vol00, Pro97], self-collson handlng s often dropped to get realtme performance. 2. OVERVIEW In ths paper we descrbe an algorthm, whch s able to anmate a pece of cloth n real-tme. The cloth can be dragged nteractvely around usng the mouse. Durng the anmaton, collsons wth the envronment and self

3 collson are handled. All partcles of the cloth are nfluenced by nternal and external forces. The novel aspect of our algorthm s, that we handle nternal forces purely by constrants. They are derved from real cloth behavor and although they gve a strong smplfcaton of realty, dfferent types of cloth can be vsualzed. External forces lke gravty are appled normally and are ntegrated over tme. Ths results n a system, whch can be ntegrated explctly and nevertheless s stable for large tme steps. Addtonally, self-collson treatment s ntegrated easly n ths system. Nearby partcles are kept away from each other by constrants smlar to the ones used for the nternal forces. 3. THE CLOTH MODEL Currently, the standard approach to the physcally based smulaton of cloth s the use of partcle systems. A pece of cloth s dscretzed nto a set of partcles. Our goal s the anmaton of arbtrary shaped cloth. Therefore we have chosen to use trangular meshes for representng the dscretzed cloth. Moreover t s possble to connect several peces of cloth to form garments wthout any topologcal restrctons. The choce of trangular meshes has some nfluence on our system, but generally all methods descrbed n ths paper can be appled to rectangular meshes as well. The edges of the trangle mesh represent the structural sprngs and the vertces are the partcles or mass ponts. Addtonally bendng sprngs are nserted between the two opposte partcles of two adjacent trangles. Snce the trangulaton s not a regular one, not all partcles have equal mass. The mass of one partcle s computed by the followng formula m = m pm adj. trangles area j 3 (1) where area j s the area of a trangle and m pm s a cloth parameter whch stands for the mass per square meter of the fabrc. The dea behnd ths formula s to assgn one thrd of the area of each adjacent trangle to each partcle. The movement of each partcle s governed by the well known Newton s equaton of moton f = m a = m d2 x dt 2 (2) where x R 3 denotes the poston of the partcle, f R 3 the force actng on t, a R 3 the acceleraton and m the mass of the partcle. To solve equaton 2 we have to determne the external and nternal forces whch are affectng the partcle f = F nt + F ext. (3) Snce the forces can by splt nto external and nternal forces, we wll descrbe frst, how we handle nternal forces. After that we wll show how to deal wth external forces and constrants. Fnally, we wll show, how to effcently ntegrate the equaton over tme n order to yeld the new postons of the partcles. Internal forces by constrants The way the partcles nteract wth each other dstngushes the cloth models descrbed n the lterature. Tremendous effort has been done to develop models whch reflect real cloth behavor [Bre94, Ebe96, Vol95, Bar98, Hau01, Cho02]. These models consder propertes lke shear, ansotropc stretch and bendng. Our model s much smpler than those, but wll reflect approxmatons to stretch and bendng propertes. Snce we beleve, that n most cases shear and ansotropc stretch can be neglected, we dd not take them nto account. Nevertheless, our model can be extended to handle these by usng rectangular meshes nstead of trangular meshes. To justfy our model we begn wth revewng shortly the most smple model a mass-sprng system whch s used n current nteractve systems [Mey01, Kan02]. Interacton between lnked partcles s handled by lnear sprngs. The resultng force s calculated by f j = k j ( x j x l j ) x j x x j x. (4) where k j denotes the sprng constant and l j s the rest length of the sprng. Usng lnear sprngs s unfortunate snce the stretch of cloth s nonlnear. It becomes extremely stff after the elongaton exceeds a certan threshold. One soluton s to take very large sprng constants and use an mplct method for solvng the dfferental equatons [Bar98, Vol01, Cor02]. Another, older approach s to use sprngs wth a small k j and to apply a post step correcton for sprngs whch are overly elongated. Ths dea s after [Pro95]. The algorthm terates over all sprngs and moves the two partcles together along ther common axs. Both partcles are ether moved by the same dstance or, n the case of external constrants, only one of them s moved. Ths was followed and appled also n the context of an approxmate mplct ntegraton scheme [Mey01]. In [Vas01] the post correcton of partcle postons was converted nto a correcton of veloctes, so that the over-elongaton never occurs. Snce our model, as descrbed later, does not use veloctes at all, we dd not follow ths approach. Another nsprng dea s after [Jak01], where each sprng s modelled as a stck whch does not change length at all. Ths results n a very stff cloth whch can by anmated fast and robust n the means of stablty. Our method heads for a smlar drecton, but takes some steps further than the pror approaches. We smply set the k j to zero,.e. we are not usng any sprngs

4 (a) (b) Fgure 2: A tabletop consstng of 1600 partcles draped on a sphere n real-tme. The coeffcents for the bendng constrants are 0.89 for (a) and 0.99 for (b). The coeffcents for structural constrants were not changed. for our computaton. Only the post correcton as proposed by [Pro95] wll take place. But now nothng hnders the cloth to collapse snce no sprngs are actve. Therefore we also add an addtonal constrant for too short sprngs. To compute the new partcle postons we determne n the frst step the drecton and length of the correcton vector by u = x j x x j x ( x j x d max l j ) (5) where d max l j s the maxmal allowed length. By replacng d max wth d mn we get the formula for compressed sprngs. Due to our dscretzaton all partcles have dfferent masses. Therefore we can not smply move both partcles by 0.5 u to fulfl the constrant because ths would change the center of mass. If the correcton vector s weghted by m 1 = m 2 = and added accordng to m m + m j (6) m j m + m j (7) x + = m 2 u (8) x j = m 1 u (9) the center of mass stays constant. Furthermore t becomes easy to add external constrants by changng m 1 and m 2 to approprate values. We refer to the next secton for detals. Fnally, the bendng sprngs are replaced by constrants as well. They are handled lke sprngs whch are too short, but another parameter s used because bendng s ndependent of compresson. In practce, values from 1.0 to 0.8 gve pleasng results. The choce depends on the desred bendng rgdty of the smulated materal. See fgure 2 for a comparson between dfferent materals. Certanly, a sngle teraton over all the constrants bendng and structural cannot be suffcent, snce the alteraton of the length of one sprng affects neghborng sprngs, whch n turn must be adjusted. But fortunately only a few teratons are needed n most cases. Ths depends mostly on the resoluton of the cloth and the tme step. Clearly, the adjustments are only local and therefore the nformaton of a changed poston s carred only to the drect neghbors n one teraton. To nform further partcles another teraton s needed,.e. the hgher the resoluton of the partcle system the hgher the number of teratons must be. We note, that the system remans stable even wth hgh resoluton and few teratons but only at the cost of vsual qualty. External forces There are several knds of external forces whch can nfluence the moton of the cloth. The most mportant are gravty, user nput and collsons wth other objects n the scene. The latter s descrbed n secton 4. Gravty s treated as a normal force affectng each partcle f g = 9.81gm (10) where g s a normalzed drecton vector whch ponts to the ground. Ths force s ntegrated over tme as descrbed n the next secton. User nput s done by grabbng some partcle wth a sutable pontng devce. By movng t around the partcle s also moved. Snce t s desrable that the partcle stcks to the pontng devce as closely as possble we drectly change the partcle poston. Because ths change has prorty over all other forces or constrants we set the mass of the partcle to nfnty. Clearly several partcles can be constraned n ths way. For acceleraton computatons we use the concept of nverse masses as descrbed n [Bar98]. The m nv, whch becomes zero when m s nfnte, are used as

5 follows a = f m = m nv f. (11) Ths concept avods the handlng of dfferent cases and therefore eases computaton consderably. For the evaluaton of Eq. 6 and Eq. 7 we smply set m 1 = 0 and m 2 = 1 f m j s nfnte. The effect s, that partcle j s not moved at all, as ntended. In the case that both masses are nfnte m 1 and m 2 are both set zero. Integraton Method The standard explct euler ntegraton yelds the followng two equatons, whch descrbe the moton of the partcles over tme v n+1 = v n + dt f n m (12) = x n + dt v n+1. (13) Ths ntegraton method can be mplemented easly and computes effcently the new partcle postons, but unfortunately t s unstable for large tme steps, when strong forces are nvolved. Snce our method replaces nternal forces by constrants and the external force of gravty s not hgh, ths s no problem for our system. Another problem are rapd changes n partcle postons caused e.g. by collson response or user nput lead to nstabltes. To cope wth ths, [Mey01] proposed to correct the veloctes after each tme step. Ths s equal to correctng t before a tme step by v n = xn x n 1. (14) dt The effect of ths equaton s that the veloctes are gven only by the partcle postons and therefore changes n poston drectly affect the velocty of the next tme step. Ths ncreases the stablty of the numercal ntegraton. Ths s not a surprse, because ths ntegraton scheme s equal to the Verlet ntegraton [Ver97] whch updates the poston wthout computng any veloctes by = x n 1 + 2x n + dt 2 f n m. (15) Ths can be seen by substtutng Eq. 12 nto Eq. 13. Ths yelds = x n + dt(v n + dt f n m ). (16) By further substtutng Eq. 14 nto Eq. 16 and rewrtng we get = x n + x n x n 1 + dt 2 f n m (17) whch s equal to Eq. 15. As descrbed n [Hua02] the Verlet ntegraton scheme s one of the best choces for problems wth low or no dampng. A dampng term can be ntroduced n the system by multplyng 14 wth an approprate value. In our system we are able to set the dampng to 1.00,.e. no dampng occurs. In order to smulate ar drag, we choose an value of 0.99 n the examples shown, snce t damps the movement of the cloth a bt, whch looks more natural. 4. COLLISION DETECTION Self-Collsons As can be seen n Fg.1, a proper self-collson handlng s mandatory for a cloth anmaton system. In general, there are two possbltes for handlng selfcollsons: Resolve them after they happened or never let them occur. Our approach belongs to the latter category. We have decded to solve the problem approxmately by not testng partcles aganst trangles and edges aganst each other, but we consder only pars of partcles. Clearly, we now have to hold the partcles a lttle bt away from each other to avod artfacts of not detected ntersectng trangles. But ths approach saves a lot of computatons. Snce we are usng a fne trangulaton and the dstance, whch we have to preserve, depends on the sze of the trangles, our approach s feasble. In order to keep the partcles apart, we smply add constrants between nearby partcles. These constrants are lke the structural ones used to avod compresson of the cloth. Due to ts smplcty a sngle test can be mplemented very effcently. In partcular, the common case, that the partcles are not nearby, s carred out by a few arthmetc operatons. Otherwse, the correcton method descrbed above s appled to the partcles. Nevertheless, a method for reducng the number of tests has to be appled, because the number of potental nearby partcles grows wth O(n 2 ). A common speedup test s to buld a boundng volume herarchy of the trangles and to test the herarchy aganst tself for collsons. Snce we are testng only partcles aganst each other the herarchy bases on them not on trangles. The structure of the herarchy does not change durng anmaton and s computed from the flat cloth n advance. After each tme step the herarchy s updated from bottom to top. The boundng boxes are made larger to take nto account the dstance the partcles should keep. Fnally the herarchy s traversed n order to fnd nearby partcles.

6 We have to admt, that our method for self-collson avodance can not compete wth the more sophstcated ones descrbed n [Pro97, Vol00, Br02], because after a self-collson accdently occurs, t can not be repared by our method. But as an approxmate method for nteractve VR applcatons, although not perfectly robust, t s extremely effcent. Furthermore, t enhances the qualty of the anmaton tremendously n comparson to pror systems, whch dd not check selfcollsons at all. Cloth Envronment Collsons We do not focus n ths work on the feld of collson detecton between the cloth and a complex envronment. For the examples shown, we mplemented a fast and smple collson detecton for spheres and axs algned boxes. Snce t s easy to calculate the dstance to these prmtves, our algorthm s based on partcleprmtve dstances and the correspondng normals. The collson response works as follows. If a partcle s detected to be closer to the prmtve surface than a gven threshold t t s set back n the drecton of the normalzed normal n. Let d be the dstance of the partcle to the surface, then r n = n (t d) (18) computes the normal component of our collson response. For an approxmate modellng of frcton we compute a tangental component r t by scalng the tangental part d t of the partcle movement vector d t = d n(d n) (19) r t = c f d t (20) where d = x n and c f s the frcton parameter (1.0 for frcton cancels all the tangental movement). The fnal partcle poston s computed by + = r n + r t. (21) 5. EXPERIMENTAL RESULTS In order to demonstrate the capabltes of our anmaton algorthm, we show several examples. In Fg. 2 a comparson between dfferent bendng rgdtes s made. By changng the value of ths parameter we are able to anmate dfferent materals. Changng the coeffcents of the structural constrants s lmted snce a value beyond 1.1 does not produce pleasng anmatons. They are the result of overly and uneven stretched trangles. In Fg. 3 a sequence of an anmated tabletop, whch s taken from the accompanyng vdeo, s shown. It valdates the soundness of our cloth model snce the anmaton looks very natural and the user, movng the tabletop around, gets the mpresson of draggng real cloth around. The mesh of the tabletop conssts of 1000 partcles and the tme-step dt s ,.e. four smulaton teratons per frame. The constrants descrbng the nternal forces were appled four tmes per teraton. The other examples were made wth the same tme step, but for a hgher number of partcles more teratons are necessary. The mplementaton was carred out n Java and uses Java3D for the vsualzaton. The system runs on a a dual Intel Xeon TM processor at 2Ghz. We dd not yet parallelze our anmaton algorthm. Only the renderng routnes of Java3D are usng the second processor. 6. CONCLUSION We have proposed an algorthm for anmatng cloth n real-tme. Our algorthm s stable for large tme steps. Ths s due to descrbng nternal forces by constrants. Only external forces lke gravty and the changes of veloctes are ntegrated over tme. Furthermore, our algorthm produces realstc cloth motons even for cloth wth low bendng rgdty. Ths becomes possble snce we nclude an algorthm, whch avods self-collsons n a very effcent but approxmate way. Artfacts due to ths approxmaton occur seldom. Nevertheless, there s room for mprovement, snce the partcles have to be kept away from each other a qute consderable dstance. Also, ncreasng the number of partcles beyond 1600 slows down the algorthm too much due to self-collson treatment. In [Pro97, Vol00] methods basng on the curvature of the cloth are descrbed, whch further reduce the number of tests to the order of actual collsons. We dd not follow these methods, snce they bear a constant overhead, e.g. for computatons of normals, whch are not needed n our partcle based approach. For anmatons of even larger meshes than the ones used n ths paper a closer look at these methods s certanly necessary. Moreover, we conclude, that our system s able to anmate nteractvely cloth of dfferent knds of materal qute well. Although our major concern was not physcal correctness the anmatons look qute natural. One open problem s the mappng of measured materal propertes of real cloth to our model. Another problem s the handlng of cloth whch can be stretched for more than one tenth of ts rest length, snce n our system no n between forces are actng on the partcles. Fnally, we are currently extendng our clothenvronment collson detecton algorthm to support more complex shapes than the ones shown n ths paper.

7 7. ACKNOWLEDGMENTS Ths work was partally supported by the bmb+f (Bundesmnsterum für Bldung und Forschung) Vrtual Try-On grant. We thank our Vrtual Try-On project partners at the Unversty of Bonn for provdng the textures used for renderng the cloth. We are very grateful to our partners at the Unversty of Tübngen for helpful dscussons about cloth smulaton. 8. REFERENCES [Br02] Robert Brdson, Ronald Fedkw, and John Anderson. Robust treatment of collsons, contact and frcton for cloth anmaton. In ACM Transactons on Graphcs (SIGGRAPH 2002), volume 21, [Bre94] Davd E. Breen, Donald H. House, and Mchael J. Wozny. Predctng the drape of woven cloth usng nteractng partcles. In SIGGRAPH 94 Conference Proceedngs, Annual Conference Seres, pages , Orlando, FL, USA, July ACM SIGGRAPH. [Bar98] Davd Baraff and Andrew Wtkn. Large steps n cloth smulaton. In Mchael Cohen, edtor, SIGGRAPH 98 Conference Proceedngs, Annual Conference Seres, pages 43 54, Orlando, FL, USA, July ACM SIGGRAPH. [Cho02] Kwang-Jn Cho and Hyeong-Seok Ko. Stable but responsve cloth. In ACM Transactons on Graphcs (SIGGRAPH 2002), volume 21, [Cor02] Frederc Corder and Nada Magnenat- Thalmann. Real-tme anmaton of dressed vrtual humans. In EUROGRAPHICS 2002, volume 21, [Ebe00] Bernhard Eberhardt, Olaf Etzmuss, and Mchael Hauth. Implct-explct schemes for fast anmaton wth partcle systems. In Eurographcs Computer Anmaton and Smulaton Workshop 2000, [Ebe96] B. Eberhardt, A. Weber, and W. Strasser. A fast, flexble partcle-system model for cloth drapng. IEEE Computer Graphcs and Applcatons, 16(5):52 59, September [Hau01] Mchael Hauth and Olaf Etzmuß. A hgh performance solver for the anmaton of deformable objects usng advanced numercal methods. In Proc. Eurographcs 2001, [Hua02] Mchael Hauth, Olaf Etzmuß, Bernd Eberhardt, Renhard Klen, Ralf Sarlette, Mrko Sattler, Katja Daubert, and Jan Kautz. Cloth anmaton and renderng. In Eurographcs 2002 Tutorals, [Jak01] Thomas Jakobson. Advanced character physcs. In Proceedngs of Game Developers Conference, USA, March [Kan02] Young-Mn Kang and Hwan-Gue Cho. Blayered approxmate ntegraton for rapd and plausble anmaton of vrtual cloth wth realstc wrnkles. In Computer Anmaton 2002, page 203, Geneva, Swtzerland, [Kan01] Young-Mn Kang, Jeong-Hyeon Cho, Hwan-Gue Cho, and Do-Hoon Lee. An effcent anmaton of wrnkled cloth wth approxmate mplct ntegraton. The Vsual Computer, 17: , [Kan00] Young-Mn Kang, Jeong-Hyeon Cho, Do- Hoon Lee, Chan-Jong Park, and Hwan-Gue Cho. Real-tme anmaton technque for flexble and thn objects. In WSCG 2000, pages , Plzen, Czech., [Mey01] Mark Meyer, Glles Debunne, Matheu Desbrun, and Alan H. Barr. Interactve anmaton of cloth-lke objects for vrtual realty. The Journal of Vsualzaton and Computer Anmaton, 12:1 12, May [Pro95] Xaver Provot. Deformaton constrants n a mass-sprng model to descrbe rgd cloth behavor. In Graphcs Interface 95, pages , [Pro97] Xaver Provot. Collson and self-collson handlng n cloth model dedcated to desgn garments. In Graphcs Interface 97, pages , [Rud02] Isaac Rudomn and Jose Lus Castllo. Realtme clothng: geometry and physcs. In WSCG 2002 Posters, pages 45 48, Plzen, Czech., [Ver97] Loup Verlet. Computer experments on classcal fluds.. thermodynamcal propertes of lennard-jones molecules. Physcal Revew, 159(1):98 103, [Vol95] P. Volno, M. Courchese, and N. Magnenat- Thalmann. Versatle and effcent technques for smulatng cloth and other deformable objects. In SIGGRAPH 95 Conference Proceedngs, Annual Conference Seres, pages , Los Angeles, CA, USA, August ACM SIGGRAPH. [Vol00] Pascal Volno and Nada Magnenat- Thalmann. Vrtual Clothng, Theory and Practce. Sprnger, 2000.

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