Efficient and Realistic Animation of Cloth with Approximate Implicit

E cient and Realistic Animation of Cloth with Approximate Implicit Method and Wrinkled Cubic Spline Young-Min Kang

Jeong-Hyeon Choi

Hwan-Gue Cho

Department of Computer Science, Pusan National University Kumjong-gu, Pusan, 609-735, South KOREA e-mail: fymkang,jhchoi,[email protected] Do-Hoon Lee

Department of Computer Engineering, Miryang National University, Korea e-mail: [email protected] Correspondence to Young-Min Kang e-mail: [email protected]

Abstract

In this paper, we propose an e
cient technique for the animation of exible objects. Mass-spring model was employed to represent the exible objects. The easiest approach to animation with mass-spring model is explicit Euler method, but the explicit Euler method has serious disadvantage that it suers from `instability problem'. The implicit integration method is a possible solution to overcome the instability problem. However, the most critical aw of the implicit method is that it involves a large linear system. This paper presents a fast animation technique for mass-spring model with approximate implicit method. The proposed technique stably updates the state of n mass-points in O(n) time when the number of total springs are O(n). In order to increase the e
ciency of simulation or reduce the numerical errors of the proposed approximate implicit method, the number of mass-points must be as small as possible. However, the coarse discretization with a small number of mass-points generates unrealistic appearance of cloth model. We propose a new technique that generates realistic details of the cloth model even though a small number of mass-points are used for simulation by introducing wrinkled cubic spline curve.

Key words: cloth animation mass spring model implicit method realistic detail wrinkled curve.

1 Introduction

For the physically-based modeling, the mass-spring model is a simple and powerful approach to representing exible objects such as cloth. There have been many techniques for simulation of exible objects Carignan92, Provot95, Terzopoulos87, Volino95], and the techniques use various models such as nite element model Celniker91], and deformable surface model Wang98]. There are trade-o relation between realism and e
ciency. Some researchers have studied on realistic representation of the exible objects Breen94, Eberhardt96], and others have tried to generate the motion of the exible object e
ciently Bara98, Desbrun99]. Among those models, the mass-spring model is the easiest and most intuitive. Various techniques have been introduced to use the mass-spring model to simulate or animate the exible objects Chen98, Desbrun99, Provot95]. Animation techniques based on the mass-spring model can be formulated as a simple ordinary dierential equation. We can easily calculate the force on each mass-point, and animate the mass-point by numerical integration of the force. Explicit Euler integration is the simplest approach to this integration procedure. However, it requires very small time steps for simulation or animation since this approach severely suers from the instability problem Kass95]. Implicit method is well-known solution to this instability problem in numerical analysis literature Nakamura91]. Recently, Bara and Witkin have exploited implicit integration method to take large steps during the simulation of cloth Bara98]. Since this method can signicantly reduce the simulation time by taking large steps, the implicit method is regarded as the best choice for the interactive animation of mass-spring based objects. However, the implicit method also has a problem that a large linear system should be solved. Although the implicit method elegantly enforces an animation process to be a stable integration, the large linear system involved in the implicit method is major obstruction to the real-time animation. Desbrun proposed an e
cient method to solve the

1

Figure 1: Snapshots of Cloth Animation linear system with the precomputed inverse matrix (i.e., precomputed lter). However, it also suers from heavy computation because the inverse matrix may not be sparse. This paper presents a fast and stable animation technique which stably updates the state of n mass points in O(n) time for the real-time animation of exible objects as shown in Fig. 1. The adaptive step size strategy or dynamic modication of animation parameters such as mass or stiness are not restricted in our model. We also considered the wrinkles for the realistic appearance, and implemented interaction between the exible object and the air in order to generate plausible motion of cloth model.

2 Mass-spring Model and Stable Integration

Mass-spring model is a simple technique for representing exible objects. Mass-spring model represents an object with mass-points and springs. The forces caused by the springs induce the movement of mass-points, and the springs can be arbitrarily constructed. This mass-spring model is very intuitive approach to the representation of cloth. By using explicit Euler integration, the mass-points are easily animated. However, due to instability problem, the simple explicit Euler integration cannot be used unless the time step h is very small. Thus, the explicit method takes too much time for producing an animation result of mass-spring model Nakamura91, Kass95, Desbrun99].

2.1 Implicit method for stable animation

Implicit method is a solution to the instability problem. By using the implicit Euler method, we can stably update the state of each mass-point as follows: vit+h = vit + Fti+h mhi (1) xti+h = xti + vit+hh

where vit denotes the velocity of the i-th mass-point at time t, and Fti+h is the force acting on the mass-point at time t + h, i.e., the force at the next time step. Similarly, xti denotes the location of the i-th mass-point at time t, mi is the mass of the i-th mass-point, and h denotes time interval between animation steps. Since this integration method stably updates the next state of mass-spring model, we can take a large step for animation. Therefore, the implicit method is the best choice for real-time or interactive animation system. However, the implicit Euler method involves Fti+h which can be approximated with a rst order derivative as follows:

Ft+h = Ft + @@Fx xt+h

(2)

2 h (I hm H)vt+h = (Ft + hHvt ) m

(3)

where Ft denotes the internal forces consisting of all the internal forces Fti on the i-th mass-point (i.e., Ft =
Ftn ]T ), and similarly xt = xt1
xt2

xtn ]T . Because @ F=@ x is negated Hessian matrix of the system Desbrun99], we will denote @ F=@ x as H henceforth. Since xt+h = xt+h xt = (vt + vt+h)h, we can rewrite the rst equation of implicit update in Eq. 1 as follows:

Ft1
Ft2


;

;

where vt+h is vt+h vt . If we can calculate vt+h , we can easily update the velocity and location of each mass-point at the next step with Eq. 1. The animation of exible objects is nally reduced to nding the value of vt+h. In Eq. 3, hHvt represents additional forces. As mentioned by Desbrun Desbrun99], these additional forces are viscosity forces and can be easily calculated as follows: ;

(hHvt)i = h

X

(i j )2E

kij (vjt vit )

(4)

;

where E is the set of spring edges between mass-points. The implicit method has a critical weakness in that Eq. 3 involves I (h2 =m)H which is O(n n)-sized matrix. Due to this matrix, we must solve a large linear system to update the state of model. Modied conjugate gradient method has been used to alleviate the computation, but it is still far from interactive animation of mass-spring model Bara98]. In order to alleviate the computational burden of the implicit method, Desbrun proposed an e
cient method which approximates the Hessian matrix H. They approximated Hij , the entry of Hessian matrix at the i-th row and the j -th column as Hij = kij , and Hii = j 6=i kij , where kij denotes the stiness constant of spring between the i-th and the j -th mass-points, and kij is 0 when the i-th and the j -th mass-springs are not linked. Then the matrix (I (h2 =m)H);1 remains constant during animation. They precomputed the inverse matrix of I (h2 =m)H, and used the inverse matrix as a force lter for the animation of cloth. This technique produces stable results with simple calculation. Their method can be expressed as follows: ;

;



P

;

;

2 ~t vt+h = (I hm H);1 Fmh (5) where F~ is the sum of spring forces and viscosity forces(i.e., F~ t = Ft + hHvt ). Their method is faster than the general techniques which exploits the implicit method. However, the inverse matrix of I (h2 =m)H is not necessarily a sparse matrix, even though I (h2 =m)H is sparse. Moreover, the adaptive time step strategy cannot be applied to the precomputed lter method, and we cannot dynamically change the mass or stiness, because the calculation of the inverse matrix requires much time. Due to these reasons, we did not employed the precomputed lter. ;

;

;

2.2 Approximation method for interactive animation

We can update the the velocity change of the i-th mass-point by considering only the linked mass-points, because

Hij is 0 when the i-th and the j-th mass-points are not linked with spring. Therefore, we can rewrite the implicit

update scheme (Eq. 3) as follows:

X

2 ~t h2 (1 h mHii )vi m (Hij vj ) = Fmi h (6) i i (i j )2E i We adopted the approximate Hessian of Desbrun et al: for the simplicity. If we assume the uniform spring constants k for all the spring-links, and ni denotes the number of neighboring mass-points that are linked to the i-th mass-point, we can rewrite the Hessian matrix as Hij = k and Hii = kni . The update formula can then be rewritten as follows: ;

;

;

P

mi + h2 kni vt+h = F~ tih + h2 k (i j)2E vjt+h i mi mi mi Therefore, vit+h can be expressed as follows: vit+h =

F~ tih + kh2 P(i j)2E vjt+h

(7) mi + kh2 ni However, we cannot calculate vit+h directly by Eq. 7 because it contains unknown vjt+h , the velocity changes of the j -th mass-points which are linked to the i-th mass-point at the next state. In order to calculate vit+h , the velocity change of the i-th mass-point at the next step, we approximate the velocity change of the j -th mass-point at the next step.

vjt+h can be expressed as follows:

When we drop the term, h2

P

F~ tj h + h2 kjl vt+h vjt+h = m + h2 (j l)2E k l j (j l)2E jl t+h (j l)2E kjl vl , we have an approximation of ~t vjt+h ' m + h2 Fj h j (j l)2E kjl

P

P

(8) vjt+h :

P

(9)

By using this approximation and assuming the uniform stiness k, we can rewrite the update formula for vit+h as follows: t 2 ~ ti h + h2 k F (i j )2E F~ j h=(mj + h knj ) (10) vit+h = mi + h2 kni where ni is the number of mass-points which are linked to i by springs, and nj is the number of mass-points linked to j . Since F~ tj is already known, we can directly calculate the velocity change of the i-th mass-point at the next step. This means that we can generate approximate motion of exible objects without solving the linear system which was major obstruction to interactive animation. Since we update the velocity change of a mass-point by considering only a small number of linked mass-points, it is obvious that our model works in O(n) time, and is faster than precomputed inverse matrix (precomputed lter) method or any general approaches to the implicit integration. Moreover, we can easily modify the mass, time step, or stiness during animation without any additional computations. These dynamic change of parameters cannot be achieved when precomputed inverse matrix is used. It is easy to modify this equation for general cases where the stiness values of springs are dierent from each other. We have tested approximate motion of exible objects, and found that our model generates stable animation results.

P

2.3 Stability of the proposed method

For the verication of stability of our method, let us consider a simple example where only two mass-points (i and j ) of the same mass m are linked with a spring. This is not a proof but will be helpful for understanding why our method is stable. Suppose that the rest length of the spring is 0, and the stiness of the spring is k, For the simplicity, let us assume that the current velocity of each mass-point is 0,0,0]T . Then, no viscosity forces are exerted on the mass-points, and the force acting on the mass-point i and j can be calculated as follows:

F~ tit = F~ j =

k(xti xtj ) k(xtj xti) =

;

;

;

;

(11)

F~ ti

;

It is clear that the location of the mass-points does not diverge when xti+h xtj+h xti xtj . Since ni = nj = 1, F~ j = F~ i, and mi = mj = m, we can calculate the location of each mass-point at the next step, (xti+h and xtj+h), as follows: j

;

j  j

;

j

;

xti+h = xti + F~ h;khm2 F~+khh=2(m+kh2 ) h t i

xtj+h = xtj

;

t i

~ ti h;kh2 F ~ ti h=(m+kh2 ) F

m+kh2

(12)

h

Let u denotes xti xtj . Then, F~ ti can be expressed as -ku, and we can express the dierence of the locations of the mass-points at the next time step as follows: ;

2 (13) xti+h xtj+h = u(1 2 (mmkh +kh2 )2 ) is less than xti xtj when 0 mkh2 =(m + kh2 )2 1. It is trivial to show that ;

;

It is obvious that xti+h xtj+h mkh2 =(m + kh2 )2 is larger than 0. Now, we have only to show that mkh2 =(m + kh2 )2 1. Because (m + kh2 )2 is m2 + k2 h4 + 2mkh2 , we can easily nd that (m + kh2 )2 mkh2 is m2 + k2 h4 + mkh2 and larger than 0. Therefore, mkh2 =(m + kh2 )2 1. j

;

j

j

;

j







;



3 Realistic Details

In this section, we propose a new technique for the generation of realistic details of the cloth. Because the approximate implicit method proposed in this paper determines the next position of a mass-point by considering the positions of limited number of linked neighbors, undesirable results can be generated when a large number of mass-points are used to represent a cloth model. In order to overcome the limitation of our animation technique, we introduce a wrinkle generation method based on cubic spline curve.

Key node Frame node Internal node

Mass-Spring Structure ( key nodes )

Frame nodes and internal nodes for wrinkle representation

Figure 2: Three types of nodes used for representing our cloth model. Black spots are the key nodes, white spots are the frame nodes, and the gray spots are the internal nodes.

3.1 Smooth Surface with Cubic Spline

As mentioned, our animation technique suers from numerical errors when a cloth is discretized with too many mass-points. Moreover, any other animation methods such as implicit integration or precomputed ltering cannot e
ciently generate the motion of cloth when it is composed of too many mass-points. In our opinion, a hybrid method that generates the motion of a small number of important mass-points with simulation technique, and calculates the locations of other many mass-points with interpolation such as cubic spline is the best way to animate the cloth e
ciently regardless to the simulation technique. We dene three types of nodes that are used in our cloth model, key nodes, frame nodes, and internal nodes. The key nodes are the nodes of which locations are determined by the animation techniques such as the approximate implicit method or any other techniques. The frame nodes are the nodes that are located between two adjacent key nodes. The locations of the frame nodes are not simulated during simulation phase but just calculated with cubic spline curves of which control points are the key nodes. The other nodes are internal nodes, and their locations are also calculated with cubic spline curves by using the frame nodes as the control points. Fig. 2 shows how the dierent types of nodes are linked to each other. By using the cubic spline curve, we can generate smooth surface even though the small number of mass-points are used in simulation phase. This enables us to generate the motion of cloth e
ciently and reduce the numerical errors in our approximate method during the simulation. Fig. 3 shows the eect of frame and internal nodes generation with cubic spline curve. We used only 100 mass-points in simulation phase for e
ciency and accuracy of our animation method (Fig. 3(a)), and generated 2500 nodes for representing the cloth (Fig. 3(b)). However, the cubic spline generates excessively smooth surface so that the result is far from realistic representation of cloth. In the following subsection, we will propose a technique that generates plausible representation of cloth when a small number of nodes are given.

3.2 Realistic Detail with Wrinkled Cubic Spline

The cubic spline curve can be used for generating the smooth surface of the cloth by calculating the frame nodes and internal nodes, but the result is too far from the appearance of real cloth. The undesirable result is due to the incapability of the cubic spline curve in expressing the detailed wrinkles. In order to represent the realistic wrinkles of the cloth, we modied our method for generating the frame and internal nodes by adding the wrinkles on the cubic spline. The result of wrinkled cubic spline curve is much better than that of simple cubic spline. Fig. 4 shows the result of the wrinkled cubic spline. Fig. 5 shows the the cubic spline curve (a), and the wrinkled cubic spline curve (b). The wrinkled cubic spline curve in Fig. 5 (b) generally follows the curve shown in Fig. 5 (a), and the segments where two adjacent control

(a) mass-spring model with 100 nodes

(b) smooth surface with cubic spline

Figure 3: Smooth surface generation with cubic spline: (a) rendering with the key nodes that are used during the simulation, (b) rendering with the frame nodes and the internal nodes calculated with spline curve.

Figure 4: The rendering result of cloth model with wrinkled cubic spline. The same number of nodes are used as Fig. 3 points are closer than a certain threshold have wrinkles. The closer the points are, the more wrinkles are generated. The wrinkles are generated according to the length of each spline segment. If the distance between two control points that generates one spline segment is shorter than the rest length of spring, the segment must have some wrinkles in order to preserve the length. Since we are not interested in exact simulation of cloth, and only plausible motion is su
cient, the strict length preservation is not important. Therefore, we implemented a simple wrinkle model which can generate plausible wrinkles. The wrinkle model creates wrinkles when the distance between two adjacent control points is shorter than the rest length of the virtual spring that links the points, assuming the wrinkle frequency is proportional to the dierence between the rest length and actual distance. Let Q(t) be a cubic spline curve segment between two control points, P1 and P2 , with 0 t 1, and Q0 (t) be the wrinkled spline curve segment between the control points. Q0 (t) is exactly the same curve as Q(t) when the distance between the control points is longer than the rest length. If the distance is shorter than the length, Q0 (t) can be expressed as follows, 



Q0 (t) = Q(t) + w(t)Asin(ft)N(t) (14) w(t) = 1=2 2(t 1=2)2 A = (l0 P1 P2 )=2 f = c(l0 P1 P2 )=l0 where N(t) is the normal vector of the curve at t, l0 is the rest length of the spring that links the points, and c is control parameter for the wrinkle frequency. When we apply this method to the cloth model, N(t) can be normal vector of the cloth surface. ;

;j

;j

;

;

;

j

j

(a) cubic spline

(b) wrinkled cubic spline

Figure 5: Wrinkled cubic spline curve.

(a) mass-spring structure

(b) wrinkle representation

(c) rendering

Figure 6: Eect of wrinkles Fig. 6 shows the eect of our wrinkle model. Fig. 6 (a) is the wireframe image of the cloth when only 100 ( 10 10 ) nodes are used for simulation, Fig. 6 (b) is the result with 2500 nodes of which locations are just calculated with wrinkled cubic spline, and Fig. 6 (c) shows the nal rendering results with the 2500 nodes. 

4 Air-interaction Model for Plausible Animation

In order to generate a realistic animation of exible thin object, two kinds of forces, drag force and lift force, must be considered. The magnitude of drag force is known as follows: FD = 12 CD  V 2 S sin  (15) where FD denotes the magnitude of the drag force, CD is the drag force coe
cient,  is the density of a uid, V is the velocity of an object relative to the uid, S is the area of object surface, and  is the angle between V and the surface. The direction of the drag force is opposite to the velocity. The magnitude of the lift force can be expressed similarly as follows: FL = 21 CL  V 2 S cos  (16) where FL denotes the magnitude of the lift force, and CL is the lift force coe
cient. The direction of the lift force is perpendicular to the direction of velocity. j

j

j

j

j

j

j

j

j

j

j

j

(a) t=0

(b) t=1

(c) t=2

(d) t=3

Figure 7: Flag in wind : 50cm 50cm size ag is tied to a pole, and wind is blowing from left to right. The velocity of the wind is 40 m=s, KD is 0:01, and KL is 0:01. Our air-interaction model generated very plausible animation of thin object. When N^ i denotes the unit normal of the i-th mass-point, and v^ i denotes vi = vi , the angle between N^ i and v^ i is =2 . Thus, N^ v^ i is sin  (= cos(=2 )). Therefore, the drag force is proportional to N^ v^ i . Since the direction of the drag force is the opposite direction of the velocity, we implemented the drag force as follows: FDi = KD N^ i v^i vi 2 v^i (17) where KD is the control parameter for the drag force. For the implementation of the lift force, we must determine the direction of the lift force. Since the direction of the lift force is perpendicular to the direction of the velocity, we determined Ui , the direction of the lift force on the i-th mass-point as follows: N~ i = N^ i
if N^ i v^ i > 0 (18) N^ i
otherwise ~ Ui = (Ni v^i) v^i Since the direction of the lift force was determined, we have only to determine the magnitude of the lift force. We implemented the lift force FLi on the i-th mass-point as follows: FLi = (KL cos  vi 2 )Ui (19) where KL is the control parameter for the lift force. The drag and lift eect caused by wind can be easily taken into account by calculating the relative velocity of each mass-point respect to the air. Fig. 7 is the animation results when the drag force and lift force are considered. The velocity of the wind is 40 m=s, KD is 0:01, and KL is 0:01. As seen in the gure, our drag and lift force model generated very plausible scene. j

;

j

j

;

j

j

;

j

jj

j

j

;





j

5 Experimental Results

j

We implemented an animation system for exible and thin objects with OpenGL and Open Inventor library on SGI Indigo2 and O2 machines with R10000 processor. The system was implemented with the techniques proposed in this paper. However, mass-spring has a limitation in representing the exible object, because mass-spring model shows super-elastic eects. Inverse dynamics process for adjustment of the super-elongated spring should be considered when the mass-spring model is used. We adopted the techniques proposed by Provot Provot95]. Our technique generated real-time animation of exible objects with hundreds of mass-points at 30Hz or 60Hz frame rate. We could also generate interactive animation with more mass-points (i.e., thousands of mass-points). We compared the results of precomputed ltering which was proposed by Desbrun et al: and those of our approximation. Fig. 8 shows that the velocity changes calculated by our model is very similar to those calculated by precomputed lter. In the Fig. 8, a virtual human moves a ag and the velocity change of each mass-point is drawn as a line segment. Two clusters of line segments represent the velocity changes calculated by using precomputed lter (top) and those approximated with proposed technique (middle). The velocity changes calculated by both methods seem similar in magnitude and direction. Thus, our model generates the stable motion of mass-spring model as the precomputed lter method does. Fig. 9 shows the rendering results of dierent details. The results shown in the rst column (a) are rendered with only 100 key nodes are used for rendering, those in the middle column (b) are rendered with the key nodes, the frame nodes, and the internal nodes with cubic spline curve, and those in the third column (c) are rendered with wrinkled cubic spline curve.

Figure 8: Comparison between precomputed lter and proposed approximation: A character moves a ag and the velocity change of each mass-point is shown as a line segment. Two clusters of line segments represent velocity changes calculated by using precomputed ltering (top) and those approximated with proposed technique (middle) Fig. 10 shows the animation result when the proposed techniques are used. The motion of the character was captured by a motion capture system, and the motion of the ag shown in the gure was generated by applying our techniques. 100 nodes were used for generating the important movement of the ags, and the location of 2400 frame nodes and internal nodes are calculated with the wrinkled cubic spline for plausible appearance of the ags.

6 Conclusion

We have proposed an approximation method of implicit integration for the stable and fast animation of mass-spring model, and showed the results produced with an animation system which was implemented by using the proposed techniques. The physical correctness was not major concern of our work, and we were interested only in fast and plausible animation of exible objects. The experimental results show that our technique produces very plausible animation of exible objects with large steps. Our technique is very stable because we exploit the ltering property of implicit method. Moreover, our method is as fast as explicit method in the calculation of the next state, since it does not involve linear system solving which is a bottleneck of implicit method. Another important advantage of our technique is that it is very intuitive and easy to implement since our technique calculates the next state with a direct update formula. Moreover, our technique allows adaptive time step, and dynamic modication of physical parameter such as mass or stiness. The most important aw of our method is that the undesirable results can be generated when a large number of mass-points are used for simulation because the proposed method computes the location of each mass-point only considering its linked neighbors. In order to solve this problem, we introduced a new technique that generates the realistic details of the cloth model when a small number of mass-points are given. The technique employs the wrinkled cubic spline which generates a curve that is similar to cubic spline curve but involves wrinkles according to the distance between two adjacent control points. By using the wrinkled cubic spline curve, we can generate the motion of cloth model e
ciently with a small number of mass-points and render the realistic details of the model with the wrinkled curve. Our technique can be applied to various animation system that requires interactive animation of exible objects.

7 Acknowledgment

This work was supported in part by the Ministry of Information and Communication of Korea: Support Project for University Foundation Research 2000 supervised by IITA.

References

Bara98] D. Bara and A. Witkin. Large steps in cloth simulation. Computer Graphics (Proc. of SIGGRAPH '98), pages 43{52, 1998.

(a) mass-spring model (100 nodes)

(b) 2500 nodes with cubic spline

(c) 2500 nodes with wrinkled curve

Figure 9: Rendering results of cubic spline and wrinkled cubic curve: (a) original mass-spring structure, (b) result of cubic spline, (c) result of wrinkled cubic spline

Figure 10: Animation results : 100 key nodes are used as mass-points for the proposed approximate implicit method, and The frame nodes and internal node are generated with the wrinkled cubic spline. The total number of nodes for the scenes is 2500. Breen94] D. Breen, D.House, and M. Wozny. Predicting the drape of woven cloth using interacting particles. Computer Graphics (Proc. of SIGGRAPH '94), pages 365{372, 1994. Carignan92] M. Carignan, Y. Yang, N. Thalmann, and D. Thalmann. Dressing animated synthetic actors with complex deformable clothes. Computer Graphics (Proc. of SIGGRAPH '92), pages 99{104, 1992. Celniker91] G. Celniker and D. Gossard. Deformable curve and surface nite-elements for free-form shape design. Computer Graphics(Proc. of SIGGRAPH '91), pages 257{266, 1991. Chen98] Y. Chen, Q. Zhu, and A. Kaufman. Physically-based animation of volumetric objects. Proc. of Computer Animation '98, pages 154{160, 1998. Desbrun99] M. Desbrun, P. Schroder, and A. Barr. Interactive animation of structured deformable objects. Proc. of Graphics Interface '99, 1999. Eberhardt96] B. Eberhardt, A. Weber, and W. Strasser. A fast, exible, particle-system model for cloth draping. IEEE Computer Graphics and Applications, 16:52{59, 1996. Kass95] M. Kass. An introduction to continuum dynamics for computer graphics. In SIGGRAPH Course Note. ACM SIGGRAPH, 1995. Nakamura91] S. Nakamura. Initial value problems of ordinary dierential equations. In Applied Numerical Methods with Software, pages 289{350. Prentice-Hall, 1991. Provot95] X. Provot. Deformation constraints in a mass-spring model to describe rigid cloth behavior. Graphics Interface, pages 147{154, 1995. Terzopoulos87] D. Terzopoulos, J. Platt, and A. Barr. Elastically deformable models. Computer Graphics (Proc. of SIGGRAPH '87), pages 205{214, 1987.

Volino95] P. Volino, M. Courchesne, and N. Thalmann. Versatile and e
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Biography


Young-Min Kang] is a Ph.D. student at the Department of Computer Science at Pusan National University, Pusan, South Korea. He received his M.Sc and B.Sc degrees in computer science from Pusan National University in 1999 and 1996, respectively. His research interests include computer animation and simulation. He is currently working on simulation and animation of exible objects, and interested in e
cient control of non-autonomous objects.


Jeong-Hyeon Choi] received his B.Sc. degree in physics from Pusan National University in 1995. He received his M.Sc degree in computer science in 2000, and is currently a Ph.D student at the Department of Computer Science at Pusan National University. His current research interests include computer animation and bioinformatics.


Hwan-Gue Cho] received his B.Sc. degree in computer science and statistics from Seoul National University , Seoul, South Korea, in 1984. He also received the M.Sc. and Ph.D. degrees in computer science from KAIST (Korea Advanced Institute of Science and Technology), in 1986 and 1990, respectively. Since 1990, he has been with the Department of Computer Science, Pusan National University. In 1994, he was a visiting scholar at Max-Planck-Institute for Informatics in Saarbrucken. His main research topics are algorithm design and analysis, and computational geometry. Currently he works for exible object animation and bioinformatics, especially phylogenetics.


Do-Hoon Lee] received the BS degree in computer science and statistics from Pusan National university, Pusan, Korea, in 1986. He also received the MS and Ph.D degree from Pusan National University in computer science, Pusan, Korea, in 1992 and 1998, respectively. Since 1995, he has been with Department of Computer Engineering, Miryang National University. He is also the director of Advanced Technology in Information and Communication Institute in Miryang National University. His research interests include virtual reality, compression, and human animation.