[DESCRIPTION] [Invention Title]

METHOD OF CLOTH SIMULATION USING LINEAR STRETCH/SHEAR MODEL [Technical Field] The present invention relates to a method of cloth simulation using linear stretch/shear model, and more particularly to a method of cloth simulation using linearized energy functions. [Background Art]

When it is needed to create an animated image of humans with a sufficient level of details and realism, whether it is in a movie, animation, video games, or virtual reality (VR) simulation, the problem of simulating dynamic movements of clothes arises. [Disclosure] [Technical Problem] Unfortunately, clothing animation has been and still remains a stressful task to the animators. One of the major reasons for the frustration is the amount of computation clothing simulation involves; it takes several days of simulation time to produce one minute long animation of a ten thousand polygon outfit. Unless one gives up the quality drastically, the speed of simulation is far from real-time. [Technical Solution]

One aspect of the invention provides a method of simulation of a cloth using an edge-based linear stretch and shear model. The method comprises: a) mapping information of geometry of material points, P ₁ P _N , of the cloth into a mesh of nodes, where N is the number of material points, and an i ^th node of

said mesh having a position x,- , a mass n\. _; and a velocity v _f ; b) calculating

a change of geometry of the mesh of nodes at a plurality of time steps using a number of equations of motion based on a force applied to the node by applying linear stretch and shear models to translation and rotation of each mesh node,

wherein the number of equations is less than or equal to 3vVS and c) updating

the position x,- and the velocity V ₁ - of the mesh nodes, wherein an in-plane

deformation of the cloth is represented by stretches along edges defined by two neighboring nodes, wherein an energy function for the equations of motion of two neighboring nodes /, j is approximated during each time step as;

*ϊ={* ^(χ .- ^χ ;H ^χ ,- ^χ ; ⁾ . where k is a spring constant connecting the nodes / and j, x _g = x ₇ --χ,- , and x ^* . is a vector that is calculated at each time step but regarded as constant when differentiating, wherein at least part of the force used in the equations of motion is determined by differentiating the energy function.

Mapping information of geometry of material points comprises: selecting a plurality of material points on the cloth, and mapping the plurality of material points into the mesh of nodes. The mesh may comprise a regular- triangular mesh and a regular-rectangular mesh. Calculating the change of geometry of the mesh of nodes at each time step may use an implicit Euler method. Dynamics of the mesh of nodes is described by AAv=b , where A is a 3N X3N matrix, and b is a 2>N Xl vector,

wherein A , corresponding to a equation of motion, Mx=F , where M and F are the 3N x3iV mass matrix and 3N Xl force vector. The δv is obtained by solving a system of linear equations, wherein the system matrix A is constant over time such that the matrix inversion is pre-computed only once in off-line.

The stretch force is obtained by a differentiation of the energy function. The x ^* . is given by a previous direction method, x ^* . where L = \ x ^* .

The x ^* may be given by an inertial edge rotation, x« =L , where a xu rotation matrix R is calculated according to a Rodrigues' formula by hω=θώ for some scalar θ , where ώ is the unit vector along ω , and wherein R corresponding to the incremental rotation hω is given by

/+sin0[ώ]+(l-cos#)[<»] , where [ώ] is the skew-symmetric matrix representing the cross-product operator S _x . o is given by an inertial

vertex translation, are constants to be

adjusted. The x ^* . may be given by x where a _m (m=0,...,M) are constants to be adjusted. Another aspect of the invention provides a method of simulation of a cloth using a triangular-based linear stretch and shear model. The method comprises: a) mapping information of geometry of material points, P _1^ P _N , of the cloth into a mesh of nodes, where N is the number of material points, and

an i ^th node having a position x, , a mass m^ , and a velocity V ₂ - ; b)

calculating a change of geometry of the mesh of nodes at a plurality of time steps using a number of equations of motion based on a force applied to the node by applying linear stretch and shear models to translation and rotation of each mesh node, wherein the number of equations is less than or equal to

SN', and c) updating the position x,- and the velocity V ₁ - of the mesh nodes,

wherein an in-plane deformation of the cloth is represented by displacements

of three neighboring vertices at U ₁ . and

U , = K ^v _* wherein an energy function for the equations of motion of three neighboring nodes /, j, k during each time step comprises;

where A , k _u , k _v are the area of the triangle in the undeformed state, it- and F-directional stiffnesses, wherein a triangle composed of three

particles having material space coordinates ^U / ⁼ [ ^W / ^V ; J , ^U ; ⁼ L ^W ^'J ' ^m ^ where x,. , x _y , and x _k are the corresponding 3D Cartesian space locations, and S is a mapping from the 3D material space to 3D Cartesian space, such that S(u,v) gives the 3D position of the material point (u,v) , wherein u and v represent diagonal axes obtained by rotating u and Faxes by +90 degrees, k- and k- represent ύ- and v-directional stiffnesses, and wherein the vector

quantities S ₁₁ , S _v , S _s , and S _$ are calculated at each time step and are

regarded constant in differentiating.

Even though the triangles have different stretch and shear strains, the strain is constant within each triangle.

The vector quantities are given by an inertial axes rotation,

s;-s;

wherein the triangle is assumed to make an inertial movement during the short duration of , and wherein S ¹ ,"* ¹ and 5" ⁺¹ are obtained by rotating S" and S" by hω . The rotation matrix R may be calculated according to a Rodrigues' formula by hω=θώ for some scalar θ , where ώ is the unit vector along ω , and wherein R corresponding to the incremental rotation hω is given by

/+sin<9[ώ]+(l-cos<9)[fi)] , where [ώ] is the skew-symmetric matrix representing

M the cross-product operator <y _χ . ω is given by ^{ω =} 2-j ^(χ m ^ω

»2=0 where are constants to be adjusted.

The vector quantities are given by an inertial vertex translation,

S _u ^* =αx ^* +bx ^* _j +cx ^* _k , and S _v ^* =px ^* +qx ^* _j +rx ^* _k , where a, b, c, p, q, and r are constant to be adjusted.

M

The x ^* may be given by ^x / ^~ / _^1 ^a _m ^x i , where oc _m \ m = 0,...,Mj are

TO=O constants to be adjusted.

The method may further comprise manipulating for each node of the mesh: a position constraints; a orientation constraints; a mixed constraints; and a static position constraints. The updating and manipulation can be performed in real-time.

[Advantageous Effects]

Various aspects of the invention provide new techniques for simulation of clothes. The disclosed techniques fix several flaws that existed in the previous real-time models. Also, the new techniques allow for several levels of quality vs. speed tradeoffs, providing a wide gamut of scalability. Therefore, the techniques of the invention provide immediate uses in real-time applications such as video games and VR simulations. Also, these techniques can be incorporated into off-line systems for previewing clothing animations before making more accurate versions of them. [Description of Drawings]

Various aspects and advantages of the invention will become apparent and more readily appreciated from the following description of embodiments, taken in conjunction with the accompanying drawings of which:

Fig. 1 is a diagram illustrating a mass-spring model for simulating a cloth according to the invention;

Fig. 2 is a diagram illustrating a rectangular mesh for nodes representing material points of the cloth, wherein the rectangular mesh comprises nodes, each having connections with neighboring nodes to form a quadrilateral shape; Fig. 3 is a diagram illustrating a triangular mesh for nodes representing material points of the cloth, wherein the triangular mesh comprises nodes, each having connections with neighboring nodes to for a triangular shape;

Fig. 4 is a schematic diagram illustrating a method to estimate of direction change of displacement according to a prior art;

Fig. 5 is a schematic diagram illustrating a first order inertial edge rotation method to predict direction change of displacement according to an embodiment of the invention;

Fig. 6 is a schematic diagram illustrating a second order inertial vertex translation method according to another embodiment of the invention;

and

Fig. 7 is a flowchart illustrating the processes of an embodiment according to the invention.

[Best Mode] Physical models for representing cloth can be classified into two: edge-based systems and area-based systems. Edge-based systems measure the in- plane deformation based on the current length of the edges, whereas triangle based systems, an area-based system, measure the in-plane deformation based on the areal deformation of each triangle defined by three nodes. Embodiments of the invention provide linearizable physical models for the stretch/shear deformations in edge-based systems and area-based systems.

In a simulation, a piece of cloth is represented with N particles (P ₁ , P ₂ , " ^• "» P _N ) 12, 13, 14, 15, 16 as shown in Fig. 1. The position and mass of

each particle are denoted as x,. and m, , respectively. A vector x , SN- dimensional vector, which is constructed by concatenating X ₁ , x ₂ , ^••• , x _N . The x,(/ = 1 to N) varies over time t . Since the position and the vector are varying with time, they may be written as X ₁ (J) and x(t) . x" and x" are the values of x,. and x at /rth discrete time step. Also, the notational convention x _tJ - X _j -X ₁ is used to represent a vector interconnecting the positions represented by x _; and x _; . An edge defined by two nodes represented by x,. and x _; can be represented by X ₁₇ .

Edge-Based Formulation of Stretch Energy and Force

The stretch energy El held on the deformed edge x, _y may be modeled by

the formula _t (1) where L _tj and k _tJ are the rest length and the stiffness, respectively, of the edge. Once the energy potential is defined, then the restorative force acting on P ₁ that is contributed from the edge x _v is given by the spatial derivative of the energy potential:

The above stems from the material' s restoration tendency and is called the internal force. The restorative force f,. acting on P ₁ from all the adjacent edges is given by the summation

where N(i) is the set of indices of the particles which P ₁ is connected to.

Explicit Formulation

To find out mechanical movement of a particle, the total force acting on the particle is needed. In addition to the internal force, P ₁ also experiences external forces such as gravity and air drag. Summation of all the external forces acting on P ₁ is represented as f ^* *' . F ₁ . is a summation of all the internal and external forces acting on P ₁ , i.e., F ₍ . = f, + f"' .

Newton' s second law states that the movement of P _{ is governed by the equation m, X ₁ = F.. (4)

Equation 4 is formulated for every (non-constrained) particle, which may be assembled into a system of differential equations

Mx =F, (5) where M and F are the 3N X3N mass matrix and 3N Xl force vector, respectively. Solving the above equation gives time-varying trajectories of the particles.

Implicit Formulation

The explicit formulation sounds intuitive. Moreover, it does not call for solving a system of linear equations, and thus straightforward to implement. However, the explicit formulation is prone to numerical

instabilities. Unless very small time steps are used, the system often diverges. The implicit formulation described below has been known to be more stable.

In comparison with the explicit Euler method shown in Equation 5, the implicit Euler method samples the derivatives at t" ⁺¹ . That is, the implicit Euler method updates the system state according to

In a rigorous sense, the time integration in this is called the semi- implicit Euler method. But, it is simply referred as implicit method.

A first order Taylor series expansion on F(x" +δx,v" +δv) is applied to get

Substituting the above to Equation 6 and eliminating δx produce

(7)

Now, the equat ion can be solved for δv , and the result can be used to compute δx = v" + δv . In the above equat ion,

so that Equat ion 7 can be written as

Aδv = b . (10)

Here, "A" is a 3 N XSN matrix, and "b" is a 3 N Xl vector.

Compared to the explicit methods, calculation of δv out of Equation 10 calls for solving a large system of linear equations. The stretch/shear model makes the system matrix A constant over time. Then, the matrix inversion can be

pre-computed only once in off-line, so that the on-line simulation can be done fast.

The above speed-up is the result of adopting a simplified (less accurate) physical model, details of which will be described later. A stretch model that causes the matrix A to become constant in the context of implicit formulation has already been disclosed by Desbrun et al. [11], which is listed below. Their simplification was based on overlooking the rotational movements, which obviously resulted in artifacts. The difference of the model from theirs is that the simplification accounts for the rotational movements. As a consequence, the disclosed method can produce conspicuously improved results.

Thanks to the pioneering work of various groups over the past decade [4], [2], [13], [8], [1], [5], [3], also listed below, cloth can now be simulated with remarkable realism. For example, natural wrinkles can now be produced using the particle model, and the robustness of the collision handling in cloth simulations has been considerably improved. Along with the improvements that have been made in animation quality, the overall simulation algorithm has been refined such that it runs at a reasonable speed. For example, producing a 30 seconds long animation of an outfit represented with about 10,000 particles takes a few days excluding the time for rendering in the prior art.

However, there are several application areas in which simulation speed should be faster. For example, in video games, animation of cloth should be generated in real-time. Desbrun, Schroder, and Barr [11] pioneered the problem of real-time simulation of cloth-like objects. To accelerate the semi-implicit method, they omitted the non-linear force components and used only the linear components in calculating the force jacobian, which resulted in a constant system matrix. Therefore, once the inverted system matrix was precomputed at the initial stage, the linear system did not need to be solved at every time step. Omission of the non-linear components could lead to artifacts. To compensate the possible errors from the ignored non-linear

forces, the angular momentum correction steps were taken at the end of simulating each time step. But those steps could not completely fix the problem; Even though the correction steps could preserve global angular momentum, they could not preserve angular momenta of local regions. The physical model also makes the system matrix constant. But, the non-linear forces are not overlooked altogether in the jacobian calculation.

Kang, Choi, and Cho [10], [9] also disclosed a simplification of the semi-implicit method that precomputes the inverted hessian matrix or avoids solving the large linear system. To achieve 0(n) time complexity, the velocity change of each cloth particle is directly updated using explicitly estimated future velocities of the nearby particles that are connected to it. Recently, Cordiner and Thalmann presented a real-time cloth animation system in [7], They classified the cloth into three categories (tight, loose, floating) based on the cloth movement pattern and adopted different approach to animate them in each category. The movement of tight/loose regions is highly dependent on the body motion, and therefore such regions may not need full 3D simulation. Geometrical techniques were used for the tight and loose regions. Qn the other hand, the particle system with semi-implicit method was used for the floating regions. Since the floating regions might take up only a small portion in normal garments, the above technique could achieve real ^¬ time animation.

Edge-Based Linear Stretch Model

An edge-based system refers to mass-spring representation (referring to

Fig. 1) of cloth in which in-plane deformation is realized by the stretches along the edges. The elementary deformable unit is an edge. A stretch energy function is disclosed for the edge between two particles P ₁ and P _} , so that the restorative force and the force jacobians can be derived from it. Since it judges the deformation by looking at only limited part (i.e., only the edges) of the mesh, edge-based system makes more sense when the topological

connectivity is kept the same at all edges. Therefore, edge-based system is adopted mostly in the context of regular meshes as shown in Figs. 2. There are two kinds of regular meshes that can be considered for cloth simulation: regular-triangular meshes as shown in Fig. 3, and regular-rectangular meshes as shown in Fig.2. When a rectangular mesh is used, shear deformation can be simulated using the above edge-based stretch model by making diagonal connections.

Conventional Stretch Model

The stretch force and its jacobians are derived from the energy function. So, the task of developing a physical model that leads to constant system matrix is reduced to finding a new energy function that has such simplifying property. Differentiation of the conventional energy function

EiλkQ x _y l -L) ² ( _H ) produces which is not linear with respect to x,. or x _; . Therefore, the force

jacobian — is not constant over time. Then, the linear system given in dx

Equation 10 should be solved at every time step.

If there had been no norm operator I ^• I in the equation, then f. would have been linear. So the problem can be rephrased as: Can an energy function that obviates the use the norm operator but its value is close to that of Equation 11 be found?

Stretch Model

The energy function which is used is

where xt. is some vector that is newly calculated at each time step but regarded constant when differentiating. There is no norm operator in the df energy. Therefore the force jacobian ^~ r ^~ is now constant . ox An immediate question that can be raised would be whether there can exist the vector quantity x ^* - such that use of the above energy function produces acceptable stretch behavior of cloth? The answer turns out to be yes through experiments.

Then, what value is used for x ^* .? To make Equation 13 match Equation 11 in the physical meaning, ideally, x^ should represent an unstretched version of x _j;/ . The situation can be expressed as the conditions: I x ^* . I = L and x ^* _y //X _y , where // means " parallel" . Equation 13 is being used for updating the value of x _tJ , from xjj to x£ ⁺¹ . Therefore, the second condition can be more explicitly phrased as: x ^* .. //x"* ^λ . Here, the problem is that x~ ⁺¹ is not known yet. One way to circumvent this problem would be to use an estimated direction of xjj. ⁺¹ .

Desbrun et al. [11] states " ... (we) simply decide to overlook the rotation, and suppose that this non-linear part will stay constant ..." . In the notation of an embodiment of the invention, the decision amounts to using ^x 'ϊ _j / ^x _(/ f° ^r the estimated direction of X _y ⁺¹ . Therefore, the decision is equivalent to using

This scheme is referred as the previous direct ion method. In the

previous direction method as illustrated in Fig. 4, x ^* is parallel to x?.. Using the direction of the previous step is a flaw. The scheme produces a side effect: the simulated cloth exhibits a tendency to maintain the old (global) orientation of x _y ', such effect can accumulate over time. This side effect is persistent even when small time steps are used. They disclose a procedure which can be taken to enforce the preservation of the angular momentum. However, what is preserved by the procedure is angular momentum of the entire system. Angular momenta of the local portions are not preserved, which would be more important for the visual realism of the simulated results. There may be two methods to estimate x ^* .. One is called the inertial edge rotation, and the other is called the inertial vertex translation. Both of these methods do not have the above kind of flaw. When time steps are taken small enough, those methods do not experience particular side effect.

The Inertial Edge Rotation: The principal idea of the inertial edge rotation (IER) method is to predict the direction of x£ ⁺¹ based on the angular velocity ω of x ₍₎ . at t",' it is assumed that x _y will continue to rotate with the angular velocity for the duration h as shown in Fig. 5. The rotation matrix R represents the estimated incremental orientation change that occurs

during |^'\£" ⁺ J . Then, the method calculates x ^* . according to

The calculation of R is done according to the Rodrigues' formula. It may be written as hω=θώ for some scalar θ , where ώ is the unit vector along ω . Then the rotation matrix R corresponding to the incremental

rotation hω is given by /+sin<9[ώ]+(l-cos<9)[ø] , where [ώ] is the skew-symmetric matrix representing the cross-product operator ώ _x .

Finally, the angular velocity ω needs to be estimated. If the angle

between x'jf ¹ and x£. is denoted asZ(X ¹ J ₁ ¹ ,X _g ) , then the angular velocity ω" at t" is approximated with

Similarly, the angular velocity ω" at f ^~λ is approximated with

Now, o , the angular velocity at t" ⁺¹ , is est imated. Using the first- order predict ion, ω is calculated by ω = ω" . (18)

Using the second-order prediction, the ω is calculated by ω = ω ⁿ + (ω ⁿ - ω"- ^ι ) = 2ω" - ω" ^~ι , (19) or using more general second-order predict ion the ω is calculated by ω = αω" + βω ^n~1 , (20) where α and β are constants to be adjusted. General higher-order expl icit predict ion of ω can be done with

M ω = ∑α _m ω ⁿ - ^m _; (21)

where α _m (m=0,...,M) are constants to be adjusted. Substituting Equation 15 into Equation 13, and regarding x ^* . as constant in differentiation, the stretch force is now linear with respect to

x. Therefore, — ⁱⁿ Equation 7 is constant. The conventional model used ox δF for representing damping is already linear with respect to v. Therefore, —— σv is also constant. Therefore, the whole system matrix A is constant.

One problem that has been overlooked so far is that treating x ^* . as constant in the differentiation can lead to an artifact. Experimentation with

the IER method reveals that the method occasionally produces vibratory results. It is predictable the artifact will be more noticeable when there is a large fluctuation in the value of x ^* .. The inertial vertex translation method described below is more robust in that aspect. 2) The Inertial Vertex Translation: Another method to estimate x ^* - , the inertial vertex translation (IVT) method, is provided. The major difference from the IER is that, instead of assuming the edges are making inertial movements, IVT regards each vertex moves independently. When there is no specific information about the vertices movements, it may be assumed that each vertex makes an inertial movement as shown in Fig.6.

The IVT method first calculates the estimated position x ^* of x? ⁺¹ by assuming that it makes an inertial movement. For example, a first-order estimation can be made by but more general first-order estimation can be made with where α and β are constants to be adjusted. A second-order estimation can be made with but more general second-order estimation can be made with where γ , δ , and λ are constants to be adjusted. General higher- order estimation can be made with

where a _m (m=0,...,M) are constants to be adjusted. The principal idea of the IVT method (compared to the IER method) does not lie in the specific scheme for estimating the new vertex position, but it lies in treating the vertices freelymovingparticles. In another embodiment, the x ^* . may be estimated with

When the IVT method is used, the vibratory artifact is reduced compared to the IER method. The reduction is more significant when the second order estimation is used. Representation of Shear and Bending Deformations

A linear stretch model for the edge-based systems has been used. In representing cloth, the other two kinds of deformation are also considered: shear and bending.

In rectangular meshes, shear deformation can be represented with the model presented in the above by making diagonal connections. For representing the shear deformations in triangular meshes, the disclosed model is not particularly suited for. However, the triangular mesh intrinsically has a tendency to come back from shear deformations due to the mesh structure itself. This is a side effect the model originally did not intend to produce. To explicitly represent shear deformation in triangular meshes, the technique presented in the triangle-based linear stretch/shear model may be used.

Bending is not explicitly represented with the model. However, when an irregular triangular mesh is used, even without mobilizing any particular bending model, the simulated results will exhibit restorative tendency from bending deformation. It is due to the artificial flexural rigidity [12]. Again, it is a side effect the model originally did not intend to produce. This side effect can be exploited beneficially in the cases when the bending deformation cannot be explicitly simulated (In real-time applications,

computation that can be allocated for clothing simulation is usually limited). An interesting aspect in exploiting this side effect is that, the irregularity of the mesh is beneficial.

When the computation that can be allocated to cloth simulation is highly scarce, then the edge-based linear stretch model can be adopted on an irregular triangular mesh, without explicitly modeling shear or bending deformation. The only springs in the model are the linear springs at the edges. The side-effects of the model described in the edge-based linear stretch model can be relied on to represent shear/bending deformation; The triangular network by its nature will create tendency to restore from shear deformation. Irregularity of the mesh will create tendency to restore from bending deformations. So, by applying the linear stretch model to a irregular triangular mesh, shear and bending behavior can be obtained as a by-product. The result may not be physically accurate, but can be useful in implementing real-time systems when speed is the utmost issue. Even in off-line systems, it adds another scalable option. Using the edge-based linear stretch model in the above is simple but very fast representation of cloth. Triangle-Based Linear Stretch/Shear Model A triangle-based system refers to interacting particles representation of cloth in which in-plane deformation is realized by the displacements of three vertices. The elementary deformable unit is a triangle. The stretch and shear energy functions for the triangle formed by P ₁ , P ₁ , and P _k are disclosed so that the restorative force and the force jacobians can be derived from it. Since it judges the deformation by looking at the (triangular) area rather than just looking at the edges, triangle-based system is less sensitive to irregularity of the mesh. A striking difference from the edge-based linear stretch model is that, in the triangle-based model, the shear deformation can be accounted for.

Material Space to 3D Space Mapping S In a triangle composed of three particles, their material space

coordinates are U ₁ . =[M ₁ -V,] , u _y . =[w _; v _y .] , U _A =[M _A V _A ] , and the corresponding 3D Cartesian space locations are x, , x _y ,x _k , respectively. If S is the mapping from the 3D material space to 3D Cartesian space, S(u,v) gives the 3D position of the material point (u,v) . A simplifying assumption is that, even though triangles will have different stretch and shear strains, the strain is constant within each triangle. Under the above simplifying

assumption, the partial derivatives ^ύ e _u - ^~ " ^d ^T ^S~ and ^ύ c _v _ ^~ ^ ^d T ^S~ can be expressed ou ov in terms of u. , u _y , u _k , x. , x _y . , x _k '•

[S ₁₁ S _v ] (28) A notable consequence of the above equation is that both S ₁₁ and S _v can be expressed as l inear combinat ions of x, , x _y . , x _k ,

S _u = ax _j +bx _J +cx _k (29)

S _v = px, +qx- _j +rx. _k (30) where a, b, c, p, q, and r are determined from the (undeformed) triangulizat ion in the material space.

Conventional Stretch and Shear Model

In previously disclosed triangle-based models, the formula which was popularly used for representing the stretch energy was [1] where A , k _u , k _v are the area of the triangle (in the undeformed state), u- and 7-directional stiffnesses, respectively. As for the shear energy, some used and other groups [6] used

where ύ and v represent diagonal axes obtained by rotating u and v axes by +90 degrees, k _n and k- represent ύ- and v -directional stiffnesses. None of the above formulations make the system matrix constant.

Stretch and Shear Model

The energy functions are

where the linear combinations given in Equations 29 and 30 are used for S _n and S _v . It is not yet described how to calculate the vector quantitiesS _u ^* , S _v ^* , Sl, and S ¹ ! that are supplied at each time step and are regarded constant in differentiating. But one thing is clear at this moment: the formulation makes the system matrix constant. Now, an immediate question would be whether there exit vector quantity 5 ^* , 5 ^* , 5- , and S^ such that use of the above energy functions will produce acceptable stretch and shear behavior of cloth. As in the edge-based systems, the answer is yes and there are two ways to make the estimations: the inertial axes rotation and the inertial vertex translation.

1) The Inertial Axes Rotation: In principle, 5 ^* and S ^* should represent unstretched version of S" _t ⁺¹ and 5" ⁺¹ , respectively. Since S" ^+l and S"* ¹ are not available, the estimations of them are used. S _n ⁺¹ and S" ^+l denote the estimations of S" ^+l and S" ^+l . In one embodiment, the unstretched versions may be calculated as:

Similarly, S- and _s£ should in principle represent unstretched version of >S~ ⁺¹ and S'* ¹ , respectively. Denoting the estimations of Si ¹⁺¹ and S'* ¹ as S~ ⁺¹ and S- ⁺¹ , respectively, S _n ^* and S~ ^* maybe calculated as: rr* _ C« ⁺¹ / 6" ⁺¹ (38)

n+1 K ¹ (39)

A simpler way of calculating S ^* ~ and S^ is to assume that they are just perpendicular toS ^* and S _v ^* . Under this assumption, the followings can be used.

How the estimations S" ⁺¹ , S" ⁺¹ , S? ^+l are made will be described below. Estimations of ^," ⁺¹ and 5-" ⁺¹ can be done in a similar manner as S" _t ^+l , S" ^+l . The methods to estimate 5," ⁺¹ and S" ^+l are described.

Assuming that the triangle will make an inertial movement during the

short duration of [^ ⁷ V ⁺ J , the 5," ⁺¹ and S" ⁺¹ can be predicted by rotating

S" and S" by hω , i.e., the _5£ ⁺¹ and S'* ^x can be calculated by

The estimation of can be done by the procedures introduced in the edge-based linear stretch model.

2) The Inertial Vertex Translation: When there is no prior information about the cloth movements, another plausible postulation to make is that the vertices will make inertial movements. The same procedures introduced in the edge-based linear stretch model may be applied to obtain the predicted position x ^* of x" ^+I . Then, their linear combinations are taken according to Equations 29 and 30 to make the estimations:

S _u ^* =ax ^* +bx _j ^* +cx ^* _k (44)

$ _v ^* =px ^* +qx _j ^* +rx ^* _k . (45)

Embodiments of the invention provide the followings. 1) Linearizable energy function for edge-based systems: For formulating the in-plane stretch energy of an edge, it is disclosed to use

(X _y -X _y )-(X _1J -X _y ) , where x ^* . is some value supplied at each time step and

regarded constant in differentiation. Conventional formulation of clothing

simulation contains the term (|x _z y ^~ C) , the derivatives of which is nonlinear with respect to x ₍₎ .. It calls for solving a large system of linear

equations. For linearization, it is disclosed to replace (|x,y —C) with

2) Inertial Edge Rotation method for estimating x ^* . in edge-based systems, which calculates x^. according to Equation 15. 3) Explicit predictions of ω : It is disclosed to use Equations 18-21 for the prediction.

4) Inertial Vertex Translation method for estimating x ^* in edge-based systems, which treats vertices as freely moving particles. The method calculates x ^* . according to Equation 27.

5) Explicit predictions of x ^* : It is disclosed to use Equations 22~26 for the prediction.

6) Using the edge-based linear stretch model for representing cloth, with shear deformation or bending deformation or both not explicitly modeled: It is a pure mass-spring model. The side effects are relied on for the restoration from the shear and/or bending deformations. The primary gain of using this simple model is the speed-up.

7) Linearizable energy function for triangle-based systems: For formulating the in-plane stretch and shear energy functions of a triangle, it is disclosed to use Equations 34 and 35.

8) Inertial Axes Rotation method for estimating 5 ^* , S _v ^* , S- , S- ^* , for the above triangle-based linear stretch/shear model, which calculates S ^* _t , S _v ^* , S ₅ ^* , S. ^* , according to Equations 36-41.

9) Inertial Vertex Translation method for estimating S _tt ^* , S ^* , , S ^* -, Si , for the above triangle based linear stretch/shear systems, which calculates S _u ^* ,

K _> S _n ^* , S- ^* according to Equations 44 and 45.

One aspect of the invention provides a method of real-time simulation of a cloth using a linear stretch and shear model as illustrated in Fig. 7.

The method comprises: a) mapping information of geometry of material points, P ₁ P _N , of the cloth into a mesh of nodes, where N is the number of material

points, and an i ^th node of said mesh having a position x,- , a mass n^. , and a

velocity v,- (SIlO); b) calculating a change of geometry of the mesh of nodes

at a plurality of time steps using a number of equations of motion based on a force applied to the node by applying linear stretch and shear models to translation and rotation of each mesh node, wherein the number of equations is

less than or equal to M (S210); and c) updating the position x,- and the

velocity v,- of the mesh nodes, wherein an in-plane deformation of the cloth is

represented by stretches along edges defined by two neighboring nodes (S310), wherein an energy function for the equations of motion of two neighboring nodes /, j is approximated during each time step as; where k is a spring constant connecting the nodes / and j, x _ϋ - x _y -χ,- , and x ^* . is a vector that is calculated at each time step but regarded as constant when differentiating, wherein at least part of the force used in the equations of motion is determined by differentiating the energy function.

In the energy equation, (x-x ^* )-(x-x ^* ) is used instead of (|x ^~ C) ,

where x ^* is some value supplied at each time step, x ^* is regarded constant in further differentiation. Calculating x ^* is based on the simplifying assumption that cloth will make inertial movements. Other method for the calculation of x ^* may be used. Although various embodiments of the present invention have been shown and described, it would be appreciated by those skilled in the art that changes might be made in this embodiment without departing from the principles and spirit of the invention, the scope of which is defined in the claims and their equivalents. This application incorporates herein by reference the following papers and publications in their entirety:

[1] David Baraff and Andrew Witkin. Large steps in cloth simulation. In Proceedings of SIGGRAPH 98, Computer Graphics Proceedings, Annual Conference Series, pages 43-54. ACM, ACM Press / ACM SIGGRAPH, 1998; [2] David E. Breen, Donald H. House, and Michael J. Wozny. Predicting the drape of woven cloth using interacting particles. In Proceedings of SIGGRAPH 94, Computer Graphics Proceedings, Annual Conference

Series, pages 365-372. ACM, ACM Press / ACM SIGGRAPH, July 1994;

[3] Robert Bridson, Ronald P. Fedkiw, and John Anderson. Robust treatment of collisions, contact, and friction for cloth animation. In John Hughes, editor, SIGGRAPH 2002 Conference Proceedings, Annual Conference Series, pages 594-603. ACM Press/ACM SIGGRAPH, 2002;

[4] Michel Carignan, Ying Yang, Nadia Magnenat-Thalmann, and Daniel Thalmann. Dressing animated synthetic actors with complex deformable clothes. In Computer Graphics (Proceedings of ACM SIGGRAPH 92), pages 99-104. ACM, July 1992; [5] Kwang-Jin Choi and Hyeong-Seok Ko. Stable but resposive cloth. In John Hughes, editor, SIGGRAPH 2002 Conference Proceedings, Annual Conference Series, pages 604-611. ACM Press/ACM SIGGRAPH, 2002;

[6] Kwang-Jin Choi and Hyeong-Seok Ko. Extending the immediate buckling model to triangular meshes for simulating complex clothes. In Eurographics 2003, short paper, 2003;

[7] Frederic Cordiner and Nadia Magnenat-Thalmann. Real-time animation of dressed virtual humans. In Eurographics 2002, 2002;

[8] Bernhard Eberhardt, Andreas Weber, and Wolfgang Strasser. A fast, flexible, particle-system model for cloth draping. IEEE Computer Graphics and Applications, 16(5):52-59, September 1996;

[9] Young-Min Kang and Hwan-Gue Cho. Bilayered approach for efficient animation of cloth with realistic wrinkles. In Computer Animation 2002, 2002;

[10] Young-Min Kang, Jeong-Hyeon Choi, Hwan-Gue Cho, and Chan-Jong Park. Fast and stable animation of cloth with an approximated implicit method. In Computer Graphics International, pages 247-256, 2000;

[11] Mark Meyer, Gilies Debunne, Mathieu Desbrun, and Alan H. Barr. Interactive animation of cloth-like objects in virtual reality. Journal of Visualization and Computer Animation (JVCA), 20001;

[12] P. Volino and N. Magnenat-Thalmann. Implementing fast cloth simulation with collision response. In Proceedings of the Conference on

Computer Graphics International (CGI-OO), pages 257-268, June 19-242000; and

[13] Pascal VoIino, Martin Courshesnes, and Nadia Magnenat Thalmann. Versatile and efficient techniques for simulating cloth and other deformable objects. In Proceedings of SIGGRAPH 95, Computer Graphics Proceedings, Annual Conference Series, pages 137-144. ACM, ACM Press / ACM SIGGRAPH, August 1995.