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Unformatted text preview: HW7: CONSTRAINED DYNAMICS OF CLOTH MOTION BY MOON-RYUL JUNG ABSTRACT. This paper is intended to help students to model the dynamics of cloth by means of differential equations. The dynamics of cloth can be derived either by Euler- Lagrange equations or by directly constructing Newton motion equations. Here we follow the second method. The students are required to describe the dynamics of cloth (described in detail in this paper) in their own language and implement the simulation of the dynam- ics. It amounts to solving the differential equations. The results of the simulation will be a trajectory of cloth configurations and velocities. The students are requested to draw the tra- jectories of sample vertices of the particle model. But visualization by means of OPENGL programming is strongly recommended for those who know OPENGL. The cloth config- uration is a very high dimensional vector, whose dimension is three times the number of vertices in the mesh representing cloth. How can you present the trajectory of such high dimensional configurations without visualizing them? 1. INTRODUCTION 2. THE DYNAMICS OF CLOTH The particle model of cloth assumes that cloth can be modeled as a network of particles as shown in Figure 1. The mass of cloth is assumed to be concentrated on the particles, and the connections between particles are modeled as springs. Among the particle models FIGURE 1. Each vertex in the particle model receives forces from the neighbor vertices, according to the connections that are allowed in the cloth model. The connections of a given vertex with the neighbor ver- tices are different depending on where the given vertex is located. In the most general case, a vertex is connected to the immediate or first neigh- bor vertices. Furthermore, the vertex is connected to the second neighbor vertices. It is assumed that the vertex is not influenced by the neighbor vertices which are away more than the second neighbor vertices. 1 2 BY MOON-RYUL JUNG suggested in the literature, that of Choi and Ko (2002) seems to be the simplest and clearest. Also, this work was exceptionally well-received in 2002 Siggraph, because of a lively animation demo and is well-known in the field of cloth simulation. This paper presents the Choi and Ko model, but uses different notions to make the presentation clearer. Moreover, the model of Choi and Ko incorporate artificial and ad hoc manipulations to the basic particle model in order to make the system matrix of the linearized motion equation positive definite. The positive definite of the system matrix is needed in order to use the conjugate gradient method to solve the linear system derived in the implicit integration method of the motion equation. But the positive definiteness of the system matrix is not required when we use other methods than the conjugate gradient method to solve the linear system. So, in this paper, As will be shown later, the configuration of cloth in this model is defined only in terms of the positions of the vertices. The particle model suggested by Breen et al usesin terms of the positions of the vertices....
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- Spring '10