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The equations of motion for a system govern the motion of the system. One of the first (and simplest) cloth models is as follows: consider the sheet of cloth. Divide it up into a series of approximately evenly spaced masses M. Connect nearby masses by a spring, and use Hooke's Law and Newton's 2nd Law as the equations of motion. Various additions, such as spring damping or angular springs, can be made. A mesh structure proves invaluable for storing the cloth and performing the simulation directly on it. Each vertex can store all of its own local information (velocity, position, forces, etc.) and when it comes time to render, the face information allows for immediate rendering as a normal mesh.
In the cloth structure seen above, the red nodes are vertices, and the black lines are springs. The diagonal springs are necessary to resist collapse of the face; it ensures that the entire cloth does not decompose into a straight line. The blue represents the mesh faces. Looking at the equations of motion:
To determine M, a simple constant (say, 1) is fine for all vertices. To be more accurate, you should compute the area of each triangle, and assign 1/3rd of it towards the mass of each incident vertex; this way the mass of the entire cloth is the total area of all the triangles times the mass density of the cloth. The gravity vector can also be an arbitrary vector; if all distance units were meters, time was measured in seconds, and we were on the surface of the earth and "y" was the "up/down" vector, (0, -9.8, 0) would be the correct "g". X(current) is just the current length of the spring, and X(rest), the spring's rest length, needs to be stored in each spring structure. F(wind) can just be some globally varying constant function, say (sin(x*y*t), cos(z*t), sin(cos(5*x*y*z)). a is a simple constant determined by the properties of the surrounding fluid (usually air,) but it can also be used to achieve certain cloth effects, and can help with the numeric stability of the cloth. k is a very important constant; if too low, the cloth will sag unrealistically:
On the other hand, if k is chosen too large, the system will be unrealistically tight (retain its origional shape.) Worse yet, the system will be more and more "stiff" the larger k is chosen; this is a mathematical term for explosive. Without careful attention to the integration method, the system will gain energy as a function of time, and the system will explode with vertices tending towards infinity.
The mass-spring model above has several shortcommings. Mostly, it is not very physically correct, so attempting to use physically accurate constants is generally unsuccessful. Furthermore it requires guessing arbitrarially which vertices should be connected to which by a spring, and choosing k such that increasing the resolution of the grid leads to a system with similar characteristics can be tricky. A more accurate model, based on integrating an energy over the surface of the cloth, consider energy terms such as:
We imagine a giant vector, S, representing every important variable in the system (position and velocities of all the vertices, although potentially there could be more degrees of freedom.) Given energy as a function of the current state of the system, E(S), the equation of motion for a single vertex at position (x, y, z) is then rather simple:
Evaluating this, however, is not so simple. Generally this derivative must be computed analytically. Suppose we attempted to compute the derivative numerically; we consider the state variable constant, reducing our energy E(s) to E(x). We then say:
But evaluating the energy E(S) takes a long time; we must iterate over all the vertices, faces, and edges, summing the energy of each one. But we might notice that the effect of x on the energy depends on a very local region (all the incident edges and faces, called the one-ring of the vertex.) So to keep our algorithm O(n) when doing the derivative numerically, we must make sure that we compute E(x) by only considering the energy of the one-ring of the vertex in question.
In general, this method can be very challenging to implement, and although it is physically much more sound, in practice the results are in some ways better and in some ways worse than the mass-spring model. This method of cloth animation, including the derivation of the energy terms, is discussed thoroughly in this paper.