As scheduled, I have implemented the work of Eberhardt [1], as discussed below. I have also done a fairly exhaustive background search on related works. At some point in time, the bibliographic data will appear on this page. I have decided to address the issue of dynamic topology (e.g. the topological equivelant of a sphere changing into the topological equivelant of a torus, etc.). Of course, this begins with implementing the meshed particle system and defining simple government for the particles (as discussed below). I think I will only consider topological changes when a surface begins to intersect itself. For now, I will just use a simple collision detection method (haven't yet decided which, it will probably just be some hack since this portion is not the crux). The big issue is how to rearrainge the connections on the mesh when the topology does change. I have focussed in my brainstorms so far on increasing Eulor number cases (sphere into torus), but I would like a solution that simultaniously deals with decreasing cases (torus into sphere). I do have a couple ideas, they need to be tried out and explored. That is the next step. Also, governing the changing of the shape needs to be addressed (I am planning on using spheres inside the solid, and moving/changing the spheres. This additional governence of the particles (based on dynamic objects which the surface cannont intersect), needs to be implemented also. I expect this will be neither trival nor too hard.
I have implemented the work of Eberhardt [1], to some extent. They use some differential equations whose details are scatterred through other papers. In looking through those other papers, I determined that they did (as is common) their differential equation work through Maple's c-code generator. I did not go to this extent, as an approximation is sufficient for my work here. I took an approach more along those of the "finite element" spirit (particularly the temperature style ones). The exact differential equations determine the next position of the particle based on its interactions with its (4) neighbors. These interactions are based on the classic spring equations (Hooke, etc.). I approximated these by, at each step, determining the next position as the average of the (4) neighbors' positions. Begining with a 250x250 mesh, arrainged in a cylindrical shape, I produced the image below. The original cylinder points are automatically generated, and can be left in this position (the differential equation approach is unstable when the initial points are too regular, so Eberhardt randomly displaces each in small amounts). Also, my approach appears initially to much quicker than the results they claim. Hence, for now, I will stick with my approach, since the interesting stuff is not related to these specifics. I possibly will build in the differential equation portion later.
