It seems to me that a central focus of the course or at least of graphics as a field is the contrast between particle-based methods for simulating fluids behavior and light (Lagrangian, ray-tracing) versus grid-based methods (Eulerian, rasterization) and maybe other constraint-solving methods. Are there any other examples of a tradeoff between simulating with particles bouncing around a scene versus simulating with a grid on the scene?
@jzhanson Yes, you start to get the feeling that the question of implicit vs. explicit representations, Eulerian vs. Lagrangian discretizations, and so forth, are fundamental themes in computer graphics and numerical methods. And that's absolutely true---that's why we teach it! ;-). As far as specific examples of Eulerian vs. Lagrangian, you've nailed the most important ones from geometry, animation, and rendering. Conspicuously missing, perhaps, is the use of Lagrangian representations in image processing. Though surely there are examples of this kind of thinking (e.g., in motion estimation), for "meat and potatoes" image processing the Eulerian representation (pixels on a grid) really is hard to beat.