page 11: A particle system as seen "from top". The dashed lines represent the rays casted from the observer through the system, the dots are particles which are hit by these rays.
page 12: A good approximation of this systems can be made simply by knowing how many particles are hit by each ray, not their exact position.
page 13: Thus it is possible to emulate this system in 2D, where the 2D movements are the projections of the actual 3D movements.
page 14: To move the particles along a direction (gray), a vector (black) can be used that is as near as possible to the original one. This leads however to very artificial looking movement.
page 15: This can be avoided by using several nearby vectors, and a scaling factors for each of them
page 16: As scaling function a gauss-function could be used.
page 17: However, here we suggest using a cosine function. This has the advantage that it can be implemented simple (and thus fast) by using the dot-product between the original vector (black) and the candidate vectors (gray).
page 18: It should be noted that we use different forces, which are evaluated individually. Thus, particles may move in very different directions in each step, allowing for fast particle spreading.
page 19: Note that the last 3 forces can be easily implemented as vectors. The first one is simply calculated as the difference in the number of particles in neighboring cells.
page 21: Examples for the different forces:
up left: input to the system, up right: diffusion
middle left: external force (note how it spreads out to the sides as well due to using the dot-product as described earlier), middle right: random
down left: these three forces combined down right: the three forces plus inertia
page 22: Comparison of our approach (left) to Navier Stokes, as implemented in Simple and Fast Fluids (Martin Guay, Fabrice Colin, Richard Egli, 2011) (right)
Note that our approach spreads out faster, although it is possible as well to get a result that looks like the one of Navier Stokes.
It seems that our approach allows more different outcomes, while at the same time being very intuitively to use thanks to blending different, very easily imaginable forces.
page 23: Note that the simulation results are collected from the viewpoints which (on average) lead to the worst results. The estimated fps assume that the simulation runs at a constant 10 times per second. The results for the iPhone are preliminary and subject to change.
Note that the simulation of NS (given as reference) is slower by 50% than our approach. More importantly, since the pressure fields (which are not needed in our approach) are typically 4 times bigger than the particle fields, the Navier-Stokes reference implementation uses 5 times the amount of data.