Mid-Late 1980s Simulator Model Results Model Results Simulator Forces Reconcile realism with control.
Model Plausible Animation Simulator Model Results Simulator Model Results Model Results Simulator Forces
Exactness? Simulator Model Simulator Model Result A Simulator Forces Result B
Plausible Animation (2) Simulator Model Results Model Simulator Model Results Simulator Model Results Model Results Simulator Forces
Three versions of physical motion Nature Model Numerics
Natures solution What really happens in the world What would really happen in the world if we tried it –Important question: Tried what? Whats the situation were asking nature about?
Model solution Might say Mathematical model A simplification of the real world –e.g. rigid body model –e.g. Newton vs. Einstein Chosen to capture interesting or relevant properties Expressed as equations of motion
Numerical solution Approximation to analytic solution of model equations Given numbers describing objects & state, returns numbers describing their motion.
Examine what we mean by the correct result What result should we be willing to accept? Why? Is there a single correct result?
Graphics models only describe an approximation Have already made a somewhat arbitrary choice No need to be too insistent on it But lets say its as good as we can get…
Numerical solution is always a cloud All values within the cloud are equally accurate Traditional view: solver computes best answer –the cloud can be made arbitrarily small –cloud converges on the correct answer. –…but is this always true?
The model may be unstable Consider a ball that lands exactly on the fence, can fall on either side Numerical cloud is disjoint Decreasing tolerance parameter doesnt cause cloud to converge. Solver chooses one side or the other arbitrarily Either side is equally correct A more honest solver would offer both sides, let us choose between them
How good are our input values? Often describe object as sphere or plane, etc. –Real-world objects are never exactly spherical or planar –Texture mapping, microfacets, etc. known in rendering to get more realistic results –Similarly we need texturing in simulation to get more realistic results
Consider input as a range/distribution Yields distribution of results If model is stable: –Results may vary slightly –But may be observable If model is unstable: –Results may vary almost arbitrarily Honest solver would offer range of results
In some sense, were saying: Because of limitations of computing… –We cant really compute Natures solution anyway –There are always many results that are equally appropriate w.r.t. model and inputs –We may as well choose the one we want
But even more: In principle we cant know inputs with analytic accuracy Natures solution isnt unique. –The real world includes instability –Random-number generators: dice –Chaos
Ultimate claim In no case can we compute a single correct solution We can therefore choose among them.