Is there a reason why we don't do some form of grid search instead of using monte carlo in most cases?
Azure
I mean since the volume of the hemisphere is about 52% of the volume of its bounding box, you'd only be throwing away about 1/2 the samples, which isn't too bad. You could also just sample from the bounding hemisphere instead (unit circle x height), which makes it more efficient to have it have 66% the volume (reject 33% of samples), although if you're already doing square roots for uniform circle (inversion) sampling you might as well just go full inversion sampling.
bpopeck
Is it possible to generalize this rejection sampling approach to surfaces more complex than the hemisphere? Could one take an arbitrary surface, make it closed (to get a volume), use rejection sampling to uniformly sample points within the volume, then project points from the interior of the volume to the original surface in some uniform way?
Seems like the last step in particular might not be feasible, but I'm not sure how else to think about sampling uniformly from a general surface.
Is there a reason why we don't do some form of grid search instead of using monte carlo in most cases?
I mean since the volume of the hemisphere is about 52% of the volume of its bounding box, you'd only be throwing away about 1/2 the samples, which isn't too bad. You could also just sample from the bounding hemisphere instead (unit circle x height), which makes it more efficient to have it have 66% the volume (reject 33% of samples), although if you're already doing square roots for uniform circle (inversion) sampling you might as well just go full inversion sampling.
Is it possible to generalize this rejection sampling approach to surfaces more complex than the hemisphere? Could one take an arbitrary surface, make it closed (to get a volume), use rejection sampling to uniformly sample points within the volume, then project points from the interior of the volume to the original surface in some uniform way?
Seems like the last step in particular might not be feasible, but I'm not sure how else to think about sampling uniformly from a general surface.