Are there any ways to sample the circle (intuitively) with only one RV?

mdsavage

@xTheBHox: It's (probably) possible to do this with a space-filling curve of some kind. For example, you could adjust a Peano curve to fill the boundary of a circle, as in this image:

Of course, this appears to have the same issue as with the uniform probability case on the left of the slide -- the sampling distribution will (probably) not be uniform. The amount of recursive computation needed to accurately compute where on the fractal curve you end up is also probably not worth the savings of generating a second random number.

ljelenak

This is a problem that was in my computational physics class that we had to solve!

theyComeAndGo

How does this paradox make sense intuitively?

Sleepyhead08

So, for our PathTracer, this circle distribution should be what our random sampler weighted by cosine should look from above, right?

dvanmali

Could someone explain what the different quantity of Epsilon measure? I know that it is the resulting probability at the point but I don't know why there exists both E1 and E2.

Are there any ways to sample the circle (intuitively) with only one RV?

@xTheBHox: It's (probably) possible to do this with a space-filling curve of some kind. For example, you could adjust a Peano curve to fill the boundary of a circle, as in this image:

Of course, this appears to have the same issue as with the uniform probability case on the left of the slide -- the sampling distribution will (probably) not be uniform. The amount of recursive computation needed to accurately compute where on the fractal curve you end up is also probably not worth the savings of generating a second random number.

This is a problem that was in my computational physics class that we had to solve!

How does this paradox make sense intuitively?

So, for our PathTracer, this circle distribution should be what our random sampler weighted by cosine should look from above, right?

Could someone explain what the different quantity of Epsilon measure? I know that it is the resulting probability at the point but I don't know why there exists both E1 and E2.