It's interesting that assigning the same probability to each theta and r value did not result in a uniform sample. But intuitively it makes sense that we would have to account for the 2nd dimension when sampling from a circle that we did not when sampling from a number line. I wonder if we wanted to sample from a sphere, we would have to take the cube root for r.
keenan
A general recipe for uniformly sampling any 2D region is given in a really nice article by Jim Arvo, "Stratified Sampling of 2-manifolds", with lots of nice examples.
It's interesting that assigning the same probability to each theta and r value did not result in a uniform sample. But intuitively it makes sense that we would have to account for the 2nd dimension when sampling from a circle that we did not when sampling from a number line. I wonder if we wanted to sample from a sphere, we would have to take the cube root for r.
A general recipe for uniformly sampling any 2D region is given in a really nice article by Jim Arvo, "Stratified Sampling of 2-manifolds", with lots of nice examples.