I feel like I can kind of see how the radius need to be square-rooted. I'm not sure if I'm thinking about it right, but say the whole circle has d density of sample points. Since we want to have that same density of sample points across any radius r, we have to square root r because as r grows, the area grows r^2 time, so if we square root r, the area would grow linearly, keeping the density of sample points the same?
I feel like I can kind of see how the radius need to be square-rooted. I'm not sure if I'm thinking about it right, but say the whole circle has d density of sample points. Since we want to have that same density of sample points across any radius r, we have to square root r because as r grows, the area grows r^2 time, so if we square root r, the area would grow linearly, keeping the density of sample points the same?