According to MathWorld, the naive way to randomly pick points inside a disk, by using two uniformly distributed variables that are polar parameters: $r \sim [0, 1]$ and $\theta \sim [0, 2\pi]$, is incorrect, leading to the distribution shown on the left.
I am trying to understand why this is. It is sort of intuitive that for a given angle, points on the edge of smaller radii will be closer together than those on the edge of larger radii.
MathWorld gives the mathematical reason as follows:
The area element given by $dA = 2\pi r \, dr$ leads to a concentration of points in the center
How does this being the area element lead to a concentration of points in the center, however?
If you select points according to uniform distributions for $r$ and $\theta$, then half your points should lie within the circle of radius $1/2$. But this disk only covers one quarter of the area of the whole disk, meaning that there are twice as many points inside this disk as we would want with a uniform distribution.
If we take the innermost $10\%$, it gets worse: The disk with radius $0.1$ covers only $1\%$ of the total area, leading to ten times the desired concentration.
In general, we want fraction $r^2$ of the points to lie within the disk of radius $r$ if going by area, but the uniform distribution gives fraction $r$.