Uniformly Distributed Random Points Inside a Circle

So I just started my blog and I already posted my first blog. Now I want to get my hand dirty by posting some math!

The problem I want to consider here is generation of uniformly distributed random points inside different geometric shapes. The simplest shape to consider is a rectangle. In this case, the generation of uniformly distributed random points is trivial. Let $l$ be the length of the square, $w$ be the width, and the point $p_c(x_c,y_c)$ be the center of the rectangle. Let $p_i(X,Y)$ be a random point inside the rectangle. Here $X$ and $Y$ are random variables. For $p_i$ to be uniformly distributed, $X$ must be a uniform distribution over the interval $\left[ l-\frac{x_c}{2}, l+\frac{x_c}{2} \right]$, and $Y$ must be a uniform distribution over the interval $\left[ w-\frac{y_c}{2}, w+\frac{y_c}{2} \right]$. Well this is fairly trivial.

Now lets consider a circle. The problem is not as trivial. We define a circle by its radius $r_c$ and the point representing its center $p_c(x_c,y_c)$. For simplicity lets assume the center of the circle is on the origin (i.e. $p_c(0, 0)$ ). The first guess that people typically give (which is wrong) is to use polar coordinates with $r$ and $\theta$ uniformly distributed over $\left[ 0, r_c \right]$ and $\left[ 0, 2\pi \right]$, respectively. The reason this turn out to be wrong is because the area of a circle with radius $\frac{r_c}{2}$ and the area of a ring (or doughnut) with width $\frac{r_c}{2}$ are not the same. What is the correct answer? (I will post the answer with derivation in a few days!)

See the answer here.