Generate a random point within a circle (uniformly)

Question

I need to generate a uniformly random point within a circle of radius R.

I realize that by just picking a uniformly random angle in the interval [0 ... 2π),

Answer 1

First we generate a cdf[x] which is

The probability that a point is less than distance x from the centre of the circle. Assume the circle has a radius of R.

obviously if x is zero then cdf[0] = 0

obviously if x is R then the cdf[R] = 1

obviously if x = r then the cdf[r] = (Pi r^2)/(Pi R^2)

This is because each "small area" on the circle has the same probability of being picked, So the probability is proportionally to the area in question. And the area given a distance x from the centre of the circle is Pi r^2

so cdf[x] = x^2/R^2 because the Pi cancel each other out

we have cdf[x]=x^2/R^2 where x goes from 0 to R

So we solve for x

R^2 cdf[x] = x^2

x = R Sqrt[ cdf[x] ]

We can now replace cdf with a random number from 0 to 1

x = R Sqrt[ RandomReal[{0,1}] ]

Finally

r = R Sqrt[  RandomReal[{0,1}] ];
theta = 360 deg * RandomReal[{0,1}];
{r,theta}

we get the polar coordinates {0.601168 R, 311.915 deg}