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

Let ρ (radius) and φ (azimuth) be two random variables corresponding to polar coordinates of an arbitrary point inside the circle. If the points are uniformly distributed then what is the disribution function of ρ and φ?

For any r: 0 < r < R the probability of radius coordinate ρ to be less then r is

P[ρ < r] = P[point is within a circle of radius r] = S1 / S0 =(r/R)²

Where S1 and S0 are the areas of circle of radius r and R respectively. So the CDF can be given as:

          0          if r<=0
  CDF =   (r/R)**2   if 0 < r <= R
          1          if r > R

And PDF:

PDF = d/dr(CDF) = 2 * (r/R**2) (0 < r <= R).

Note that for R=1 random variable sqrt(X) where X is uniform on [0, 1) has this exact CDF (because P[sqrt(X) < y] = P[x < y**2] = y**2 for 0 < y <= 1).

The distribution of φ is obviously uniform from 0 to 2*π. Now you can create random polar coordinates and convert them to Cartesian using trigonometric equations:

x = ρ * cos(φ)
y = ρ * sin(φ)

Can't resist to post python code for R=1.

from matplotlib import pyplot as plt
import numpy as np

rho = np.sqrt(np.random.uniform(0, 1, 5000))
phi = np.random.uniform(0, 2*np.pi, 5000)

x = rho * np.cos(phi)
y = rho * np.sin(phi)

plt.scatter(x, y, s = 4)

You will get