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

Such a fun problem.
The rationale of the probability of a point being chosen lowering as distance from the axis origin increases is explained multiple times above. We account for that by taking the root of U[0,1]. Here's a general solution for a positive r in Python 3.

import numpy
import math
import matplotlib.pyplot as plt

def sq_point_in_circle(r):
    """
    Generate a random point in an r radius circle 
    centered around the start of the axis
    """

    t = 2*math.pi*numpy.random.uniform()
    R = (numpy.random.uniform(0,1) ** 0.5) * r

    return(R*math.cos(t), R*math.sin(t))

R = 200 # Radius
N = 1000 # Samples

points = numpy.array([sq_point_in_circle(R) for i in range(N)])
plt.scatter(points[:, 0], points[:,1])