Uniform random point in a disk

For a point uniformly distributed in the unit disk, its polar coordinates are independent with densities

f_{R} (r) = 2 r 1_{[0, 1]} (r), f_{Θ} (θ) = \frac{1}{2 π} 1_{[0, 2 π)} (θ) .

(1)

Table of contents
- Origin in a triangle of three radial random points Uniform random point in a disk

Origin in a triangle of three radial random points

 0  0

For three independent points from any rotationally symmetric planar distribution with nonzero radii, the origin lies in their triangle exactly when their three angles do not all lie in one semicircle. Three independent uniform angles have this property with probability