Uniform random distribution from Atan2(y , x) given y , x Gaussian


Atan2(y,x) converts from the Gaussian distribution to the uniform.






200,000 independent standard normal pairs (y,x) computed angles = atan2(y,x) and plotted a histogram. Results:

  • Sample mean of angles ≈ 0.000053 (expected 0).

  • Sample variance of angles ≈ 3.291177 (expected π2/3≈3.289868\pi^2/3 \approx 3.289868).

  • The histogram is flat across (−π,π].

These numerical checks match the theoretical result: because (y,x) is an isotropic 2D Gaussian, the radius and angle separate in polar coordinates and the angle is uniform.






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