• Introduction
• 2D
  • Uniform disk
  • Directions
    • Uniform Unit Hemisphere
    • Power Cosine Weighted Unit Hemisphere Sector
    • Power Cosine Weighted Unit Hemisphere Cap
    • Cosine Weighted Unit Hemisphere
  • Normals
    • Beckmann Distribution
    • Phong Distribution
    • GGX Distribution

Cosine Weighted Hemisphere

The probability of each sample is weighted by its cosine with the normal.

Here is the result:

function cos_hemisphere(){

    const u_1 = Math.random();
    const u_2 = Math.random();
    
    // Math.sqrt(u_1) works the same way
    const theta = Math.acos(Math.sqrt(1.0 - u_1));
    const phi = u_2 * 2.0 * Math.PI;

    return [theta,phi];
}

Algorithm

Generate points weighted by the cosine hemisphere.

1. Choose uniform numbers $u_1, u_2$ in $[0,1]$
2. Calculate $\theta = \cos^{-1}(\sqrt{1 - u_1} )$
3. Calculate $\phi = 2\pi u_2$

Optionally convert to cartesian coordinates with:

$$x = \sin\theta\cos\phi$$

$$y = \sin\theta\sin\phi$$

$$z = \cos\theta$$

PDF Used In Monte Carlo Integration

$$p(\omega) = \frac{1}{\pi}\cos\theta$$

$$p(\theta,\phi) = \frac{1}{\pi}\cos\theta\sin\theta$$

Derivation

This is a special case of the power cosine weighted hemisphere cap with $\theta_{max} = \frac{\pi}{2}, n = 1$