Diffuse reflection equation

Diffuse reflection occurs on rough (dull) surfaces. Incident light is scattered equally in all directions, so the surface looks equally bright from every viewing position. The colour depends on the surface material and the light, not on the viewer position.

Lambert's cosine law. The intensity of the diffusely reflected light is proportional to the cosine of the angle θ between the unit surface normal N and the unit direction L to the light source (the surface receives less light per unit area when it is tilted):

I_diff = kd · IL · cosθ = kd · IL · (N · L), for 0 ≤ θ ≤ 90°

• kd = diffuse reflection coefficient (0 to 1)
• IL = intensity of the point light source

        L  N
         \ |
          \|θ
   --------+---------- surface

If N·L < 0, the surface is facing away from the light and I_diff = 0.

Ambient light. A surface not directly lit still receives light scattered from the surroundings. This is modelled as a uniform background intensity Ia, and the reflected part is ka·Ia (ka = ambient coefficient).

Total diffuse reflection with ambient light

I_diff,total = ka · Ia + kd · IL · (N · L)

Often kd = ka is taken for simplicity, giving I = kd (Ia + IL·cosθ).

With distance attenuation f(d) = 1/(a0 + a1·d + a2·d²): I = ka·Ia + f(d)·kd·IL·(N·L).

Example: ka = kd = 0.5, Ia = 0.2, IL = 1, θ = 60°: I = 0.5·0.2 + 0.5·1·0.5 = 0.10 + 0.25 = 0.35