Phong illumination model expression

The Phong illumination model (empirical) computes the intensity at a surface point as the sum of ambient, diffuse and specular components.

1. Ambient light: constant background light, equal in all directions.

I_amb = ka · Ia

(ka = ambient reflection coefficient, Ia = ambient intensity.)

2. Diffuse reflection (Lambert's cosine law): light is scattered equally in all directions; intensity depends on the angle θ between the surface normal N and light direction L.

I_diff = kd · IL · cosθ = kd · IL · (N · L)

(N and L unit vectors; use max(0, N·L).)

3. Specular reflection: the shiny highlight seen around the direction of perfect reflection R. With φ the angle between R and the viewing direction V:

I_spec = ks · IL · cos^n φ = ks · IL · (R · V)^n

where R = 2(N·L)N − L, ks = specular coefficient, n = shininess exponent (larger for shiny surfaces). Using H = (L + V)/|L + V|, (N·H)^n can replace (R·V)^n.

Total for one point source

I = ka·Ia + IL [ kd (N·L) + ks (R·V)^n ]

With distance (attenuation): the light intensity from a point source decreases with distance d:

f(d) = min( 1 / (a0 + a1·d + a2·d²), 1 )

I = ka·Ia + f(d) · IL [ kd (N·L) + ks (R·V)^n ]

For several light sources i = 1 … m:

I = ka·Ia + Σ f(d_i) · IL_i [ kd (N·L_i) + ks (R_i·V)^n ]

Example. ka = 0.3, Ia = 0.4, kd = 0.5, ks = 0.4, n = 10, IL = 1, f = 1; N·L = 0.8, R·V = 0.9:

I = 0.3·0.4 + 1·[0.5·0.8 + 0.4·0.9^10] = 0.12 + [0.40 + 0.4·0.3487] = 0.12 + 0.40 + 0.139 = 0.659