Lambertian Surfaces

A Lambertian surface is referred to as a perfectly diffusing surface, which adheres to Lambert’s cosine law. This law states that the reflected or transmitted luminous intensity in any direction from an element of a Lambertian surface varies as the cosine of the angle between that direction and the normal of the surface. The intensity I_θ of each ray leaving the surface at an angle θ from the ray in a direction perpendicular to the surface (I_n) is given by:



Therefore, even if the luminous intensity decreases with a factor cos(θ) from the normal, the projected surface decreases with the same factor; as a consequence, the radiance (luminance) of a Lambertian surface is the same regardless of the viewing angle and is given by:



It is worthwhile to note that in a Lambertian surface the ratio between the radiant exitance and the radiance is π and not 2π:



This equation can be easily derived. Suppose we place an infinitesimal Lambertian emitter dA on the inside surface of an imaginary sphere S. The inverse square law [Equation (5.15)] provides the irradiance E at any point P on the inside surface of the sphere. However, d = D * cos θ, where D is the diameter of the sphere. Thus:



and from Lambert’s cosine law [Equation (5.22)], we have:



which simply says that the irradiance (radiant flux density) of any point P on the inside surface of S is a constant.

This is interesting. From the definition of irradiance [Equation (5.9)], we know that Φ = E * A for constant flux density across a finite surface area A. As the area A of the surface of a sphere with radius r is given by:



we have:



Given the definition of radiant exitance [Equation (5.10)] and radiance for a Lambertian surface [Eqation (5.23)], we have:



This explains, clearly and without resorting to integral calculus, where the factor of p comes from.