# Why do Lambertian Reflectors Privilege the Observer?

Lambert's Cosine Law for Lambertian Reflectors states that the radiant intensity $I$ (watts per square meter per steradian) of reflected light as a function of the angle $\theta$ from the normal to the surface of the reflector is $I(\theta)=I\cos\theta$. Because of this law, uniformly-lit diffusely-reflecting planar surfaces (which approximate perfect Lambertian Reflectors) also appear uniformly-lit to an observer, which is to say that the radiant intensity of light reaching the observer from such a planar surface is constant no matter where on the surface the observer looks.

So far so good. There is, however, a deep mystery lurking beneath this simple principle, because it would seem that it would be far more parsimonious for the radiant intensity of light reflected by a Lambertian Reflector to be constant for all $\theta$ (i.e. have no angular dependence) at the "cost" of having the observer perceive an angular dependence to the radiant intensity of light reflected by a uniformly-lit planar Lambertian Reflector. Why is the observer "privileged" in such a manner by Lambert's Cosine Law?

• Sorry I don't understand your question. Do you mean why a lambertian diffusor looks uniformly illuminated? or what's exactly the question? Sorry I don't get you. Commented Sep 18, 2017 at 19:08
• Well, for a start, what accounts for Lambert's Cosine Law? I assume it's not just an axiom but can actually be derived from more basic principles. What are those principles? Commented Sep 18, 2017 at 20:19

## 1 Answer

This has to with the Jacobian for solid angle:

$d\Omega = d\phi\sin{\theta}d\theta=d\phi d(\cos{\theta})$

so that the azimuthal coordinate is uniform in $\phi$ and the polar coordinate is uniform in $\cos{\theta}$, not $\theta$.

• What you are basically saying is that the solid angle is not constant over equal increments of $\theta$ in the spherical coordinate system. I agree. But this misses the point. What Lambert's Cosine Law states is that the watts per square meter reflected by a Lambertian Reflector per unit steradian is not constant, and varies with the cosine of $\theta$. So this has nothing to do with solid angles per se because those are already taken care of by the definition of radiant intensity itself. Commented Sep 18, 2017 at 13:17