Editing Lambert's cosine law (section)

==Details of equal brightness effect==

[[Image:Lambert Cosine Law 1.svg|thumb|Figure 1: Emission rate (photons/s) in a normal and off-normal direction. The number of photons/sec directed into any wedge is proportional to the area of the wedge.]]
[[Image:Lambert Cosine Law 2.svg|thumb|Figure 2: Observed intensity (photons/(s·m<sup>2</sup>·sr)) for a normal and off-normal observer; ''dA''<sub>0</sub> is the area of the observing aperture and ''d''Ω is the solid angle subtended by the aperture from the viewpoint of the emitting area element.]]
The situation for a Lambertian surface (emitting or scattering) is illustrated in Figures 1 and 2.  For conceptual clarity we will think in terms of [[photon]]s rather than [[energy]] or [[luminous energy]]. The wedges in the [[circle]] each represent an equal angle ''d''Ω, of an arbitrarily chosen size, and for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge.

The length of each wedge is the product of the [[diameter]] of the circle and cos(''θ''). The maximum rate of photon emission per unit [[solid angle]] is along the normal, and diminishes to zero for ''θ'' = 90°. In mathematical terms, the [[radiance]] along the normal is ''I''&nbsp;photons/(s·m<sup>2</sup>·sr) and the number of photons per second emitted into the vertical wedge is {{nowrap|''I'' ''d''Ω ''dA''}}. The number of photons per second emitted into the wedge at angle ''θ'' is {{nowrap|''I'' cos(''θ'') ''d''Ω ''dA''}}.

Figure 2 represents what an observer sees. The observer directly above the area element will be seeing the scene through an aperture of area ''dA''<sub>0</sub> and the area element ''dA'' will subtend a (solid) angle of ''d''Ω<sub>0</sub>, which is a portion of the observer's total angular field-of-view of the scene. Since the wedge size ''d''Ω was chosen arbitrarily, for convenience we may assume without loss of generality that it coincides with the solid angle subtended by the aperture when "viewed" from the locus of the emitting area element dA. Thus the normal observer will then be recording the same {{nowrap|''I'' ''d''Ω ''dA''}} photons per second emission derived above and will measure a radiance of

:<math>
I_0=\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}
</math>  photons/(s·m<sup>2</sup>·sr).

The observer at angle ''θ'' to the normal will be seeing the scene through the same aperture of area ''dA''<sub>0</sub> (still corresponding to a ''d''Ω wedge) and from this oblique vantage the area element ''dA'' is foreshortened and will subtend a (solid) angle of ''d''Ω<sub>0</sub>&nbsp;cos(''θ'').  This observer will be recording {{nowrap|''I'' cos(''θ'') ''d''Ω ''dA''}} photons per second, and so will be measuring a radiance of

:<math>
I_0=\frac{I \cos(\theta)\, d\Omega\, dA}{d\Omega_0\, \cos(\theta)\, dA_0}
=\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}
</math>  photons/(s·m<sup>2</sup>·sr),

which is the same as the normal observer.