Lambert's Cosine Law - Details of Equal Brightness Effect

Details of Equal Brightness Effect

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 photons rather than energy or luminous energy. The wedges in the circle each represent an equal angle dΩ, and for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge.

It can be seen that the length of each wedge is the product of the diameter of the circle and cos(θ). It can also be seen that 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 photons/(s·cm2·sr) and the number of photons per second emitted into the vertical wedge is I dΩ dA. The number of photons per second emitted into the wedge at angle θ is I cos(θ) dΩ 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 dA₀ and the area element dA will subtend a (solid) angle of dΩ₀. We can assume without loss of generality that the aperture happens to subtend solid angle dΩ when "viewed" from the emitting area element. This normal observer will then be recording I dΩ dA photons per second and so will be measuring a radiance of

$I_0=\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}$ photons/(s·cm2·sr).

The observer at angle θ to the normal will be seeing the scene through the same aperture of area dA₀ and the area element dA will subtend a (solid) angle of dΩ₀ cos(θ). This observer will be recording I cos(θ) dΩ dA photons per second, and so will be measuring a radiance of

$I_0=\frac{I \cos(\theta)\, d\Omega\, dA}{d\Omega_0\, \cos(\theta)\, dA_0} =\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}$ photons/(s·cm2·sr),

which is the same as the normal observer.

Read more about this topic: Lambert's Cosine Law

Famous quotes containing the words details of, details, equal, brightness and/or effect:

“There was a time when the average reader read a novel simply for the moral he could get out of it, and however naïve that may have been, it was a good deal less naïve than some of the limited objectives he has now. Today novels are considered to be entirely concerned with the social or economic or psychological forces that they will by necessity exhibit, or with those details of daily life that are for the good novelist only means to some deeper end.”
—Flannery O’Connor (1925–1964)

“Different persons growing up in the same language are like different bushes trimmed and trained to take the shape of identical elephants. The anatomical details of twigs and branches will fulfill the elephantine form differently from bush to bush, but the overall outward results are alike.”
—Willard Van Orman Quine (b. 1908)

“The doctrine of equality!... But there exists no more poisonous poison: for it seems to be preached by justice itself, while it is the end of justice.... “Equality for equals, inequality for unequals”Mthat would be the true voice of justice: and, what follows from it, “Never make equal what is unequal.””
—Friedrich Nietzsche (1844–1900)

“Writing is to descend like a miner to the depths of the mine with a lamp on your forehead, a light whose dubious brightness falsifies everything, whose wick is in permanent danger of explosion, whose blinking illumination in the coal dust exhausts and corrodes your eyes.”
—Blaise Cendrars (1887–1961)

“The sensation of seeing extremely fine women, with superb forms, perfectly unconscious of undress, and yet evidently aware of their beauty and dignity, is worth a week’s seasickness to experience.... To me the effect [of a Siva dance] was that of a dozen Rembrandts intensified into the most glowing beauty of life and motion.”
—Henry Brooks Adams (1838–1918)