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
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
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:
“Then he told the news media
the strange details of his death
and they hammered him up in the marketplace
and sold him and sold him and sold him.
My death the same.”
—Anne Sexton (1928–1974)
“If my sons are to become the kind of men our daughters would be pleased to live among, attention to domestic details is critical. The hostilities that arise over housework...are crushing the daughters of my generation....Change takes time, but men’s continued obliviousness to home responsibilities is causing women everywhere to expire of trivialities.”
—Mary Kay Blakely (20th century)
“It is not only their own need to mother that takes some women by surprise; there is also the shock of discovering the complexity of alternative child-care arrangements that have been made to sound so simple. Those for whom the intended solution is equal parenting have found that some parents are more equal than others.”
—Elaine Heffner (20th century)
“For pain is perhaps but a violent pleasure? Who could determine the point where pleasure becomes pain, where pain is still a pleasure? Is not the utmost brightness of the ideal world soothing to us, while the lightest shadows of the physical world annoy?”
—Honoré De Balzac (1799–1850)
“Man is endogenous, and education is his unfolding. The aid we have from others is mechanical, compared with the discoveries of nature in us. What is thus learned is delightful in the doing, and the effect remains.”
—Ralph Waldo Emerson (1803–1882)